Plotnitsky: …

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]]>**Plotnitsky**: I would like to suggest, by way of a preliminary framing of this interview, two subjects: the first, more general, is “mathematics and narrative in mathematicians’ dreams,” and the second is a particular instance of this general theme, Robert Thomason’s story. Were concrete details of the story, say, those related to Tom Trobaugh’s biography—Who was he? Did his life and work have anything to do with mathematics?, and so forth—important for Thomason’s dream? Are they important for your argument concerning the relationships between mathematics and narrative, and among mathematics, narrative, and dreams? Or is the narrative itself—the* event*—that defines the dream most crucial: the appearance of a figure that makes a pronouncement in a dream that occurs at a certain juncture of one’s quest for a proof of a difficult theorem? The pronouncement ultimately leads Thomason to his proof, even though it was, mathematically, a false statement, but, importantly, related to a crucial aspect of the proof.

**Harris**: I discovered this paragraph in 1991, leafing through the Grothendieck* Festschrift*, and it struck me as one of the most amazing things I had ever seen in a mathematics article. I was trying to understand what Thomason had proved, reading the main result, then I saw this. A friend, a K theorist, confirmed to me that this is indeed the key point in the proof; Thomason was not just being sentimental, it was the key point. This stuck with me just because it was a striking story, unusually expressed, that with economy of means manages to convey a considerable pathos. The specifics are of interest because they grab the readers’ attention. For this particular project I’ve been thinking along the lines of trying to understand what a mathematical idea is. There is also a very personal reason for my interest in mathematical ideas. I also once discovered a very different sort of idea in a dream: I was given a solution to a problem I’d never previously considered. I’m not satisfied with the logicians’ account of mathematics, particularly insofar as this account leads, it seems to me inevitably, to the replacement of mathematicians by androids. From that point of view the specifics of the story are not so important. It’s the fact that the dream identifies very explicitly the key step in the argument, and one can ask what it means for it to be the key step.

**Plotnitsky**: That Trobaugh’s statement is incorrect gives a particular feature to Thomason’s story: a narrative of somebody coming, in a dream, and making a statement that is wrong, which nevertheless enables one to do something right. How do you read this part of the dream? Does it reflect the fact Thomason does not have the right answer to his problem as yet, even in his unconscious? Or is being “wrong” before getting it “right” a common part of the process of mathematical proof, and hence part of our narratives of this process, which may manifest itself in a dream?

**Harris**: Thomason explains how he knows it’s the wrong statement, because he had already explored this among many other wrong turns. What Trobaugh did according to Thomason’s version of the story is to pick out the wrong turn that could be set right and to insist that he look at this point and understand why it didn’t work. What he discovered was that it was exactly the wrong turn whose wrongness had an explanation.

**Plotnitsky**: In other words, Trobaugh tells him what his unconscious already knows: it “thinks” that this may be a point at which something right may emerge. Dreams scramble, reshuffle life, but also reflect something that is deeply true in or about it, as Freud indeed tells us. There appears to be a particular loop-like aspect to the process of mathematical proof: one finds oneself on a trajectory that *seems* to lead in the wrong direction and one abandons it; but one might also need to return to the same point in order to get where one needs to go. Thomason’s dream appears to me to be part of the story of this return, and the reason for this return is that one’s unconscious continues to revisit such junctures. It no more “trusts” us on being wrong than on being right, and continues to recheck everything. Could something like this occur in an android’s thinking? If not, is it because androids don’t make mistakes, or don’t have an unconscious of the human type (which defines our dreams), anymore than they have consciousness? Can it be that the presence of both and the difference between them are most crucial for human thought—narrative, mathematical, or other—which is perhaps the question of your title? If so, to what degree is the difference between us and androids defined a) by our narrative thinking and b) more generally, by conjunctions and disjunctions of such things as consciousness and the unconscious, reality and dreams, logic and narrative, truth and error, and so forth?

**Harris**: Well, I don’t want to prejudge androids’ strategies…

**Plotnitsky**: But these are androids that *you* create in your paper. You are their creator. So, perhaps, you can judge them.

**Harris**: True, but I want them to be as faithful as possible to the pre-existing narrative of mechanization of mathematics. It seems too simple, but I could imagine that an android has a search strategy, like an automatic theorem prover. It goes node by node and somehow is able to put a metric on the distance between each node and the goal. Then upon calculation it turns out that the first node was closest to the goal all along, and that doesn’t seem at all conducive to story telling. In a formal sense, it has the same structure as the return to the abandoned direction. But the reason that you return to the abandoned direction is not that you have done a calculation and found that all other conceivable directions are less satisfactory, but rather because you see the other direction in a new light, you see it has possibilities that you hadn’t imagined; there’s a turning you would have not perceived before. So I would be perfectly happy to say that an android is not capable of that, not capable of a flash of enlightenment, as a result of an unconscious or conscious process. It often happens that you meet another person who says your problem is a special case of some other branch of mathematics you hadn’t considered before. That’s not what happened to Thomason. He returned to this idea and focused on it, he had an emotional reason to focus on it. It’s perfectly natural, being who he was, that he could have carried out the step suggested by Trobaugh to identify the obstruction. This is within his repertoire, you could imagine that he abandoned it for emotional reasons, because it didn’t look promising and later returned to it for another emotional reason. I wouldn’t pretend to have a theory of such things.

**Plotnitsky**: Would you maintain, then, the difference between the way androids (as you conceive of them) think and the way human mathematicians think?

**Harris**: I’m not making such a strong claim, because you know, time could prove me wrong. I would say that in order for the android to tell ** us** what it’s thinking it has to be able to tell stories. The android can find its own route. This is what I’m talking about, and really of course the purpose. You have read

**Plotnitsky**: Yes, I have.

**Harris**: So you know that at the end it turns out that the narrator’s job was not to teach the computer to read, it was to teach himself to teach. In the same way, it’s quite likely that my story will turn out not to have been about the androids at all but about the human. If the narrative framework allows me to shed some light on what it means to follow a proof, then I’ll be satisfied.

**Plotnitsky**: Do you believe, however, that narrative is part of human mathematical thinking, specifically in the construction of a proof?

**Harris**: I had not entertained that possibility before I was invited to this conference but it did seem very fruitful in this particular instance. I looked back at the account of the proof and I was very pleased to see that I could narrate the proof of Lemma 5.1. It was at that point that I decided that I was satisfied, up until then I was not sure it was going to work. It certainly does not suffice to characterize mathematical thinking by any means, but I believe it can help to understand why mathematicians actually do think and communicate with each other.

**Plotnitsky**: Do you think that an android could think in the same way? Could it be that the main reason that human mathematicians do not think like computers is that they are not only mathematicians, that “nonmathematical” aspects of life are part of human mathematics?

**Harris**: The proof strategies that I’ve found in my sources are based on quantitatively reducing some sort of abstract distance from the goal. They are not structural in any sense I can determine. There is no room in them for the topological obstruction that a mathematician would perceive. Can mathematicians program computers to deal with obstructions? I can’t rule that out but in that case the androids start to think like we do.

**Plotnitsky**: “Obstruction” plays a curious role here. The word carries two types of meaning. The one comprises specific mathematical concepts of obstruction, including in topology, which are not necessarily obstructions in the everyday sense of overcoming some obstruction created by life, which is the second type of meaning of obstruction. Both, however, appear to play their roles in your paper. Is it possible that the double meaning of obstruction has helped Thomason to arrive at his proof? In other words, was he helped by the fact that a certain mathematical concept of obstruction key to his proof was linked to the everyday meaning of obstruction? This would be very human, “all too human,” as Nietzsche said, would it not? It would be very hard to create an android that thinks like this.

**Harris**: As Teissier pointed out, René Thom said that computers don’t know pleasure they also don’t know frustration; and it’s an interesting question whether an android would become a better mathematician if it were capable of frustration, if it were capable of realising its time is limited, or its memory capacity is approaching saturation. I find helpful elucidating “obstruction” in terms of human intuitions, primitive intuitions of obstruction, perhaps completely illegitimate from a philosophical point of view.

**Plotnitsky**: They are not illegitimate.

**Harris**: I found it a useful way to talk and it’s something that humans have in common, not only mathematicians. If this section on obstructions can be read by someone who’s not a mathematician, it proves that we have something in common. If we could translate what it means not to be able to go somewhere (a computer can do mazes but it doesn’t panic when it reaches a dead end), if an android could be taught to understand that passage in the way human can, well … it’s the same thing we said before. Frustration itself is a kind of obstruction and the intuition of obstruction can also be traced back among other sources to movement in space and being unable to reach one’s goal. This is a complex intuition that forms on the basis of experiences. The other point is that Thomason’s an expert, he has experience with analogous kinds of problems and can perceive the difference between the problem he’s facing and the other case suggested as an analogy by Trobaugh’s ghost, only because he has this experience.

**Plotnitsky**: As I indicated earlier, to me one of the main lessons of your paper is that what makes human mathematicians different from computers and perhaps androids, especially when it comes to complex concepts, such as those of K-theory, is, again, the fact that they are *human*. They are not only mathematicians. What does it mean to be an expert android mathematician versus an expert human mathematician, who has the *experience* of proving difficult theorems?

**Harris**: Thomason learned that coherent sheaves extend and the android can possibly be told that coherent sheaves extend but if the android were then to know to ask the question why perfect complexes don’t extend, it would become more like Thomason, more like the Ghost I guess. What is it that enables… it’s even impossible for me to ask the question without using visual, spatial metaphors: *extends*, *to see the analogies*. I can’t express these concepts in set-theoretic language. What is it about analogies that can’t be expressed in set-theoretic language?

**Plotnitsky**: This capacity for spatial thinking appears to be a defining (evolutionary?) part of our thinking, as Apostolos Doxiadis’ paper suggests. Could androids be taught to think spatially?

**Harris**: I think it’s much harder than teaching them to think algebraically. Some mathematicians are very combinatorial, and it would not surprise me if logic and combinatorics were formalizable, Gowers a combintorialist, he’s not surprised that he predicts that computers will be able to solve kinds of problems, he even lists some of the intuitions he would build in to the computer and of course it would become more like him, but it would still be an android. Geometric thinking, I guess, is traditionally the most difficult to approach philosophically and certainly to formalize logically. Something is missing when you formalize a continuum even though you can work with something unsatisfying. I’m somewhere in between, Thomason is definitely a topologist…he thinks spatially in a way I can’t, maybe that makes me more android-like than Thomason

**Plotnitsky**: Spatiality, then, seems to be a significant point, which links your paper to several other papers in this volume. It also relates to the title of your paper as an illusion to the title of Philip K. Dick’s novel, *Do Androids Dream of Electric Sheep?*, which deals with spatial dreaming. Sheep are about movement in space. I wonder if you want to say more about our human capacity to geometricize even algebraic situations, which allows us to bypass a great deal of computation.

**Harris**: I’m too much bound with that to be able look at it from the outside. One of the things I’m working on right now is trying to understand an ongoing construction geometrically because I’m unable to follow the calculations. There are people doing extremely difficult calculations and I’m unable to understand where they come from. I’m satisfied that I understand a problem when I understand it geometrically. But I don’t know how to convey that. I was just talking to Chris Meister and he was asking what it means to understand, and for a mathematician, that’s a very good question. For certain kinds of questions, my understanding seems to need a geometric setting, but other people see these things differently.

**Plotnitsky**: Although difficult to define, intuition appears to be crucial for distinguishing human mathematical thinking from whatever machines do mathematically. Do androids have mathematical intuition? Do computers have mathematical intuition? Or at least, do they have something that works like mathematical intuition in humans? We might recall that the German word for intuition is Anschaulichkeit, which relates to visualization.

**Harris**: I think I went out of my way to credit the android with some sort of intuition.

**Plotnitsky**: But would it be spatial intuition? As I said, the original title of the novel, *Do Androids Dream of Electric Sheep?*, reflects, interactively, both a capacity to dream and a capacity for spatial visualization, and both capacities appear to be essentially related. Would you like to comment on your title in relation to the title of the novel?

**Harris**: When I came up with the title some years ago when I was thinking of writing about this I had seen the movie but not read the book and now I have. The movie and the book are both very good. The book is deeper in some ways, in ways I haven’t really been able to work through. The theme in both cases is understanding what it means to be human, and in that sense the parallel with he purpose of this essay is clear. I don’t think there’s much more than that. The allusion fits the circumstances of Thomason’s dream very well, I thought, but I wouldn’t say its deeper than that.

**Plotnitsky**: Do you want then to explain your concept of “android” a bit further?

**Harris**: I define an android as a realization of a logician’s dream of mathematics. An android is a fictional automatic theorem prover, and it’s fictional because this is a conference about mathematics and narrative and so this is a fiction and also because the logicians’ make the hypothesis that such a proof could be (or should be) automatic.

**Plotnitsky**: But how would an android do it? If you say that an android is a computer, then I can form a conception of how computers prove theorems. Do you have in mind a computerized automatic theorem prover or some other kind of automatic theorem prover?

**Harris**: Computers shouldn’t be limited by what we imagine them to be… So I suppose that historically this arises in the context of the speculative meta-fiction of a computers or some human-computer hybrid as our evolutionary successors. The android is a way for us to talk about this. The android may or may not need us to talk about it, this may be our problem. It is our limitation that we need to tell stories, the android is intrinsically a logical being, but for us to grasp it, we need to tell some sort of story about it.

There are people who say that the universe is a computer and is proving all the theorems without our knowing it. Edward Fredkin said that physics is the result of the universe’s computation and I think he thinks along the same lines. What theorems this universal computer chooses to prove is it’s own business but in a sense it’s already been anthropomorphized. Ignoring the futurists if we try just going back to the logicians and follow their reasoning, the implications of their view of mathematics, then either it happens by itself and we don’t talk about it, or if we start talking about it somehow it has to be anthropomorphized and then we can communicate with computers because they give us the results. Or we can try to understand , they can do us the favor of explaining to us what they did. That’s another way Roy can spare us, or fail to spare us, Roy could just decide to go on proving theorems but without telling us about them…

**Plotnitsky**: But human mathematicians would not be automatic theorem provers, right?

**Harris**: Of course there are people who believe that materialism requires them to say that if people are more or less like computers…

**Plotnitsky**: This is not what you think, however.

**Harris**: No, no!

**Plotnitsky**: What, then, makes humans *non-automatic* theorem provers?

**Harris**: I don’t have the last word, just some things I’ve already said, being spatio-temporal creatures with all that entails.

**Plotnitsky**: Perhaps we can look at it from a different perspective, by noting that mathematics doesn’t simply consist of theorem proving. Of course your case is about proving a difficult theorem, and perhaps a computer could even prove it as well (although I doubt it can). Could, however, a computer or android create something like K-theory?

**Harris**: Of course, would it have any interest in doing so? If we give the computer enough resources…

**Plotnitsky**: Are you now talking about the computer-like android?

**Harris**: Yes, if you give the android enough resources it can certainly create K theory, but in order to take it further we would have to keep giving it more constraints.

**Plotnitsky**: What are the resources required for the creation of something as complex as K-theory? As I said, I can imagine that you could have a computerized proof of certain theorems of K-theory, but I would be hard pressed to imagine one could in a foreseeable future create a computer that could create K-theory. What is your thinking about the creativity aspect of human mathematicians? And what is the role of narrative in creating mathematical theories rather than only proving a theorem?

**Harris**: Well, there are the precursors of K-theory I mentioned. They come from various sources and to see what they have in common is to combine them into a common narrative. This was the work of 15-20 years starting with Grothendieck’s proof and going through Atiyah and on to Quillen. I don’t know that history very well, I certainly don’t know what was going through their minds.

**Plotnitsky**: Perhaps K-theory is similar to Kronecker’s “Jungendtraum” program, as Barry Mazur discussed it in his paper. Could an android generate this type of program or is the capacity to do so a manifestation of something that is our own, uniquely ours perhaps (leaving aside “human” or human-like androids, like perhaps those of Dick’s novel).

**Harris**: There was the context of generalized cohomology theories, one could recognise that K-theory is a generalized cohomology theory, and all of that of course deals with obstructions. One of Grothendieck’s contributions, one that was actually mentioned when he got the Fields Medal, was to introduce the relative notion into algebraic geometry and this was maybe the first instance of that. This is not really answering your question, which has to do with narrative. I can certainly say that K-theory could not have developed so quickly without the narratives that went along with it. Take the narrative of analogy, the motivation for Thomason’s paper in particular, as he explains in his introduction, was that Quillen’s K-theory was too rigid somehow; its framework did not allow the proof of the localization theorem with all the richness one is entitled to expect from a generzlised cohomology theory. That is a research program built into a narrative. Within this, there are micro quests but they are all embedded in the larger quest to prove the localization theorem and in this way complete K-theory as Thomason thought it should be completed. He had to give up something; he had to limit himself to certain kinds of algebraic varieties, and one reason his paper is so long and took so long to write is that he has to show that his framework applies to a sufficiently interesting class of varieties. Based on my own experience in writing mathematical papers there are a few fundamental ideas and then there’s all the filler, a few turning points then the filler. You can’t keep your bearings writing all the filler unless you have a narrative in mind. Now that’s us, I don’t know whether androids would need it in the same way.

**Plotnitsky**: That seems to be a very central point, a tremendous non-androidal point, made by you as a human mathematician and as a human being. You just brought together history, narrative, and mathematics in a way that may be explained in a general, nonmathematical language, since the technical details of the mathematics involved are not essential here. You said that K-theory is a theory that is like other theories of *that type*. Let us say, it is *like* other *cohomological* theories. Therefore, one should expect from this theory a similar set of structures to be generated; and, unless we bring these structures together within the overall architecture of K-theory or several K-theories, these theories are not going to develop in the right way. This architecture directs our thinking concerning such theories, and this type of thinking is not androidal (on your definition) because it doesn’t explore what you can calculate from a concept but what you derive by analogies. There is a narrative guidance or gradient here that, I think, is very difficult to expect from an android or a computer. The story of Thomason’s dream is a story of that part of mathematical thinking, which is especially difficult to computerize for several reasons: first of all, the mathematics itself involved, that of K-theory; the historical and narrative constructions involved due to both the history of mathematics and the personal experiences of a particular mathematician, such as a purely human story of a dead friend who appeared to the dream. In Dick’s novel androids are humanized; it is about their humanization, and the death of an android, as a friend or a lover, is part of the story of this humanization. I, thus, come back to the idea that the main reason that human mathematics works the way it does is because human mathematicians are also not mathematicians; they are human, “all too human,” in other respects. I think that your previous answer gave a picture of how Thomason thinks, via the conjunction of many different things: a (quest) narrative perhaps embedded in his proof; a narrative of the history of mathematics; an analogical narrative; a personal narrative; a narrative of obstruction, in which these narratives come together, and so forth. To me, this is a deep and beautiful point of your paper.

**Harris**: I just want to mention that I did explore the notion of K-ness as an intuition. This is mysterious to me, it’s something I can’t really define. You would know never to submit certain kinds of articles to the journal *K-Theory*, but others would make sense, I don’t know that I could convey K-ness as an intuition to non mathematicians, whether this has to be developed in the course of trying to solve many different kinds of problems and then seeing certain features that keep reappearing in different guises, or whether it’s something that belongs to all human beings.

**Plotnitsky**: But you have just conveyed the point of *what it does*! You didn’t convey how it does it because that this is very complicated, but you just conveyed the point concerning what the theory has accomplished. It’s very difficult to convey to a lay person what is at stake in K-theory or in something like Wiles’ proof of Fermat’s theorem. But one can convey why it is important and what is the formal structure of the history or narrative of what has happened here, what may be called the (human) *mathematical situation* and a (human) *mathematical event*. Now, one of the key features defining a good tragedy according to Aristotle is a complex (rather than simple) plot, and specifically the role of chance there. One of the great statements of *Poetics* is: “It is probable that improbable things happen [in a tragedy].” On the one hand, because of his dream, Thomason’s story contains an element of chance, and yet, on the other, as Freud would argue, this dream may not have been by chance. This interplay of chance and necessity (invoked already by Democritus) is not the way computers operate.

**Harris**: I had not thought of that, but I see two different issues. First what makes Thomason’s story compelling can certainly be analyzed in terms of the aesthetic criteria of the *Poetics* or otherwise. I haven’t tried to identify what makes it compelling, apart from pathos. There’s clearly a chance element, it seems the dream came to him by chance.

**Plotnitsky**: It is not entirely clear that it was chance; it could, as we discussed earlier, have been the product of the causal logic of the unconscious.

**Harris**: But maybe chance is for Aristotle really in the working of the unconscious as well. Then it may mean to recognize the work of the unconscious as chance as well in that sense. Then what makes a good proof is the same as what makes a good story. And that I didn’t explore. And it’s very interesting.

**Plotnitsky**: This brings me back to the main theme of this interview and, in my reading, of your paper—that mathematics is a human project and product, and in order for androids to dream of proving theorems in their sleep or in general of proving theorems in the way we do, it might be that androids must become human.

**Harris**: That certainly is implicit in the choice of the title. It’s interesting that Philip K. Dick does not explain how his androids became human or the problems in making them. Deckard in the book undergoes a sort of conversion experience by realizing that he can’t distinguish humans from androids. I haven’t read some of Dick’s earlier books in which the androids are less appealing and more of a threat. Here they are not a threat they are just trying to survive. The problems are evoked but not explored in any depth; it’s assumed that the difficulties have been solved. So it’s very philosophical, but what would it mean to be a human faced with this situation? My article deals with an earlier philosophical stage: what problems must be solved in order to make such a confrontation possible in the first place, in mathematics?

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]]>David…

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]]>**Corfield**: You talk about wanting the best organization of a narrative. How does this apply to your story of the prophetic vision of Kronecker? In the discussions today you said at one point that if you told this story either in its nitty-gritty historical detail, or if you put it in a sort of Bourbakian-crystal form, then you wouldn’t have been able to tell the story as you did.

**Mazur**: Well, if I pitched it in a Bourbakian way, I would have thereby chosen a very particular audience, not the audience that I was interested in addressing—at least in this piece. If I pitched it as a history of the intellectual milieu of the mathematicians at the time, I wouldn’t have managed to tell the story at all, partly because I would have gotten distracted. Kronecker is a complex individual, unsavory in many ways; and it is very possible that elements of his make-up, his biography, connect in some essential way to his Jugendtraum. To tell that story responsibly would be a wholly different project. I want to talk about “Kronecker’s vision”—somehow disembodied, and how it—the vision— was taken up by generations of mathematicians that followed Kronecker; how that vision has a development all its own.

**Corfield**: Something that came up during questions: can there be a kind of disembodied actor of that sort? What kind of thing can this dream vision be? One thing that occurred to me, and you talk about this on page two, something one often associates to the notion of a person at least is the notion of responsibility. So I was rather intrigued to see that you talk about responsibility here. You say “all dreams come with responsibilities”. So what happens though when it—the dream— becomes disembodied? What happens to that dimension of responsibility with a disembodied idea. I mean, if the vision itself is an actor, does it have a responsibility?

**Mazur**: I’m imagining a history of ideas that is—in effect—historical fiction. Start with an idea; follow the idea as it develops. The word it, of course, hides a large conceit, for mathematicians and not a disembodied it are the agents for this development; that’s why it is fiction. But sometimes, a useful fiction.

**Corfield**: I am still intrigued though about the relation between that kind of story and the historian’s history. I mean the particular way historians of science work. This has changed in the last 20 years or so. Is there a sense in which there’s an element of the truth of mathematics they cannot capture with their resources?

**Mazur**: Yes I think that there’s a type of truth—and a type of audience—that traditional history of mathematics will not capture. One might want to capture the flow of the idea in a non-personality and non-Bourbakian way thereby explaining certain aspects of the idea to a different intellectual audience than the traditional accounts could possibly get to.

**Corfield**: OK. Regarding the notion of disembodiment, Shafarevich, when he says that mathematics is like a symphony written by a single composer, has taken that a stage further.

**Mazur**: Oh yes! But if we were to pursue this thought, and the personality of this thinker (Shafarevich)…

**Corfield**: We would be distracted …

**Mazur**: Totally distracted. This is a good example.

**Corfield**: Yes OK, let’s move on to loftier things then. You quote Grothendieck, writing at the outset of his grand project about the novelty of the abstraction it requires: “It will be difficult, no doubt, for the future mathematician to gauge this new effort of abstraction—perhaps tiny, all told, in comparison with that exercised by our fathers when habituating themselves to the Theory of Sets.”

The issue here is retrospective and prospective views. One is trying to write, always constructing a framework for how the future will perceive…

**Mazur**: How the future will perceive you or me, just as we have perceived the past. We have had difficulty understanding the complexities of the mission of our forefathers. Because it’s all now so very very plain to us. And the future will have difficulty understanding how difficult our contemporary visions are to us.

**Corfield**: Right, it’s a sort of ratcheting process almost.

**Mazur**: Yes.

**Corfield**: Which brings me to the next point on Michael Friedman. You’ve read the book, The Dynamics of Reason?

**Mazur**: No I haven’t read the book.

**Corfield**: OK, but there is a really interesting thing there about the difference between a retrospective rationality and a prospective rationality. The interesting idea of the future easily making sense of the past comes into it. But that’s not all. There is a prospective rationality as well where one has an eye to the future. It’s harder grittier work in a way. For Friedman, with quantum mechanics it never quite happened, so that’s in some sense a failure, whereas the hard work of people like Helmholtz and Poincaré did the prospective work for Einstein. Didn’t they?

**Mazur**: Yes.

**Corfield**: Freidman’s notion of retrospection, is definitely the telling of a story. And it’s more the telling of the sort of story it sounds like you want to tell, rather than nitty gritty history. He talks about constitutive principles. Consider the passage from the Newtonian paradigm to the Einsteinian. In the Newtonian one you need a mathematical language like the calculus that provides you with a way in which you can state various constitutive physical principles such as the laws of motion. And then within that framework you’ll have various facts. And one of the facts is the equivalence of inertial mass to gravitational mass. As time goes by, there will be some tensions and difficulties in that setup. But as it stands, for a reasonable amount of time at least, it can’t be falsified. They are not all on the same level.

**Mazur**: Yes absolutely.

**Corfield**: You can falsify the facts and you’ll try to adjust various principles to keep the main principles intact.

**Mazur**: That’s right.

**Corfield**: Right, which is rather like what you are saying. But it will come to a point when something isn’t fitting anymore and then a new mathematics is needed, like the tensor calculus, which will allow the expression of new principles, general relativity say, and then you have this lovely turn around where what were facts before can become principles now. Mass, inertial mass, gravitational mass. And what were laws before can become now maybe just approximate facts. He goes on to say in that book that you don’t get this in mathematics. The difference between mathematics and physics is you don’t get that change, from something being a fact to something being a principle.

**Mazur**: But I do.

**Corfield**: I know exactly. You are right. I have an example. You had some axioms of topology and algebra and a bunch of homology theories emerged, Cech, singular, etc. Later on, with a new constitutive language, if you like, category theory, you can set up the Eilenberg-Steenrod axioms and then what was a fact, that Cech homology was a homology is now no longer a fact. I think it’s the one that fails, isn’t it?

**Mazur**: Yes, but it has made its comeback in étale cohomology.

**Corfield**: Right. This is related to Peter Galison’s question: “Does your sense of template actually go against Kuhn’s notion of paradigm, which is a little bit too strict and rigid’?

**Mazur**: It doesn’t quite go against it: it is rather that pliable templates are the ones I am particularly interested in. Templates that you use for purposes for which they weren’t initially designed, and where they actually don’t quite work as intended—but they do “work” in some unexpected way.

They’re pliable, they keep changing. But it also is how they are changing that you come to understand as obliquely offering the answer to the problem you are attacking. So it’s Kuhnian in that sense. It’s all mutability. There’s no revolution. There’s no break.

**Corfield**: Is there a cut-off point between stretching and revolution, like an earthquake, a general build-up of pressure and a sudden sliding of the plates? Is it really the difference between mathematics and physics that it appears that there are these sudden shifts in physics, and mathematics actually seems a little more flexible in that sense?

**Mazur**: At least the examples I am holding in my head at the moment in mathematics are more flexible. I seem to be attracted to these examples. There must be similarly Kuhnian examples in math as well.

**Corfield**: But things are shifting. I think that even in history of science people are tending to avoid the word revolution now. I was at a conference recently and that certainly was the case. Even people who study the industrial revolution don’t call it a revolution anymore.

I was interested in your account of the taxonomy of storytelling. For example, origin-stories, purpose-stories. How sharp is the distinction or how does one distinguish between an origin-story and a dream?

**Mazur**: The origin can be external to mathematics. For example, Archimedes being asked by some tyrant to figure out how to weigh gold. That’s an origin.

**Corfield**: Is that not a purpose-story?

**Mazur**: Let me see if I can find a good example of a purpose-story… Can I take the Dirichlet principle as an example? You want to understand diffusion of heat on the boundary. That’s the purpose. For example on the disk, you express the function describing heat distribution on the boundary by a Fourier series and you use the radius as the extra parameter that turns the Fourier series into a power series and it works. And then come various elaborations of that, allowing you to develop a kind of Dirichlet principle. But it’s not a mold, it’s not a template. And its purpose was internal—so to speak—to the context of the problem. Your initial aim was to understand heat diffusion. That is a different sort of aim than the aim to please a tyrant.

**Corfield**: That’s an interesting story. To people like Riemann, it didn’t need proving because it obviously must be true. If I set this electric charge on the boundary of a conducting surface, the electric potential will form the solution. But in a way that does carry on playing quite a strong role doesn’t it? It’s there in Klein; he actually talks about electricity doesn’t he?

**Mazur**: Yes, he plays on the analogy. He says that in order to understand a lot of Riemann’s analytic work you have to keep in mind the electrical analogy.

**Corfield**: You talk about a seed for a dream. So is this actually a little origin-story in a way? “To my mind, the seed for Kronecker’s dream is in Gauss’s expression for square roots of integers as trigonometric sums.”

**Mazur**: To start with the simplest such Gauss sum,

√2 = |1+i|,

is delicious, because if you see it as an algebraic statement it is indeed the beginning of the grand project of giving explicit expressions for abelian algebraic extensions. This project encompassed —as it expanded—Kronecker’s dream, and went beyond that to what is now largely dealt with by the Langlands program; while if you see it as a geometric statement in the complex plane it is indeed the beginning of the grand project of geometry, in that we may interpret the displayed formula as giving us the length of the diagonal of the unit square. In sum, you can think of that formula as being the “beginning,” at least in one sort of narrative, of lots of mathematics.

**Corfield**: And Kronecker isn’t really aware of this?

**Mazur**: I bet he was. He wouldn’t bother to say it. But you can say it. And I think of this as the first trigonometric expression of a radical. Find the next. And now you are off.

**Corfield**: Retrospective storytelling isn’t it?

**Mazur**: It’s a retrospective story.

**Corfield**: So an origin story must be almost historically accurate?

**Mazur**: Yes I imagine that I would still want it to have some element of accuracy in it.

**Corfield**: You see you are very close to rational reconstruction.

**Mazur**: Yes but a self-conscious rational reconstruction. I love Lakatos.

**Corfield**: Right, Lakatos didn’t see it is a bad thing. It was never negative. Kuhn says he was telling fairy tales. But in a sense you are happy with them being fairy tales.

**Mazur**: I am happy with them being fairy tales because there is something lost in comprehension if you don’t tell them. For example, this Kronecker story has never been told in the way that starts with the sentence “In the beginning √2 = |1+i|.” Nevertheless there are a number of good books wholly devoted to Kronecker’s deam. The title of one of them is Hilbert’s Twelfth Problem: A Comedy of Errors. And the type of history it gets into is not personal nitty gritty but rather textual nitty gritty, or the anxiety-of-influence-type nitty gritty arising when nobody really understands what anybody else is doing. That’s a wonderful book! Then another book on the topic is entitled Kronecker’s Jugendtraum, period. Only my graduate students after five years could read that book. But that’s not the audience that I want to explain this idea to. There is a very specific sweep of an idea in Kronecker that I believe is suitable, and useful, for the audience I’m looking for. I want to write about the evolution of the use of the word ‘explicit’ and ‘constructive’ in a certain context and I want to show how pliable these words are, how related they are to what we naively mean by explanation, and how subtly they can change. And you can’t tell all this except by employing this type of historical fiction: it’s not quite a rational reconstruction in that I don’t claim it to be history. I claim it to be just useful… it’s a parable. Maybe it’s a parable.

**Corfield**: Lakatos actually was comparing methodologies. Of the historically minded philosophers, the one whose notion of rationality captured as much of the real history as possible and that saw as little irrational as possible was the one—from Lakatos’s perspective— who had the better theory.

**Mazur**: Lakatos said that? I thought he was all for very carefully recording the back and forth of error and correction.

**Corfield**: No. There is a good sort of back and forth in the world of ideas on that level. It’s not nitty-gritty back and forth. I mean you can talk about back and forth in your type of story to some extent.

**Mazur**: I could.

**Corfield**: I think you are looking at it very similarly then.

**Mazur**: Is that right? I’ve got to reread Lakatos.

**Corfield**: Not so much Proofs and Refutations. Perhaps the later work. The Methodology of Scientific Research Programs in his collected works. You were talking about purpose-stories with the external influences on the content, in terms of the effect on mathematics. I wonder how black and white that is. You know some mathematicians don’t see this great gulf between physics and mathematics. And even someone like Atiyah, when asked which mathematician he most identified with, said Hermann Weyl. Strange because Atiyah only belatedly got into physics. So that’s curious, as though there was an almost pure mathematical core in mathematical physics.

**Mazur**: Yes, well that’s a big issue I think. Mathematical physics. Why is there nowadays such a clean break between mathematics and physics? Why isn’t there a continuum, and we simply understand that there is?

**Corfield**: You will get some physicists who will accuse mathematical physicists of just being mathematicians. Somebody will see someone like Witten just like that. String theory crystallizes this problem at the moment of ‘what on earth are they doing?’ And for some people, like Jaffe and Quinn, there is this nasty intermediate ground where they pretend to be mathematicians, they haven’t got though the background of a proper mathematician.

**Mazur**: Oh I think they’re OK.

**Corfield**: There were certainly worries expressed of people…

**Mazur**: One way of thinking about all this is—with the rise of String Theory— there is just a third field emerging, which can’t be considered traditional mathematics, nor traditional physics, but rather something new. I mean, that’s a possibility. Physics ultimately depends on its anchor of experiment and mathematics ultimately depends as its anchor on proof. String theory seems to have cut both anchors, and yet is alive and flourishing; it depends on some formidable intuition — some kind of inner resource— whose roots seem mysterious to me, inaccessible; nevertheless I don‘t doubt for a second that they—the string theorists—are going in the right direction. And moving so fast.

**Corfield**: Atiyah talks about mathematical confirmation of string theory.

**Mazur**: Amazing confirmations. There are a large number of rational curves of degree five on a smooth quintic threefold and the mathematicians thought they knew this number (it is over twenty million) but got it wrong. But the physicists, via string theory— well, mirror symmetry—predicted the right number. And they not only got that number right but broadened the ambition; they said, in effect: “Oh you mathematicians, your view of the subject is so parochial. There are all these numbers out there left to be computed, enumerating curves of given genus and degree in various algebraic varieties, and this is what we predict them to be! Get to work.”

**Corfield**: From the mathematicians’ point of view doesn’t this mean that there must be a core mathematical idea embodied in what the string theorists are doing? Mustn’t there be a theory there, so that one hundred years down the line we will look back and understand?

**Mazur**: I think that may be true but that’s not what drives them at the moment. Some platonic vision. They see something that mathematicians don’t see. I have no idea how they see it but they do.

**Corfield**: But do mathematicians have a belief that there must be mathematics behind what they are doing? I mean there must be something they somehow latch on to intuitively.

**Mazur**: Yes but I don’t want to say ‘and therefore it’s not a subject.’ I think String Theory’s a subject and if the mathematicians fifty years from now will be able to translate it into their language that’s great. But at the moment it is a subject.

**Corfield**: I think that Atiyah’s reaction was that we can look after ourselves, we mathematicians. They’ve got a source of great ideas. Let’s just live with that. I mean it’s brilliant, that’s wonderful for mathematics.

**Mazur**: Yes, it’s wonderful.

**Corfield**: All these worries that the standards will slip and so on.

**Mazur**: I don’t agree to that.

**Corfield**: Somebody suggested the idea that you can get people stomping through a field in a rather ad hoc kind of way, not a very principled kind of way, and then you leave this great mess behind and nobody wants to go in there anymore. A bit like doing bad archeology, you know.

**Mazur**: OK, you destroy the fields.

**Corfield**: Right. Whereas the careful, more meticulous… but you know, this isn’t an archeological site, I mean you can go revisit surely.

**Mazur**: You can go revisit this, and also no one had quite visited the String Theory fields before. String theorists are discovering these archeological fields. So that alone is priceless.

**Corfield**: Are they providing templates of some sort?

**Mazur**: They are moving so fast that I don’t know.

**Corfield**: There are probably templates to be extracted from what they do eventually?

**Mazur**: Yes.

**Corfield**: Later on you give a rephrasing in slightly more modern vocabulary than Kronecker himself might have used. So in a sense that’s your type of story telling there, the vision?

**Mazur**: Yes. I have cut myself loose from Kronecker—the guy who was so mean to Cantor —and…

**Corfield**: A historian might worry about anachronistic language

**Mazur**: Yes, all this is achronistic.

**Corfield**: But that’s great for you…

**Mazur**: Exactly. If I were writing History (with a capital H) I would never do this.

**Corfield**: Right and as long as one is very explicit about the genre that you’re writing in, that’s fine. One of the historians of mathematics, Ivor Grattan-Guinness, has suggested ‘Why don’t we just call them different names? We’ll call what we’re doing ‘history’ and, we’ll call ‘heritage’ what a mathematician might write, that kind of smoothing the path, that telling of the story of how we are where we are today and where we are going…

**Mazur**: That’s a possibility but I don’t think that is what I am interested in. I am not interested in heritage. I am interested in explaining something.

**Corfield**: Well it’s the past as it affects the future more. You have to see what he actually says that heritage is…

**Mazur**: I see he doesn’t really mean heritage…

**Corfield**: No he doesn’t really mean heritage.

**Mazur**: Heresy maybe?

**Corfield**: Yes, you could probably think of a better word for it but you wouldn’t object to it being labeled differently?

**Mazur**: Well, historical fiction would be good.

**Corfield**: OK, so now we can come onto this part about ‘explicit’. And I am sure this word does change, so you say “a companion word to ‘explicit’ is ‘constructive’ with its own vast history, and perhaps a more expressive description of Kronecker’s hope is that one might explicitly construct algebraic number fields”. What struck me there was thinking about Weyl and his book Algebraic Number Theory. Do you know his chapter called ‘Our Disbelief In Ideals’?

**Mazur**: Yes. He is deeply Kroneckerian there.

**Corfield**: He was closely identified with Brouwer at one point through the 1920s although he had probably broken away from that when he wrote Algebraic Number Theory.

**Mazur**: I have the impression that, at times, Weyl expressed more strongly finitistic sentiments than you would ever find in Brouwer. Brouwer was perfectly willing to talk about the infinite. It’s not that Hermann Weyl was unwilling to talk about it; he just didn’t feel he had to. He could express anything he wanted in finitistic terms.

**Corfield**: Right, so Kronecker’s vision is carried on by him, is alive in him in a sense?

**Mazur**: He thought so. Clearly in that chapter. Do I think so? It’s an unreadable chapter.

**Corfield**: I did try to read it. Because I did a chapter in my book on contrasting Dedekind and Kronecker as the successor to Kummer, which is an ambitious thing to do.

**Mazur**: Yes it is!

**Corfield**: And I probably bit off far more than I could chew. But I was quite interested in Weyl’s appraisal of the two strands. Actually all the reasons he gives against Dedekind aren’t very good reasons. He accuses Dedekind of being ad hoc and actually using Kronecker’s ideas. Dedekind did get around to using Kronecker’s ideas. And then Emmy Noether had fixed things by then anyway.

**Mazur**: So the whole issue is: is Weyl faithful to Kronecker? He is certainly faithful to one of Kronecker’s aims, which is the explicitness of things. Is he faithful to Kronecker’s way of making it explicit?

**Corfield**: Right. Then you go on to talk about explicitness. You’ve got these three E’s on the go: explicit, explanation and economy. And in your explanation of explicit you actually start using words like ‘economical’. I want to talk about the idea that explanation involves economy as well.

**Mazur**: It does.

**Corfield**: So in a sense they are all linked…

**Mazur**: I came to believe that explicit is just a little flag you sometimes wave: you say it if you think you have explained something in terms you think of—vaguely—as concrete. As long as you believe that, that’s your felt experience; OK for you. But it’s just a tag which says ‘I believe I have explained it.’ I think that’s the way mathematicians use this word, at least some of the time. I hadn’t realized that until I wrote this paper.

**Corfield**: But then you’ve got to say that the word ‘explain’ is even more important. Does it contain more than explicit?

**Mazur**: I think it does because sometimes you explain things and you say ‘I’ve explained it but it’s not explicit.’ An example being the type of “either e+π or e.π is not algebraic” sort of statement.

**Corfield**: Not explicit, right. So perhaps ‘explicit’ is explaining in a certain way.

**Mazur**: Explaining in a certain way, yes. Explicit is an “I know it when I see it” sort of thing.

**Corfield**: OK, so let’s talk a little bit about explanation, which as I mentioned in the discussion is an enormous topic. It has been treated extensively by philosophy of science, for example, there’s a Wesley Salmon volume of four hundred pages. It’s just enormous. A very simple breakdown of different theories of explanation in science: there was the camp that saw unification as being this critical part of it.

**Mazur**: Oh really?

**Corfield**: Yes, Michael Friedman and Philip Kitcher were identified with this. So electricity and magnetism being unified into electromagnetism is an example. That’s one school, the other school is: understanding the causal mechanisms that are behind the scene is the key. Let’s illustrate this. You ask the following question to people:

‘There you are in an airplane and you are holding a helium balloon and it’s accelerating down the runway, which way will the balloon go?’

Did you know that the helium balloon will tilt forward as you accelerate down the runway? I mean in the horizontal direction. OK, so, if you apply general relativity, the unification explanation of this is: if you are accelerating it’s the same as being in a gravitational field. Imagine that you are lying on your back holding your balloon your balloon would want to go upwards, yes? So that ought to be the same as going forward in the plane.

**Mazur**: OK.

**Corfield**: That’s the unification picture. And the causal picture is that, as you accelerate, there are more air molecules at the back of the plane so a pressure differential though the plane…

**Mazur**: OK, very good…

**Corfield**: So there is actually higher pressure providing a force sufficient to accelerate the balloon faster than the plane. There is a battle between those two schools, and Wesley Salmon wanted a unification of the two.

Then there was a third strand as well which actually rather picks up with what you are doing now, that explanation is relative to an individual. This is Bas van Fraassen, who says there is very much a pragmatic dimension to explanation. I mean really all someone is doing is that they are asking a why question, ‘Why this in contrast to that?’, and they will always have in mind a set of contrasts.

**Mazur**: Why x rather than z?

**Corfield**: And all the explanation is doing is arguing from what they already know.

**Mazur**: If I use the word “template” would you say that—in Wesley Salmon’s vocabulary—that it’s an attempt at a unificational explanation?

**Corfield**: Yes.

**Mazur**: Since my tale starts with an individual’s spark, it also sounds like the third one. But then it takes off and as template it becomes a unificational principle. That seems to be the story I am telling.

**Corfield**: Right. Kitcher was motivated by mathematical examples as well so that would fit nicely that he comes from the unification end. I wonder, can one make any kind of story from the causal account? Do you know Paolo Mancosu’s book on the philosophy of mathematics in the Seventeenth century? Aristotelianism was still alive then. Mathematicians were using the word causal. To give an example, there was an idea that a reductio proof did not get at the cause.

**Mazur**: That’s right. There is this hatred of reductive proofs. Reductio proof versus cause, that’s interesting. And that is prominent in the seventeenth century.

**Corfield**: It’s about getting the concepts the right way round. And if one had them the right way round one wouldn’t have to resort to this nasty non-constructive (as we now see) double negation sort of proof. It seems to be about that Aristotelian idea of getting the concepts the right way round.

**Mazur**: So you’re telling me that for people studying explanation, there is unification, there is the causal and then as you say there is the individual pragmatic spark of the why. Somebody’s asked why.

**Corfield**: It doesn’t have to be the original somebody. It could just be anybody now asking you: why did the car radiator break? Because the temperature went down, it froze and it burst.

**Mazur**: This third way, the why, doesn’t give you a format for how to answer. I could do it by unification…

**Corfield**: Right, in a sense it doesn’t seem like a third alternative in a way, does it? Is it purely a personal thing whether something is a good explanation or a bad explanation?

**Mazur**: Oh it is definitely a personal thing. If you tell me something and I say ‘it doesn’t explain it for me’, I am the last arbiter.

**Corfield**: What happens if you later come to see ‘Oh that was a good explanation’? You can do that can’t you? ‘If only I had realized…’.

**Mazur**: Well, there must have been something the explainer hadn’t realized, some trigger that should have been there, but wasn’t. The trigger might have been the student’s own prior preparation. And, even so, it stands as a bad explanation. I mean if you try to explain something not knowing what preparation the person has to whom you are explaining that something, it’s your fault, it’s not the person’s fault.

**Corfield**: Right but I am wondering if there are two aspects there. There is this sort of ordering-of-concepts type aspect to explanation. And then there is what a particular person whose clicking into this arguing knows, how their prior knowledge can mesh with this current explanation.

**Mazur**: OK. The ordering let’s call Cartesian because, after all, most of Descartes’ ‘Rules to the direction of the natural intelligence’ is exactly that. And a good deal of Discourse on Method is—in effect—that. If you order things correctly everyone will understand, and they will not need intuition. Descartes is an anti-intuitionist in that sense. In effect, he keeps whispering: “My analytical geometry means that you don’t need geometric intuition”. He doesn’t put it that way, but he puts it in terms of grand philosophical principles. So if you carefully go from step one to step two to step three you can abolish these ridiculous leaps of flashes and intuition.

In order for an explanation to be an explanation, it has to explain something to somebody. And the somebody is the arbiter. At least for the non-explanation. The body can say ‘yes you have explained it to me’ and then half an hour later say ‘oops, you have pulled the wool over my eyes.’

**Corfield**: Can’t it work the other way? ‘I didn’t think that you had explained it to me, but now of course I see.’

**Mazur**: Well it could. But I still think something was wrong if it didn’t “take” the first time.

**Corfield**: Then you move on to proof = explanation + guarantee.

**Mazur**: Yes, that was the equation which I borrowed and I am happy to find similar `equations’ all over Klein’s psychology of mathematics. In the Göttingen library there are beautiful handwritten notes of Klein’s that record lectures given in his seminar: someone in his seminar on the psychology of mathematics gives such an equation.

**Corfield**: I suppose we are just referring back to what you said about Descartes actually: if you had the explanation properly right it would already contain its guarantee wouldn’t it, because you would see it?

**Mazur**: Yes.

**Corfield**: There is a lot of interest isn’t there in the all these papers, Michael Harris and Tim Gowers from earlier, about androids. And you end up actually in this section talking about “the point I am making is an obvious one, desire pure and simple is often the main motivator for explanation”. So we are sort of getting back to the subject in a way, the bodied person with their desire which the computer does not have. I am wondering “why are people evoking these computers all the time?” I did some work myself on computers and they are dumb animals and they have no sense of the ‘why’, there is no desire there and there is no sense of judgment about what they do.

But a computer can be used as a device, can’t it, to try and understand the human?

**Mazur**: Yes.

**Corfield**: I suppose the whole point of the conference, in a way, is that humans are the only beasts that have this kind of narrative notion of rationality.

**Mazur**: That’s right.

**Corfield**: So, I am wondering if that’s linked in a way.

**Mazur**: I think it is linked.

**Corfield**: That a computer is dumb because it does not have a sense of narrative. Could it be?

**Mazur**: It certainly does not have desire. It doesn’t want to know. We want to know—and we have a keen sense of what sort of thing it is that we want to know—and that is a big difference.

**Corfield**: And the wanting to know, does it involve a story-like element? Because you must then conceive of yourself as knowing. You know you don’t know and you know you could know and you know you will be able to look back at yourself now and see that you are better or your understanding is much better.

**Mazur**: Yes, and also you have a past and you are just impelled to know because of something in your literary or mathematical biography. If mathematics is doable entirely by machine, what is being done is the guarantee. And eventually there has to be a subject to whom all these guarantees – truths are being explained. I guess what is worrying, for people who are talking about the androids, is that the guaranteeing mechanism might be overwhelming the rest of the equation. One almost sees a tiny bit of this in the ‘four color’ problem where the computers have guaranteed it…we are told!

**Corfield**: And you can react to that and say ‘OK, well if we never get a good proof, then it just wasn’t an interesting problem.’

**Mazur**: Or we could say we are out of the mix.

**Corfield**: This is Doron Zeilberger’s position. He is delighted if we find something that isn’t humanly provable. The other approach is to say ‘well that just isn’t interesting if it doesn’t fall into this explanatory accessible to humans.’

**Mazur**: I think it is interesting for what it is, which is: inaccessible. But the explanation of the ‘four color problem’ is quite different from the explanation of why base times the height over 2 is the area of a triangle. We have a closer inner involvement in the latter (we “see” it) and we don’t have it in the former (we’re merely “guaranteed” it) .

**Corfield**: Can one say that if a problem doesn’t provide a mechanism or way into producing these rather beautiful explanatory sorts of theories then it just an uninteresting problem?

**Mazur**: I don’t believe this.

**Corfield**: Oh really?

**Mazur**: I mean there is a movement afoot to make a curious equation between beauty and truth. I don’t know who started this. von Neumann says that mathematics has not completely done its job until it’s beautiful. Now that is OK, I like that. But then there are also people who proclaim that ‘beauty is the beacon of truth.’ Just follow the beautiful and you will get to the true.

**Corfield**: And you disagree?

**Mazur**: I don’t know. I would say that anyone who says that beauty is the beacon of truth has to work to explain it. Because most of the time in mathematics we are after understanding something and if we have to get to it by the ugly truth and if that is all there is, I would say we would do it. And to say that we follow the beautiful because we are looking for the truth is an incredible metaphysical proclamation of faith and it should be labeled as such.

There is the Isak Dinesen ditty “Straight is the path of duty/ curved is the path of beauty./ Follow the straight line, thou shall see/ the curved line ever follows thee”. So it is possible that we just do our thing and—my gosh!—every time we look back we see that we have done something beautiful. That, I am happy with.

**Corfield**: OK, as in the retrospective direction?

**Mazur**: Yes the retrospective direction. Yet again we have done a beautiful thing but all along we have just been following understanding. So it is a little gift, it is a bonus. We turn back and we see ‘Oh wow, we have done something beautiful here.’ But to say in a doctrinaire way, ‘we follow the beautiful and then we will get to…’ I don’t know.

**Corfield**: But just as the explanation of ‘explicit’ changes through time, the notion of beauty in mathematics mustn’t it, aesthetic sense and sensibilities, change?

**Mazur**: Absolutely. That’s the key. For example negative numbers they were hated, they were ugly.

**Corfield**: There is a retrospective telling of the kind of stories we were talking about. Some mathematical fictions are a way of narrating the past so they seem like the following of beauty.

**Mazur**: Well it is clear that Cardano was doing something strange when he wrote exhorting his readers to ‘dismiss mental tortures‘ and plough on with a certain computation. That’s not following beauty.

**Corfield**: Yes. So I wonder if one presented this, how they would respond the people that say ‘we should follow the beauty to get to the truth’? If you presented that example to them what would they say?

**Mazur**: I am currently writing a brief thing about this; I don’t yet know.

**Corfield**: There was a beauty there that they didn’t know about? Strange thought.

Another thought I had, of course you can take the further step and not equate the guarantee with the computer. There always must be a guarantee for a person. It’s our formal system. It’s a question of responsibility again. We are putting it on the computer but of course it is our responsibility.

**Mazur**: Absolutely.

**Corfield**: I was talking to Michael Harris yesterday about Tom Hales, and his being slightly irked by this comment which appeared in the Annals of Mathematics, this rider saying subject to such and such.

**Mazur**: Yes. It is a very strange thing for editors to do. The rider surely implies something larger than the editors intended, because it suggests that they personally have a very high degree of confidence in the contents of all the other papers that don’t have riders attached. What they should have—qua editors—is a high degree of confidence in their choice of referees. All of them.

**Corfield**: So why not flag every journal article up as being subject to…? So there is a question of where the responsibility lies.

**Mazur**: Yes there is.

**Corfield**: You talk about explanation-futures as in a sort of stock market of the mind;

**Mazur**: like pork belly futures.

**Corfield**: Yes, you have sold them typically before you have even touched them.

**Mazur**: Before they existed.

**Corfield**: You get these funny stories of people who forget to offload them, and then the call is made upon then and they’ve actually got to get hold of these pork bellies. So what is an example in mathematics?

**Mazur**: Oh it is all over the place. Well the example is that lots of the time when you begin to study a concept, you use it vastly before it actually has explanatory power for you; only in the course of using it many times does it have explanatory power. Now how did it get that explanatory power if it didn’t have it before?

**Corfield**: Didn’t have it for you or didn’t have it for anybody?

**Mazur**: For you. Here is an example. We have many analogues of it in mathematics. You tell the doctor that you have this disease in the muscles and he says ‘oh you have myalgia’. And you feel you have been explained something. In a sense you have, and in a sense you haven’t. It is a perfect example of what I would consider to be an “explanation future”. That is to say, there is a little post there that has been rooted in the ground of your mind and you can hang things on it. Already you know that it has a name, that your disease has a name.

**Corfield**: I can go join the myalgia sufferers club for example.

**Mazur**: For example. I can google it. But it is not an explanation. And a lot of mathematics, an immense amount of mathematics is that way.

**Corfield**: It sounded so from what you have written. So in that case the body of knowledge exists, it’s just that you don’t have it?

**Mazur**: Sometimes the body of knowledge exists and you don’t have it, sometimes…

**Corfield**: It doesn’t exist?

**Mazur**: Absolutely. Algebraic K-theory (in a particular setting) I thought was that way for a while. I’m referring to the K-theory of special linear groups over arbitrary fields, and over the rings of integers in quadratic imaginary fields in particular. Now the K-groups are wonderful, and purport to tell you something about certain central extensions of those special linear groups, and to give you serious control over their generators and relations. But at the time of its origin not a single interesting K-group had been computed in this context. K was a place-holder for a future theory of these special linear groups. At the time, all we had was the setup of such a theory: we hadn’t powered it up yet. Nevertheless K-theory still had explanatory power because you knew that eventually somebody would fill in some of—or maybe all of—whatever is needed to be filled in, to make it pay. And the funnier thing is that it was already rooted in your mind, waiting to offer up explanations. Explanations not in existence yet, just explanation-futures. Promissory notes.

The Shafarevich-Tate Group until Kolyvagin’s work was—in a particular aspect—similar: take a homogeneous cubic equation in 3 variables, and ask ‘does it have a rational point or not?’ One asks this question with a glimmer of hope that some sort of local-global principle might prevail. What I mean is this: if the equation were a quadric rather than a cubic, and if it had a rational point over every complete field, then it would have a rational point (the local-to-global principle—also called the Hasse Principle—holds for quadrics). But if it’s a cubic, it doesn’t necessarily satisfy this very convenient, and somewhat mysterious, local-global principle. (The equation 3X3+4Y3+5Z3 = 0 is a famous example, due to Selmer, of a cubic having no nontrivial rational point, and yet having points in every completion of the rationals.) So, given a homogeneous cubic equation in 3 variables that has points in every completion of the rationals, you define a certain gadget called the Shafarevich-Tate group related to it; the main mission of this Shafarevich-Tate group is to provide an answer as to whether or not your equation has a rational point. This move—done by Shafarevich and Tate—is already extremely clarifying! Yet for twenty years the Shafarevich-Tate group had never been computed in a single example. Despite this, the Shafarevich-Tate group remained throughout that period, enormously explanatory. Now, happily, we can compute a few of these Shafarevich-Tate groups, so the promissory note is beginning to pay off, big.

**Corfield**: OK, that makes clear to me how you see explicit and explanatory as so close. Possibly, do you think if we had Grothendieck here, would he not see it the same way or was he somebody who ever cared about concrete calculations?

**Mazur**: It is absolutely true that he seemed not to feel the need to make specific numerical calculations.

**Corfield**: In Mykonos you told us about some work you do on computers. There are quite a few mathematicians you know who probably wouldn’t use computers. I couldn’t imagine Atiyah or people like that grunging over their respective calculations.

**Mazur**: Did you know that Euclid’s Book Seven on number theory doesn’t have a single number? Scholars have made a career on coming down one way or the other regarding the question of whether there were diagrams in early manuscripts of Euclid, but nobody mentions that there isn’t any actual number in his book on number theory. So Euclid is a Grothendieck, in this way.

Diophantus has a completely different take than Euclid. Diophantus is a numbers guy. He’s interested in posing a general question and then saying: ‘well here are some solutions to it,’ not even saying that they are all the solutions, or —it seems—worrying much about getting absolutely ALL solutions. Sometimes you don’t even know what species of solutions he’s asking for, because he presents you with solutions of it that are not integers; they are surds. “Oh Diophantus,” you want to shout at him “you’re happy with surds as solutions for this problem, well why then didn’t you consider surds as solutions for the previous problem?”

**Corfield**: So there’s a weird sort explicitness there isn’t it, where he hasn’t actually specified how to be explicit.

**Mazur**: He hasn’t specified the range of where his solutions are, and also the order of the problems. Now I don’t know—and I don’t think anybody knows—the origin of the text enough, to say whether or not these are just the manuscripts we’ve gotten, it’s what has come down to us; maybe the problems were in some more systematic or rational order before and now they are just scrambled. But they’re very, very scrambled problems. Here’s a game you can play: Take Diophantus’s Book 2 and read the first seven problems and guess what the eighth is. You’ll lose this game, surely, because one problem will be about biquadratics that would be a serious issue for any modern mathematician to obtain all of its solutions, and then the next will seem to depend only upon commutativity of addition. etc. You don’t see a progression. Also you don’t know what his audience is, why he’s doing this. I don’t think there is a secondary literature to help us much here. If you think of Euclid, second century commentaries are there in volumes and the commentarists are also very loquacious.

**Corfield**: Is that because that’s been taken up and woven into philosophical ideas, e.g., neo-Platonism?

**Mazur**: Well, you’re absolutely right.

**Corfield**: Diophantus was never taken up in that way.

At an international congress of mathematicians Weil gave a talk ‘History of mathematics: why and how’ and came out with this line that really only mathematicians could write the history.

**Mazur**: Yes but you still have to both be able to read the texts, and actually read the texts. Weil had complete control of all the languages and complete control of all the mathematics. Nevertheless… I think there’s such a progressivist thrust in all his historical writing, that he was only interested in what the texts anticipated and not that interested in what the texts were saying.

**Corfield**: You were talking about mathematical fiction, the story, the sweep of ideas, must it involve close reading then?

**Mazur**: My mathematical fiction does not involve close reading. It doesn’t—in essence—involve reading at all. It is historical fiction, and a different genre from what Weil claimed to be doing when he wrote history. You have to label these genres.

**Corfield**: When you’re doing your close reading work, you want to call that history?

**Mazur**: No, I want to call it reading, period. In fact I think reading (period) is very undervalued nowadays. An older breed of scholars—I don’t mean 19th century, I mean just after the 2nd world war—really read things, I mean really read things. This activity seems to be a bit less popular now.

**Corfield**: So we’ve got the mathematical fictions, we’ve got close reading…

**Mazur**: reading…

**Corfield**: and then there’s history of mathematics. So I guess that we should keep these genres separate.

**Mazur**: Yeah, I think one has to, I mean not because they are necessarily separate, but because humans only do one thing at a time well…if that…

**Corfield**: I was intrigued by this comment “It is ridiculously unfair to liken such an “explanation-future” (as X learning that a particular disease—known to X only by its symptoms—has a standard technical Latin name with Greek roots) to a mathematical formula—e.g., such as Gauss’s formula, the one cited in the footnote above that expresses a square root as a linear combination of roots of unity. It is unfair because, except for formulas that we label tautologies, any mathematical formula that equates one thing with some other thing is (if correct) valuable and is prima facie explanatory on some level or other.” Some people might think that there are contingent mathematical facts out there. For example, the 4th triangular number happens to be one more that the 3rd square number (10 = 9 + 1), but that’s not in the slightest way interesting, that’s just a fact.

**Mazur**: Yes right, I see. You call them contingent?

**Corfield**: I tried to do that in a paper of mine, picking up from Poincaré who says that the important facts of mathematics are those as in the physical sciences which lead to generals laws.

**Mazur**: That’s interesting. That’s a stance isn’t it? The sentence of mine that you quoted. It never occurred to me that it is a stance that needs real argument to support it, until you pointed this out. Maybe it’s not true that the important things are only important because they lead to general things. You use the word ‘contingent’. Do you really want ‘contingent’?

**Corfield**: Coincidence or something like that.

**Mazur**: Maybe ‘coincidence.’

**Corfield**: In physics there are some physical facts that don’t fall under the remit of physics. I mean some neutrino, from the sun, is passing through your body at the same time as a photon is doing some other thing. There are some facts and no way I can put them into any sort of package or explanation, any form of deduction from general laws. Surely most mathematical facts, according to some sort of measure, must be boring inexplicable, happenstantial …

**Mazur**: Happenstantial: I like that.

**Corfield**: That have no kind of general theory around them at all.

**Mazur**: You know John McKay. Well what I consider, and I’m sure what he considers, his greatest mathematical discovery, is the equation 196883+1 = 196884. Famous, and far reaching…

**Corfield**: Yes, that leads to a monstrous future I’d say. That’s a good example of a non-happenstential fact.

**Mazur**: I think that’s my vote for the most non-happenstantial mathematical fact.

**Corfield**: To start with, you can’t imagine what it is pointing to, but you know there has to be a big story behind it.

**Mazur**: That’s right, yes.

**Corfield**: OK, back to the notion of ‘explicit’ being relative, you say “until you the judge decide upon the format and the vocabulary, that what you will count as explicit you have no way of gauging who is the victor.”

**Mazur**: The sixteenth century guys had a formula for finding the roots of every cubic polynomial. And if the polynomial has 3 real roots the formula is a closed expression that—in some formal sense— solves the equation for getting the roots. But the formula will generally not give the sixteenth century mathematicians the actual 3 roots because at that time one didn’t know how to take cube roots of complex numbers in general. So here they have a beautiful closed form expression which purports to be a way of—what we would say is—explicitly solving the equation, yet which gives them not even an approximation to the real solutions of the equation. On the other hand we have Newton’s method, just a century afterwards, which gives a beautiful sequence of approximations converging to the solutions of the equation. And yet there’s something a little strange about saying that Newton’s method is explicit, because it’s not—by any criterion—explicit. You have to make an arbitrary choice even just to begin the game. So it’s a funny business. You have to say what you mean by explicit. From a certain point of view, Newton’s method would be an explicit solution and the 16th Century formula would not; and from another point of view, the formula would be the explicit solution and Newton’s method would be not.

I describe, in my essay, how Gauss solves a fifth degree equation, and how Klein solves the general fifth degree equation. Klein has a beautiful thing where he uses the j function. Are they explicit, these solutions?

**Corfield**: It’s becoming clear to me how you see explicit as a sort of sub part in a way of explanation.

**Mazur**: I didn’t see it before I wrote this essay.

**Corfield**: You talk about Descartes’ Treatise ‘Rules to the direction of the natural intelligence’, “I assume that Descartes was thinking, in analogy, of the degrees that occur in polynomial equations that cut out curves in “his” Cartesian plane”. He pairs them doesn’t he, one and two, three and four, etc. He called them genders. He’s debating with Clavius about the nature of mechanical curves. In that book by Donald Gillies, Paulo Mancosu has a chapter on precisely that debate.

**Mazur**: Oh yes, Descartes is making the analogy: degrees of thought/degrees of a polynomial; also in “Discourse on Method” even though there’s no mathematics there. There is mathematics in the appendix, but if you read it with a mathematical sensibility you see it.

**Corfield**: Right, but of course one can correct this. Descartes had an understanding of this step-by-step complexity, which then gets reformulated later on. You said that Galois has a different way of doing this, “we have an alternative way of gauging the difficulty of irreducible polynomial equations which I will hint at in this footnote”.

**Mazur**: Yes, it’s the complexity of the Galois group, whether it’s soluble or not.

**Corfield**: You say “Just as in Gauss’s formula, where any square root of an integer can be expressed as a linear combination of roots of unity…, after Kronecker-Weber we know that any abelian algebraic number can be expressed as a finite linear combination”, and then talk about Hilbert’s reaction to the result.

**Mazur**: Hilbert is wildly enthusiastic…

**Corfield**: So someone transcribed the lecture, did they?

**Mazur**: His assistant transcribed every word he said. Hilbert’s talking about the exp(2πiz): “Es ist wunderbar. Es ist ein geschenk von Himmel.” He says this a few times, and just keeps going on with great amazement that you can get all abelian extensions from a single analytic function.

**Corfield**: I remember having a discussion with some mathematicians about the notion of a miracle. Is there such a sense of a miracle in mathematics? It always sounds like he thinks this is a miracle.

**Mazur**: He thinks this is a miracle…

**Corfield**: We were touching on this idea of laws in mathematics. The original notion of miracles was something that went against the laws of nature, so…

**Mazur**: That’s good: I like that.

**Corfield**: Are we seeing that here?

**Mazur**: I don’t know; it never occurred to me.

**Corfield**: I remember in that discussion, André Joyal coming up with the fact that all you have to do is add the square root of minus one onto the reals and you’ve got algebraic closure.

**Mazur**: and that’s a miracle, yes.

**Corfield**: And someone said, but there’s a general story in terms of algebraic closures that you can put that into, as though they were trying to give a unificatory kind of explanation of this fact. But there could be an example where unification isn’t really explanatory in a sense.

**Mazur**: Yes, but I can see how someone might say: just adjoin this single tiny square root and you will have solved all polynomial equations; how marvelous!

**Corfield**: So do we, later on down the track, do we have a kind of explanation of why this works, the thing that’s fascinating Hilbert here, why this…?

**Mazur**: Oh yes. Hilbert must be beginning to understand Kronecker’s Jugendtraum in his own way; he is also evolving his own version of class-field theory. But the budding vision is not just for the rationals: the exponential function alone is good for the rationals and forms the bench-mark for the type of answer one is seeking. Then Kronecker asks about quadratic imaginary fields—generalizing the template that works for the rationals—- and the answer that he hits on is that you use the j-function instead of the exponential, and a few other functions like the j-function. Very few functions still, so its quite economical, and you get a uniform way of constructing all of the extensions of this quadratic imaginary field.

**Corfield**: OK, so that rather ties into what we were talking about origin-stories and templates, doesn’t it?

**Mazur**: Yes.

**Corfield**: Sometimes, you know, what appeared wonderful at one point, later on, is just seen as something quite obvious. Is Hilbert imagining you’ll never get to the point where you’ll look back and say “see this is actually quite boring”…

**Mazur**: We don’t think it’s boring even now, and it’s 80 years later.

**Corfield**: Because it’s still like a live metaphor, rather than a dead metaphor, it’s still the template, the seed… ?

**Mazur**: Yes. Kronecker’s dream so far, has only worked for quadratic imaginary fields and what are called complex multiplication fields, the collective noun phrase describing, basically, the fields for which his dream can be made—with effort—to work like a dream.

**Corfield**: Is that non-falsifiable?

**Mazur**: No, it’s quite falsifiable and it’s quite a beautiful thing but it gets more and more complicated and yet the goal gets larger and larger; it’s a perfect example where the template is beginning to be molded around larger and larger goals.

**Corfield**: But you talk about non-falsifiability of a template because it adapts to the new conditions it needs.

**Mazur**: That could be. I want templates that almost work perfectly but don’t work perfectly. And that ‘imperfection of the fit’ will change the very notion of explanation. I think Kronecker’s Jugendtraum does this sort of thing, because it doesn’t quite fit and yet it fits enough so that I can re-organize my thoughts so as to still keep it as a guide.

**Corfield**: The contrast to that may be in what Andre Weil talks about. What is most attractive to mathematicians, these analogies, the “furtive caresses”. And then when you achieve the unification it becomes cold and as though it’s lost its sparkle. Is that in a sense because the template you found to have unified the fields doesn’t go through such transformations? Where the template doesn’t have to bend very much as you shift through a field, perhaps later then you see that initial spark as not so interesting.

**Mazur**: Weil says “as the Gita teaches us, knowledge and indifference are attained at the same moment”. I don’t believe it. I mean if you look at Dennis Sullivan, he seems to be continually excited by the simplest topological truths. The Jordan Curve Theorem, I’m sure that if he thought about it this moment, he would be as ecstatic, maybe not as ecstatic as Hilbert, but as ecstatic as he was when he first heard of it. Take a theorem that’s classical, that Dennis must have taught say a 1000 times, and every time he thinks about it there’s still this sparkle.

**Corfield**: Not as though it’s giving him new ideas. It’s just there.

**Mazur**: It’s there! And Weil claims not to be that way.

**Corfield**: Is that linked to the Bourbakian crystalline imagery. Aren’t their texts cold?

**Mazur**: Oh, possibly.

**Corfield**: How do you ever say when a dream ends, when it’s been exhausted?

**Mazur**: When you wake up?

**Corfield**: I talked to Peter Galison after the dinner and we discussed Wheeler’s approach, all these free floating ideas on the one hand, and then Bourbaki on the other. You could say that they are just two sorts of activities which will happen at different times in the course of mathematics, and just because you’ve set down a kind of ordering of the concepts at one point doesn’t mean that in future things won’t change. Is there some sort of sense of irreversibility that once some things have been organized one way you’re never going to look back and think that was a wrong move?

**Mazur**: I think that there are wrong moves.

**Corfield**: There can be wrong moves, but are there right moves and that will permanently be right moves?

**Mazur**: a good question.

**Corfield**: How can one even know that?

**Mazur**: I suppose you can have faith that this or that template is something that we will always have with us in some form or other.

**Corfield**: You cannot imagine if humanity carries on, let’s say it does for a 1000 years, you can’t imagine in a 1000 years time they’ll look back and say “Oh Langlands that was just such a big mistake.”

**Mazur**: No no.

**Corfield**: If they do say that, they would have gone wrong somewhere.

**Mazur**: Yes.

**Corfield**: It’s sort of unfalsifiable in a way.

**Mazur**: It’s unfalsifiable, exactly. I certainly think it will be seen as a good step.

**Corfield**: Did you say that we’ve only just begun?

**Mazur**: Yes.

**Corfield**: I mentioned this to John Baez who said “Well, we’ve always only just begun.”

Why begin at a certain point? Why begin with Kronecker’s dream when you were putting the seed in Kronecker’s dream back with Gauss?

**Mazur**: I was putting it back even further, to Pythagoras!

**Corfield**: So in some sense it’s the whole of the history of mathematics.

**Mazur**: Well, lots of the history of mathematics.

**Corfield**: I suppose we ought go back to ‘character’ taken from the narratologist’s point of view. In response to your usage, they said: imagine if they—the narratologists—came to a maths conference and told you “when we talk about groups we define it our way”. What would your reaction be?

**Mazur**: Touché! Instead of my mis-using the word character, then, let me revise my essay and use the phrase the agent of the tale. Now we do have in literature inanimate or non-organic objects acting as—let me not call them characters, but rather— agents of the tale. That’s easily dealt with as long as they are labeled as characters by the story. Orhan Pamuk’s novel ‘My Name is Red’ is an example. The book is many things at once and it would take a long time to say what it is, precisely, but every chapter begins with “My name is ____” and is narrated by the eponymous thing. I say thing because it is often some miscellaneous thing; sometimes it’s an object, sometimes it’s the murderer, for the novel is also a mystery story. Sometimes it’s the murderer labeled as murderer and you have no idea who it is. Sometimes it’s a character who might be the murderer. Animals. Colors. Inkwells too. Somehow this conglomeration of agents manages to tell a story. In fact, these are probably not interesting characters from the narratologists’ point of view, nor from my point of view, because they’re actually labeled characters. But there are other works of fiction, where unlabeled agent-like entities are at play. In the Fisherman’s Wife Tale, as told in Virginia Woolf’s ‘To the Lighthouse,’ it’s the sea that grows more and more turbulent, it changes color like those octopuses that change color. The sea in the told fairy-tale within the tale has emotions… and all the while the ambient Hebrides weather is somewhat in control. On top of this, the narrator is not quite a consistent-voiced being, but rather a kind of collage of voices. The narratologists have to give me a term for this collective agent that—that has an outer inconsistency and an inner consistency—and that voices the story.

**Corfield**: That’s an interesting question you put, to the extent that one deals with a person you imagine there is a certain consistency and when you see realizations of them in different contexts, you imagine there’s an integral core or something.

**Mazur**: An integral core of identity. And indeed in the collage of voices there is.

**Corfield**: That justifies you seeing it as a one entity.

**Mazur**: And you see that in Kronecker’s dream!

End of interview

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]]>McLarty: …

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]]>**McLarty**: You talk about MacIntyre contrasting three versions of inquiry: encyclopaedic, genealogical, and tradition-based. But are these really rivals?

**Corfield**: That’s the title of the book, isn’t it? *Three Rival Versions of Moral Enquiry*. They are rivals in that they embody different notions of rationality. One timelessly seeks timeless principles, another seeks something timeless but achieves this recognizing its historical situatedness, and the last one recognizes nothing timeless, seeing everything as necessarily historically situated.

**McLarty**: But do these have to be rivals?

**Corfield**: Encyclopaedic history gives a fairly smooth, glossed-over picture of the progress to the present, where the past is really just a prelude to the present. To the extent that there was any obstacle in the past, you raise it just to point to the way people behaving rationally overcame it.

**McLarty**: Well in history as in mathematics, sometimes they didn’t think of stuff yet. It’s not that there was an obstacle, it is just that things hadn’t gotten done yet.

**Corfield**: Typically when talking about why somebody’s contribution doesn’t get recognised, this history has that flavour to it that it is only an irrationality that was standing in the way. There is no notion that actually you have to have spent a long time living in a tradition to be able to understand somebody’s step as something progressive.

**McLarty**: But, for example, Lagrange can tell a lot about the roots of a polynomial even when he cannot actually find them, but he does not think of the Galois group yet. Most people do not want to say some failure of rationality stopped him. He just did not get that far. Galois did.

**Corfield**: No, but if you carry the story on you want to explain why Galois is not received very quickly when he does introduce the groups. Could we say that Galois himself is in a tradition or that he is trying to begin a tradition?

**McLarty**: Oh, he sees himself in a tradition.

**Corfield**: So, those not following him, to what extent are they thoroughly immersed in the tradition that he sees himself to be part of?

**McLarty**: He feels the people that don’t follow him are failing that tradition. They should have understood what he showed them. Cauchy out of pride refuses to recognize Galois’s contribution, even though it is in exactly the tradition. So there is an obstacle. An obstacle of pride. These people are unwilling of recognising a valuable new idea.

**Corfield**: This seems quite compatible with tradition-based history. One needs the notion of the virtues and Cauchy is failing according to one of the virtues, so he is not a good member of the tradition in one sense.

**McLarty**: It also fits with an encyclopaedic enquiry. I don’t believe Galois’ version of it exactly, but if you did then it would fit with an encyclopaedia enquiry.

**Corfield**: Just because you can tell two rather similar stories that means we might not have been able to split those two versions of inquiry in that case, but it doesn’t mean that we can’t find cases that wouldn’t split them later on.

**McLarty**: They are different, but I am asking, do they need to be rivals? Now clearly, the logical positivists think theirs are rivals to any Nietzschean kind of explanation. They certainly say if they’re right then he’s wrong. And of course, like you said, their epistemology makes him wrong.

**Corfield**: There are occasions when MacIntyre is going to invoke a genealogical history. For him there was never an actual defeat of Aristotelianism. Rather the Aristotelians failed in some sense, failed to follow their tradition. So he gives a genealogical explanation of why alternative forms of moral thinking have arisen in its place, right up to emotivism in the early twentieth century.

**McLarty**: Is he also happy to give or at least to draw on encyclopaedic versions?

**Corfield**: Is he being unfair in a way? He is allowed to avail himself of genealogical history when it supplements his. But he has the idea that the genealogists can’t avail themselves of anything like the tradition-based notion of history because they will always see aspects of tradition as a masking of power relations. But is that fair to them?

**McLarty**: Well if you want to say it of Nietzsche you are going to have to say he was kidding himself when he wrote on Uses and Abuses of History, because he says there are these different uses. There is memorial history, where you really are memorialising the great achievements of the past, but not necessarily as depicting them leading to us now. We may have fallen away from them but they are the great achievements of the past. And surely he must have some kind of concept of a dry history which is just going to tell you what actually happened.

**Corfield**: Collingwood calls this chronicling.

**McLarty**: Then a genealogical conception where you undercut things by discovering where they really came from.

**Corfield**: But then does he have any sense that the one can tell a story of progress as understood within some kind of body of thinking?

**McLarty**: Well this is what makes it a pressing question to me because it appears to me that Nietzsche finally doesn’t believe the memorialising history. He depicts it as one function that history can have, but he doesn’t in fact respect it at all. He really does only respect this deconstructing genealogical extreme. Yet, he depicts four uses which all exist.

**Corfield**: But no-one’s claiming they don’t exist.

**McLarty**: But did he have to also take this extremely invidious position that only one is the right one? On the face of what I would like to say well of course they are not rivals, “Let one hundred flowers bloom”, but on the other hand the positivists did consider themselves enemies of phenomenology and Nietzscheanism.

**Corfield**: Is it philosophically possible to claim that there are *good* histories? What do we mean by “good” at that point. What is a good history?

**McLarty**: Well I wouldn’t like to try to do the history of twentieth century math without Dieudonné’s books. On a first reading they’re fairly encyclopaedic, but then they arrive at this *telos *of Grothendieck, so are aiming at one polemic purpose.

**Corfield**: Well that would be encyclopaedic if it is dismissive of the past as something we cannot learn from. Is he saying we have everything we need now by following Grothendieck?

**McLarty**: He is very happy with all of twentieth century topology and algebraic geometry. Each of his histories comes up to Grothendieck and stops there, and says we’re not going to get to this. The history of differential and algebraic topology is a history of wonderful ideas we still use, but it’s encyclopedic.

**Corfield**: Does he believe you must be trained to some extent to appreciate that these are good ideas? I mean it is not something open to any rational person, is it?

**McLarty**: The thought is you should be able to train yourself by reading it, and then you’ll be someone who will be able to judge this history. This is the classical encyclopedic understanding that you can train yourself by reading it.

**Corfield**: The Encyclopaedia Britannica is a great example of precisely this idea of aiming at the common man, or at least the rational man. You may distinguish people by class, or perhaps by gender, those who can and those who cannot achieve this level of rationality, but there is a large audience.

**McLarty**: So professional historians like to dislike Dieudonné’s histories because they are too dry, too progressivist. And yet, who would want to be a historian of twentieth century mathematics without those books to go look at?

**Corfield**: To a large extent historians of mathematics, especially genealogical ones, haven’t gone for the twentieth century. I know one historian who was interested in how they were going to depict Wiles. The idea he came up with was to question the represention of Wiles as the lonely hero in his garret and unmask it, but not to include anything of what he was thinking.

Leo Corry has moved to the twentieth century, hasn’t he, with his book on structures. Now how would we want to think of that?

**McLarty**: To me what he achieves in that book is fairly encyclopaedic but that’s not the only vision Leo has for the history of mathematics.

**Corfield**: Does he suggest that there is almost an irrationality in Bourbaki’s reluctance to accept category theory?

**McLarty**: Yes, and he is critical of some people being too quick to take an image for the body of mathematics. People should have been able to see in the first place that this was just an image which wasn’t going to work as body.

**Corfield**: And yet he was the one at the last conference who was wanting to make a big distinction between the historians’ history, people who didn’t see any forms of necessity in the course of events, and mathematicians’ stories. These he ran together with mathematical fiction, such as Uncle Petros, writings we don’t judge critically for accuracy, where generally we don’t suspend disbelief. We don’t go reading Uncle Petros and say of some event that it couldn’t possibly have happened, because we’re engaging with a work of fiction. He said that you should approach the mathematicians’ stories in just the same way. He took us back to Aristotle’s ideas of how there is a story-telling where one gets away from the nitty gritty details of what actually did happen and one tells a tale of what *should* have happened. This is something the modern historian is trying to get away from, these Whiggish histories. It was as though he was presenting himself within the genealogical line.

**McLarty**: In the *Structures* book he does constantly distinguish between what the mathematicians are doing in the bare sense of what they are doing and how we should think about that. On the other hand, I would still say that the real achievement of the book is fairly encyclopaedic because it is entering into a vacuum that needed at least an encyclopaedia as a start. Maybe I am just projecting my own values onto it but I don’t see how you can begin to do a twentieth century math if you don’t at least have an encyclopaedic history, and then you can proceed to be more critical and evaluative.

**Corfield**: Perhaps issues are clearer when you look at histories of moral thinking. You see different schools and they seem so far apart. It is not hard to believe that if you are not, say, an Aristotelian then there is a lot of baggage you have to learn if you are to understand their way of thinking. MacIntyre gives the example of the English when they come to Ireland bring along with them their long history of property rights. Their legal system confronts an Irish culture which doesn’t have these concepts. There is no neutral standpoint from which you can assess things. You are either in one camp or you are in the other. These two cultures are so far apart that they can’t understand each other – they’re incommensurable. Perhaps in mathematics instead of talking about different languages we should say they use different dialects.

**McLarty**: Certainly within the twentieth century you don’t have schools of mathematics that flatly can’t understand each other. In a way though I wonder if we are dealing with a kind of thing that Kuhn talks about sometimes. When the English come to Ireland neither the English nor the Irish, at least not the leaders of these communities, are deeply immersed in the studying of conceptions of property. But when we look at the Princeton topologists they are deeply immersed in the study of topological theorems. They have discussed this to the limit of their ability already, that’s how they got where they are. The English had not discussed conception of property. They were not interested in discussing conceptions of property.

**Corfield**: But then they would have reported back cases to be heard to the experts back in England who would then settle the dispute.

**McLarty**: But even those experts are going to be subject to massive practical pressure, leading to mistakes on understanding conceptions of property. For example who will have the property? Who is going to get to eat what grows on that field? Whereas in math we are dealing with people that have already argued the issues out, a lot. I am not saying that any debate has ever really exhausted all the possibilities but they have been debating exactly these issues a lot. So a disagreement that remains after both sides have been working on the same problem for a while is a hard disagreement to settle.

**Corfield**: Yes. MacIntyre certainly doesn’t believe that necessarily there is going to be a resolution, by any means. But one ought to be very aware of the conceptual problems within one’s own field and be at least aware of the fact that the other camp may be able to have an insight into it too, and make sense of it in their own terms.

But, as we were saying, in mathematics perhaps we are just talking across dialects rather than whole conceptual systems. In my book, in the discussion on groupoids, there are not exactly schools involved, but there are individuals who are saying “Groups are fine. They capture symmetry. I don’t like what you’re doing – it’s not giving me anything new. Groupoids are an unnecessary elaboration”. There is clearly a frustration on the side of those who are proposing groupoids, like Ronnie Brown. “For years and years we have been banging on about the good things we can do with groupoids and it is never ever enough for these people on the other side”. Grudgingly they might admit that something might be useful or convenient, but it doesn’t go to the essence of the matter. There is no neutral court where we can say “You’ve satisfied this and this criteria so you’ve won and you haven’t won”.

It’s not as sharp as the Irish- English kind of difference, which from the native Irish perspective must have seemed bizarre. The same with Native American Indians when the Europeans came and imposed this completely foreign system. How on earth could they understand what was going on?

**McLarty**: Maybe that’s one way of putting things – the ease of considering alternatives. You talked earlier about ways other things could have happened. I have not really absorbed everything the Princeton school thought about topology. And I sure don’t feel in a very good position to find some other way it could have been done. I am supposed to invent an alternative to early twentieth century topology, as a historian considering it could have gone some other way? Whereas it is terribly easy to think of other conceptions of property besides the English and the Irish in that situation. Partly because neither of those groups had it as one of their major projects to come up with conceptions of property. So it’s not hard for us to think of alternatives. But could Poincaré have done something else than Analysis Situs to solve his problems? Am I supposed to come up with a plausible alternative?

**Corfield**: The genealogists say things could have been otherwise, but I suppose it is not up to them to come up with the alternatives.

**McLarty**: But in issues like the property debate it is terribly easy to come up with those alternatives, whereas in mathematics it is hard to come up with alternatives as to what happened.

**Corfield**: So let’s think what do they appeal to? The Bloor school were interested in the Intuitionism debate. Brouwer could have won. Mathematics might have opted for a different logic. Maybe that’s easy just to say that someone could have won.

**McLarty**: Yes, but then how do you explain that in his lectures in the fifties Brouwer is saying that he really doesn’t have an adequate proof of the fundamental theorem of intuitionist analysis. Brouwer evidently didn’t find this alternative way. But you can ignore that and just say that there was an alternative way.

**Corfield**: Right. Bloor goes right back to two plus two equals four. How could that have gone differently? Lakatos gave the example of adding things in containers, so that when you add two containers it makes a difference to the sum.

**McLarty**: Chemists certainly know this kind of thing. You add two gallons of alcohol to two of water and you get maybe 3.5 gallons of liquid. This is a perfectly clear well known fact. We don’t normally consider it an alternative to two plus two equals four. So you could say that it is an alternative to two plus two equals four, but it would sound silly to most people.

**Corfield**: And yet they think they’ve got an easier task in mathematics than in something like science. One would imagine that the world is a big constraint on science. Could physics have gone very differently? It is hard to imagine. Whereas in mathematics there is an emphasis on freedom. If Cantor had been shot when he was 5, would we have any thing like Cantorian set theory?

**McLarty**: Yes, because of Dedekind.

**Corfield**: It would have looked a little bit different.

**McLarty**: Yes, but Cantor set theory was different from Zermelo’s.

**Corfield**: It’s a funny game to play. What about geometry?

**McLarty**: Aristotle asks what could we do if we didn’t have the assumption on the parallels. He says we could not prove the diagonal of the square is not commensurable without the assumption on the parallels. Well that didn’t mean that anybody ever developed a geometry without it at that time. It’s not clear whether Aristotle meant you could actually do geometry without that, or that you would fail to do enough geometry if you had not thought of the parallel postulate.

**Corfield**: I noticed in a history and philosophy of science department I was working in that the historians were always surprised that the philosophers asked counterfactual questions of history. What if such and such had happened, would something have been different. The historians would express amazement that one could think that way, because what happened happened. There is a certain historian who doesn’t play this game of ‘things could have gone differently’. This is a straightforward historian who is just trying to illuminate the times in which certain developments took place. But I think that there is that brand of historian that has a more philosophical mission behind what they are doing.

**McLarty**: Well, Shapin and Shaffer, they declare we are going to see how this science could have gone otherwise.

**Corfield**: It’s all determined by what’s going on politically in the seventeenth and eighteenth centuries.

**McLarty**: So I’m left really wondering how rival these things are. On the one hand, I have an impulse, why can’t we do them both? But then the practitioners think that they can’t do both. The logical positivists were intolerant of the existing genealogical account in the Nietzsche – Heidegger tradition.

**Corfield**: But we don’t have any examples of people that do straddle these different versions, do we?

**McLarty**: Well, on a lower level, surely. Shapin and Schaffer had immersed themselves in works of many historians who disliked Shapin and Schaffer’s approach. And I hope they felt that some of that was profitably. They are going to value those works of history whose methodology they don’t share. Just in the lower level empirical sense that they read it and valued it.

**Corfield**: So that’s the meta-question you want to put them. So we’re hearing how histories are much better now. In history of science they’ve learned the mistakes from the past. They re much more subtle. But what does it mean to say they are doing better history? How do you construe that notion of ‘good’ or ‘better’ unless you yourself have a notion that you belong to a tradition that is improving and you’ve overcome the obstacles of the historians of the past. Otherwise, why can’t we ask what they are trying to do, what they are trying to achieve? Just some dominance in the field of the history of science? They must have some sense that they are doing something better.

**McLarty**: Another thing I was wondering out of the paper and the talk. You know everybody is against Platonism.

**Corfield**: Yes, whatever that means.

**McLarty**: How much of that is because of this Aristotelian alternative? How much of this is because they like Aristotle more than Plato because of the virtues in Aristotle that MacIntyre has in fact talked about. It is not just that the nominalists are against Platonism and the structuralists are against Platonism and the modalists areagainst Platonism. Everybody is against Platonism.

**Corfield**: Apart from Penelope Maddy in her early days. But we can ask whether this is the Platonism of Plato, as you discussed in your paper that Glaucon is in fact closer to modern Platonism.

**McLarty**: Can we explain this wave of dissatisfaction with Platonism, not by attributing it to those people having learned Thomistic Aristotelianism, because they haven’t, but can we explain it in fact against their will by saying they are recognizing the flaws that Aristotelian Thomism has articulated?

**Corfield**: So invoking notions of the adequacy of the mind to its objects.

**McLarty**: And certainly you could take that reading of Benacerraf’s paper about we can’t know mathematical objects. Precisely he is pointing out that the mind can’t be adequate to the kind of mathematical objects that he believes Platonists are describing. He wouldn’t describe it in terms of adequacy of mind but we can say that is exactly what he is complaining about.

**Corfield**: Behind this MacIntyrean notion there is some form of realism, of the virtues, not as eternal objects but as something one’s mind can become more adequate to. Is that option really available to the people that are unhappy with Platonism?

**McLarty**: But we as Thomists want to absorb the thoughts of all these other groups.

**Corfield**: We should want to explain the frustrations they are feeling. And to show them their lack of resources. Their only alternatives in the form of realism are abstract objects or things existing spatio-temporally. But, if abstract, how do we have contact with them? If spatio-temporal, where are they? Their alternatives derive from causal theories of knowledge and these are not Aristotelian notions.

There you go. Aristotle provides you with more resources because he has a richer notion of cause, some parts of have been abandoned. This is precisely the sort of exercise we should be engaging on. Does analytic philosophy of mathematics recognise it has a problem or is it happy with its account? Are your Hales and Wrights happy?

**McLarty**: Well they certainly recognise they have problems. One of their books is about seven successive attempts to explain what proper names are and they find problems with each one. So they are aware of problems. But it is perhaps not *the* awareness of *the* problem that you or I might think they should have.

**Corfield**: Right, so we can set ourselves the task of showing that we have the resources to explain the frustrations that they are experiencing, and this doesn’t mean that we think that they should recognise our explanation of their failures. But after a time, when their frustrations are never released, they will have to think about the ways other people view their story, and whether they have the resources to explain their problems.

**McLarty**: So each field of analytic philosophy certainly recognises that it has problems. But you have this Aristotelian Thomistic view that we actually know something more systematic about these problems. They are not local to each field.

**Corfield**: There is a large distance between their way of thinking and ours. Don’t you feel that frustration all the time? You want to say “Surely you can see that things aren’t going very well for you. Look at the questions you’ve been led to ask. You must be missing something.” But yes, to be consistent in this line of thinking, one ought to do just what you say and explain the pattern of their failures.

**McLarty**: Well there is this MacIntyrean project to engage with all those alternatives, at least to your own satisfaction, if not to theirs.

**Corfield**: It may be a long process, not an easy process. Perhaps I’ve been guilt of employing encyclopaedic thinking myself, imaging that they would see what I see as obvious, believing that any rational person should recognise that they are going wrong because they are not dealing with any content of mathematics.

**McLarty**: Well that brings me to the content of mathematics. Are there objects that mathematicians can conform their minds to?

**Corfield**: How to treat the notion of object? In moral thinking when the mind is becoming more adequate to its objects, we include things such as the virtues. They’re certainly not like tables. It is clear from the way the mathematicians describe their experience, it feels to all intents and purposes as though they are getting hold of something, they perhaps do not have at the moment. Their minds are not adequate, they know there are questions they do not understand about certain situations and they expect that some time in the future they will be able to tell us the story of a field that will make sense of the problems they felt before.

One has to be careful about the question of object there. I like this idea that Michael Harris had. Hacking had this notion that in science there is no question that particles exist. Not quite like tables do, you can’t buy them from Ikea, but by the time you get to be able to control them, you can use them to do other things, in experiments you can shoot them at balls of niobium and look to see the charge you put on these balls, and see where you’ve got fractional charges.

**McLarty**: Some of them can be shot. Others, like mesons, almost never exist except virtually.

**Corfield**: Right that is a further twist to the thing, isn’t it? For Hacking, “If you can spray them, they’re real” was the famous statement. But as you said there is a lot more to be said about particle existence. But Michael Harris’s point was “Look at mathematical ideas, you can steal them, so they must exist.” What happens when somebody writes a paper and you say you just used my idea. What does that mean to say “that’s my idea you used”? A tricky question, and yet there is quite a robust sense of what that means.

So maybe one should approach this notion of object through the grammar of the way you talk about the thing. It doesn’t have the same grammar as for tables. If I walk off with the table it’s not there for you, but if I walk off with your mathematical idea you still have it. There’s clearly a difference.

**McLarty**: One approach which attracts me sometimes is to say “Sure there are objects in mathematics, they are space and number”. They are not groupoids. Groupoids are an attempt to become adequate to those objects, which turn out to have a lot more in them that you might think. You think space is only three dimensional. Yes, but there are also sets of five particles in space. And in the most naive sense the space of their configurations is fifteen dimensional. We are not talking about some alternative universe. So that the objects really are just space and number.

**Corfield**: That is Pierre Marquis’ line in one paper, isn’t it? That mathematics can be divided between the machines and the objects. Algebraic machinery that is going to do the measuring, like a homology theory which measures aspects of a space. And that seems like a neat picture, but then you go and do tricky things like collect these algebraic objects into their own spaces, and start investigating them as spatial things. It becomes rather less clear what exactly is the thing studied by something else, and what is the thing itself. You can get this reversal effect sometimes where you start studying the machinery as a mathematical entity. It’s not clear that the status is a permanent one.

**McLarty**: But he is interested in a different division. I am suggesting that really just space and number are the objects and all the rest is a means of becoming adequate to them. Whereas he wants to draw a divide where a lot of the objects are the things to study and some of the others are the tools to do so.

**Corfield**: He certainly has space in the things to study.

**McLarty**: But even there, I don’t think he means just the space around us.

**Corfield**: I see, so you literally mean the space around us.

**McLarty**: Now, this space has in it surfaces. It has families of surfaces. Now we have an infinite dimensional space which we have discovered in the three dimensional space. But those infinite dimensional spaces are the mind’s attempt to become adequate. Those aren’t the objects.

**Corfield**: So you think all mathematics can be phrased as some attempt to grasp space and number. So what is number then in this sense? Number is something to do with discrete entities? And then what is an imaginary number?

**McLarty**: Well we have this whole history of making our minds adequate to space and number, and you can see imaginary numbers arising in that.

**Corfield**: As tools to make our mind more adequate, not things to be studied.** **

**McLarty**: To me what are imaginary numbers? Well Feynman tells us what the unit imaginaries are. They are positions of a clock hand. Of course they are. We see this all the time.

**Corfield**: And if minus one is understood in terms of half a turn, then the square root of minus one is a quarter turn.

**McLarty**: Yes, we see clock hands make half a turn. We see electric waves go half way through an entire phase. It just happens all the time. And these are complex numbers.

**Corfield**: We can clearly make visual pictures of large chunks of mathematics, but I remember someone trying to teach me Borel sets. And when asked how he pictured Borel sets, he said, “I don’t really”.

**McLarty**: Some people won’t. But Borel sets come out of Fourier’s attempts to describe temperature.

**Corfield**: Right, for anything you are always going to be able to tell a story that leads up to it.

**McLarty**: A true one! Take the standard conjectures. Here’s Grothendieck saying, “you’ve given up on a simple vision and if you had not given up on it you might have achieved it”.

**Corfield**: So you haven’t fully allowed your mind to become adequate.

**McLarty**: Yes. We would now have a far simpler proof of the Weil conjectures if only people had had faith that they could prove the standard conjectures. Grothendieck is saying they didn’t make their minds adequate. Because they didn’t look for enough simplicity.

**Corfield**: Ok , but does that tie to the space and number idea?

**McLarty**: A lot of the time it’s not that there are objects the mind has got to conform itself to. The mind creates these things. Like the way we invented radio but also the way we invented the Dewey decimal system to classify books.

**Corfield**: But isn’t that more arbitrary? There it could have gone a hundred different ways, couldn’t it?

**McLarty**: Yes, but you see this with group versus groupoids. Of course anything you can do with groupoids you could have done with groups. But that doesn’t make it arbitrary these arguments that this problem really does need groupoids.

**Corfield**: But then nobody is going to say of two library systems that one is the good, the right way of classifying books. One may be better, but do people think they are approaching the best form of classification system?

**McLarty**: Well it’s a lot more complicated a problem. To take a simpler problem than groupoids, whether we should use matrices, it was a big question in the late nineteenth century. People said “I’m not going to use matrices because it’s really just these equations”. Clearly you have a free choice to do it either way.

**Corfield**: There are some people, like Alan Connes, who are ‘realist’ about these things. You can tell you are on the right track because amazing and surprising things will happen if you follow a better path, and if you don’t follow the path you won’t meet these amazingly unexpected effects.

**McLarty**: But this realism to me it doesn’t make the decisive issue on matrices. Nobody said you can’t use matrices and nobody had any application of matrices that you couldn’t have done without using them.

**Corfield**: But won’t there come a point when the explosion happens, so that if you hadn’t gone down the path of using matrices, something would be blocked off to you? It would come to a point where you can’t follow those who did go down the path of matrices.

**McLarty**: There won’t come a point where absolutely you can’t follow, because we know a translation. And then yes the opponents of matrices will say that conceptually you *couldn’t* have taken that path without matrices which means you *shouldn’t* have taken that approach. The history of math is full of this – “Sure they’ve proved this theorem but they don’t really understand it because they proved it with tools I don’t like”.

**Corfield**: Sure, but how do we judge that comment? Aren’t you being judgemental about that line? Aren’t you’re thinking they were wrong to say that?

**McLarty**: I was putting it that way when I said it, but surely you can be right or wrong about saying that a certain concept is good or not. But on the level of the question we are talking about now, it isn’t that there were objects the mind had to conform to. The mind is conforming better to itself, by using this method. The object is no more a matrix than an abstract linear map or a bunch of equations. The mind becomes more adequate to itself by thinking of this problem in matrices and this one in linear transformations.

**Corfield**: Can we tell a similar story about the virtues, that our minds are not becoming adequate to pre-existing objects?

**McLarty**: Hegel certainly would say increasing understanding of the virtues isn’t getting better and better at understanding something that was in the world without thought. Of course, for Hegel there is nothing in the world without thought. It is a matter of coming better and better to understand yourself.

**Corfield**: By objects is not necessarily meant things that existed independently. You don’t have to limit yourself to these by any means.

**McLarty**: But except, when you talk of your mind having objects to conform themselves to, that makes it sound like the objects were there before your mind set about conforming itself. And now it is up to your mind to conform itself to them.

**Corfield**: When we look back in time and see the trail the mathematicians took, you want to say that there was a kind of reality that they were engaging with, that we actually have a better grasp on now. And we can see them grappling with this problem and we start to see some of their thinking becoming what looks to us more adequate to some kind of structural possibility.

**McLarty**: But to some extent when I do that kind of exercise, it’s not that there is this reality that they are grappling with, it is that they’ve brought a conceptual apparatus to bear, and they’re grappling with their conceptual apparatus. Wonderfully. I am not saying that they messed up, but that that’s what mathematics is about. It is about dealing with its own address to these problems. And sometimes dealing with tremendous skill. And sometimes failing of that to some extent.

**Corfield**: One big issue is this notion of perfected understanding. What is one to make of that?

**McLarty**: And what role does that play for MacIntyre?

**Corfield**: It’s not something one can know one has achieved. Here I am now and I think that my understanding is better than someone’s twenty years ago. And I don’t just think this from my own perspective, but in some objective, timeless sense. So, what do I mean by that? I can’t just mean that the people in the future will look back and say there is a story we can tell which takes us from the original person, through me now, on to us in the future. I can’t think that and then think that some time further in the future than that, people will reverse their opinion and think that that wasn’t the correct order in terms of seeing an improvement of understanding. At any point there are some aspects of partiality in anyone’s understanding. Later on the understanding of a particular point will be considered inadequate. I am not saying this just about the understanding of a piece of mathematics, but also about the understanding of previous understandings that they were in some sense inadequate. But unless one has this regulative ideal of perfected understanding then what can it mean to think that one has an improved understanding?

**McLarty**: So does MacIntyre talk about particular cases of perfected understanding?

**Corfield**: No, he says that people at some point thought they had achieved perfected understanding and were proved to have been wrong.

**McLarty**: Well this is the state of projective geometry after Veblen and Young. There were these various problems that people had been addressing for 60 years. Veblen and Young write a two volume book, and they settle some open problems, substantially all the open questions, without creating any new questions. And the subject in fact withered for a little while. What brought it back was doing it over finite fields, a radically new context.

**Corfield**: So that raises the question, is it the same when it’s taken to this new context?

**McLarty**: But also you could say our understanding of classical projective geometry really has not moved much beyond Veblen and Young. They really produced a pretty perfect understanding of those problems. Now we’ve found some allied problems.

**Corfield**: We can’t say that we know they have achieved this, but we have an understanding of what it would be to show that they didn’t have a perfected understanding. The understanding of that changes in time. Because one may think one can’t conceive at the moment of any way of showing that they are partial but in the future people have a better understanding of that partiality. So we’re far from that principle of some analytic philosophers that if you know something, then you know that you know it.

**McLarty**: Yes, certainly that kind of thinking isn’t going to help with this.

**Corfield**: The question here is can one have this theory, this tradition-based theory of rationality, without this regulative ideal of a perfected understanding.

**McLarty**: Well I am not against this idea, I just don’t have it yet.

**Corfield**: Again our understanding of what it would be would clearly change in time. Generically, all you are going to say is in contrast to this notion of being shown to be partial. And with the understanding that what it is to show something as partial can change. We know examples from the past of people who believed themselves close to something perfect, for instance those trying to remove the blemish in Euclidean geometry – if we had a clear fifth postulate we would have perfected geometry. We now see from Hilbert’s perspective, that even if they had succeeded, they would still have a partial view in some sense. So we certainly have a conception of people discovering that people in the past who had some notion of perfection in fact failed.

**McLarty**: I am not supposing that MacIntyre some silly naïve idea here.

**Corfield**: There is an old Thomistic account of the understanding, but then that takes us to theology, and I am not sure where I want to go with that, where we discuss the angels’ understanding. For Aquinas this is different from humans’ understanding, and in another way from God’s understanding.

**McLarty**: I suppose in Aquinas that’s not something we can use in any way. We just would have no contact with angels’ understanding, I would guess.

**Corfield**: We certainly can never have a perfected understanding of angels’ understanding, but we can talk a bit about it. There is a rich theory there, but I don’t know how much of a risk it is to buy into MacIntyre, and what the costs are of adopting his resources to help us here. How much of Thomism must we adopt if these resources help us to a philosophy of mathematics where we are paying more attention to the content of what mathematicians think, studying how their thinking has changed and yet we don’t get taken down some relativist genealogical line? What resources do we have, which philosophical framework provides us with this opportunity to do what we think we ought to be doing, which is studying the changes in mathematical thinking? I don’t know who else can help us here, but as I said, possibly it leads us to some sort of metaphysical theology. Are we going to be taken down that line?

**McLarty**: I was thinking of Hilbert’s endorsement of I forget, he says some French mathematician, who says a mathematical theory isn’t complete until you can explain it to the first man you meet in the street. So in that sense Veblen and Young produced an angels’ understanding of projective geometry. Maths professors they still couldn’t explain it. Well if they were on the street they would need a good bit of patience.

**Corfield**: Do you think that was the motivation for some of his writings, when he writes with Cohn-Vossen.

**McLarty**: Yes, his book *Geometry and the Imagination*. And he does projective geometry in it. At least the point-plane duality and the projective theory of conics. Again you know if I walked down the street here most people would probably lose interest, but he takes great results and makes them really transparently available.

**Corfield**: Of course if somebody doesn’t have the desire to learn then they can resist.

**McLarty**: I think MacIntyre gives access to that kind of thing. I think he raised some issues that you do well bringing them into mathematics. I think this is a very good framework. But so that makes me want to see a pursued much more concretely.

**Corfield**: I think by bringing MacIntyre into philosophy of mathematics it could help us make contact with philosophers who haven’t even thought that mathematics could be of any interest to them whatsoever. They might think that there are some technical problems that people are trying to sort out which have no bearing on what they’re doing. But there are some moral theorists, interested in virtue ethics, we could make contact with.

**McLarty**: What is this comparison with virtues ethics?

**Corfield**: Very similarly to the case of mathematics, with theories of meaning which analyse statements with logical tools, then unless you rewrite moral statements, it looks like you are going to be committed to what they refer to. So if you want to say, for example, “Murder is wrong” or “Courage is required of us”, if your notion of reality doesn’t allow such things as courage to exist, then you are going to have to do some rewriting. There is a dissatisfaction with this approach and a desire to revive a richer discourse of the virtues.

**McLarty**: What I know of virtue ethics is from Nussbaum’s *Fragility of Goodness* where the point is not to defend the claim that courage is required of us. The point is to depict virtue and courage in such a way that people will want them. The point of virtue for her, as I take it, is not a bunch of claims of what you ought to do. It is depictions of things that when you have seen them you will want to do.

**Corfield**: And to go back to Aristotle, to be in a position to see that they are desirable takes a certain upbringing and moral environment.

**McLarty**: Yes, you cannot teach ethics to men who have not been brought up well.

**Corfield**: So there is a flavour of that when MacIntyre suggests with the notion of tradition in science, that one must have been brought up in a certain way to understand what the problems are in science. This resembles the Kuhnian idea of the profound transformation that takes place in you as you go from a young person to the trained scientist. There’s a heck of a lot of baggage and ways of seeing the world that you’re taking onboard, to be able to parse scientific activities.

**McLarty**: Does that mean that there can’t be philosophy of these things, because there can only be philosophy by the people who are in these traditions?

**Corfield**: Like, as a moral philosopher, one must be part of a moral community.

**McLarty**: And this is what Penelope Maddy says in her current book. The second philosopher pursues questions of methodology and mathematics because she is pursuing mathematics and would not otherwise. It is only because she is in this community. So now we’ve got that Maddy and MacIntyre agree.

**Corfield**: For Maddy, they are just trying to distill methodological principles out of the practice, aren’t they? She doesn’t want to use the word ‘philosophy’.

**McLarty**: In her new book *Second Philosophy* she wants to not back off the word ‘philosophy’.

**Corfield**: But is there still the idea that you’ve almost got to help the mathematician out, who is inclined to go off on some foggy metaphysical quest?

**McLarty**: Well, you should be a mathematician yourself if you are asking these questions, and you should approach them from the point of view of efficacy towards your goals. The way we choose correct method in mathematics is by being mathematicians, and asking what will help us achieve our goals.

**Corfield**: Right, but she made the point in her previous book that mathematicians themselves sometimes become first philosophers, and so one has to help them by showing them there is a lot of gas going on there, and that they should stick to mathematical reasons.

**McLarty**: As I look at it in the current book she is less interested in helping out those people. Those people are fine by her if they are proving theorems, she’ll learn their theorems, insofar as she is a mathematician. But when she is trying to answer questions of methodology her standard is which method will advance my goals.

**Corfield**: In the previous book she talked about the goals as not being up for question, which seemed a rather bizarre point of view. I got the impression she was saying that once they are established, it wasn’t the methodologist job to question them. She argued against the axiom V = L in set theory by relying on some pre-determined goals and showing us that V = L doesn’t help us towards those goals. But it wasn’t clear to me from her earlier work how the goals were established.

**McLarty**: But how could you approach that from MacIntyre’s point of view? She is here referring to a community of set theorists, a tradition of set theory. And this community will change its goals from time to time to some extent. But not by stepping outside and saying whether this is a good goal or not.

**Corfield**: But for MacIntyre there ought to be the possibility that, because of some form of partiality of view, even if they are modifying their goals, they will still experience frustration. And it is possible this may be explicable in the terms of another group. Take someone like *Angus* MacIntyre who claims that there are limitations within the set-theoretic viewpoint, that it’s not going anywhere. It’s run out of steam.

**McLarty**: It’s not that he is saying you shouldn’t base model theory on Zermelo-Frankel set theory. He is saying you shouldn’t conduct you work in model theory with a lot of concern for set-theoretic issues. You should be looking at higher level issues of, the way I would put it, mapping and symmetry, things suggested by algebraic geometry in particular.

**Corfield**: So, if there are some people who are constraining themselves from his point of view, there is a claim of partiality. Now, they don’t have to recognise Angus MacIntyre’s claim, do they? But from the other MacIntyre’s point of view, his approach should start to achieve important new things.

**McLarty**: Well Angus feels that his approach *does* achieve the goals of current model theorists. The field has moved in this direction. He is not against the goals these people had. He does feel to some extent that some people are sticking with older methods where they shouldn’t be.

**Corfield**: Oh, but they agree on goals?

**McLarty**: Yes, largely. There is differences in detail. How important is it to classify all the models of a given theory? There is plenty of disagreement over this among plenty of model theorists. Of course it is nice to have classification theorems, it’s not nice to waste your time. So how much effort do you put into which problems? And in particular, MacIntyre feels that model theoretic techniques of a kind that grow out of the honest history of model theory might help us understand Weil cohomologies and for example address the standard conjectures. But this is going to be a much higher level model theory not much concerned with set theoretic questions about these models. It is going to be concerned definability questions, symmetry questions.

**Corfield**: But I am wondering whether the resources provided by the other MacIntyre help us understand what is going on there.

**McLarty**: I’m drawn to say that we can understand this question you had about Maddy, what enables these standards sometimes to change and sometimes not. Well Maddy says the mathematicians in that field, it is not much of a gloss to say the community pursuing that subject. That maybe her phrase ‘the people working in this field’ we should understand in Alasdair MacIntyre’s terms of ‘the craft community’.

**Corfield**: Is it possible to say that Angus MacIntyre perceives a group of people who maybe are stuck in one place? Who aren’t coming fully his way.

**McLarty**: Yes, he is published.

**Corfield**: In which terms can he understand that they’re stuck?

**McLarty**: Well, he does it historically. He looks at Tarski’s work and he says everybody agrees that these are great theorems, but MacIntyre argues that what really matters about the decidibility of elementary geometry is not quantifier elimination *per se*. It’s the special feature of that theory that led to that like o-minimality. So we should look at aspects of that theorem which are underappreciated and say those are the ones that we need to continue expanding. But he certainly wants to say that o-minimality very directly is a part of the heritage of Tarski’s work. He wants to say “Don’t look at completeness. Look at the very simple classification of the definable objects that let him proved that completeness or decidability”.

**Corfield**: So he can from his perspective tell a very good story of how we get from Tarski to where he wants to take us now.

**McLarty**: Yes

**Corfield**: I just wonder what he thinks about someone who doesn’t quite buy into his story.

**McLarty**: Much of that story is fairly straightforward factual history. Indeed these are things that they were successively proved. Then there is the value component. And you talked about that. You said it is not just a matter of stories because you get into value judgements. And then he wants to say that what is underappreciated is that this is the value of that old Tarski result. Everyone agrees that this did happen and everyone agrees that the result is valuable but they don’t see that this is why.

**Corfield**: Right, but I am intrigued in that “they don’t see”.

**McLarty**: Or don’t sufficiently understand.

**Corfield**: Can we see this in a tradition-based light? I mean you’re not going to adopt the encyclopaedic view and say “You irrational people over there for not seeing what I can see”. But you might say “I think you can be brought to see what I can see”, but not instantly. I could start explaining some of the frustrations you’re actually feeling over there.

**McLarty**: Yes, and one of the kinds of opposition that I think he runs into is not people saying, “oh you’re wrong about my frustrations, I wouldn’t like to prove the standard conjectures”, it’s “I am not sure you will prove the standard conjectures”. No doubt a lot of my frustrations would be relieved if that worked. I am just not sure that that’s going to work.

**Corfield**: This is a nice case study.

**McLarty**: But this is what I would like to see, more of really applying these MacIntyrean ideas very specifically to issues. It’s a problem I’ve often had with Maddy, I read some of her earlier work as saying that philosophy of mathematics shouldn’t be interested how concepts change in mathematics. But then philosophy of mathematics can’t address Saunders Mac Lane because most of his career was about concept change. Well, Maddy did not agree with my evaluation. But certainly there is a problem there with Maddy, at least to me and to you, of where does she get the idea that set theorists want *this*. Where does the ‘this’ come from? One way to address that would be to understand set theorists as a tradition and say “If we’re not sure how she explains how they got to their goals, let’s see if we can.” And draw on a lot of her comments and say “Right, her arguments of how they got there make sense to me, because if I look if I look at this as a community in MacIntyre’s sense, that is what I would expect.”

**Corfield**: And we ought to try to observe whether she is experiencing frustration herself in her role as methodologist.

**McLarty**: I like that way of putting it, because my take is that she really could have relieved some of her frustrations by looking more broadly, instead of looking at set theorists as a community. That’s not to say she thinks that set theory is all that matters in mathematics. But don’t just look at set theorists as a specific community, look more broadly. Of course, they are a community in a transparent sense, they have meetings.

**Corfield**: She’d be a very useful person to consider. Because who else after all is there who works in this way? What else has she worked on? The articulation of goals in nineteenth century geometry.

**McLarty**: Yes, she is interested in comparing V = L to the parallel posture, or maybe to some denial of it. People say that since Cohen we know that V = L is independent. When we found out the parallel posture was independent we said, “Well, there are three geometries”. You’ve got the parallel posture, you’ve got denial of non-intersecting lines and you’ve got lots of non-intersecting lines. Now, we’ve got two set theories. One with V = L and one without. She wants to say “No. When you understand the goals of the two communities, you see that the goals of the geometers were well met by saying that there are three axiomatic geometries, whereas the goals of set theorists will not be met that way.”

**Corfield**: So we could wonder where she has got the goals of geometers.

**McLarty**: Although she doesn’t use this terminology of tradition, we can understand her account that way, even if we don’t like what we understand. One of the criticisms I would make there is that she is looking at a very sweeping picture of geometry and a very detailed picture of set theory.

I think this gets to a kind of thinking you were talking about that MacIntyre wants to give inquiry a historical but not a relativist idea of the goals. Well, Maddy is clearly interested in historical accounts of the goals because she writes historical accounts, and Maddy is explicitly not interested in relativist ideas of these goals.

Angus MacIntyre is a nice allied case, because Angus is close to the set theorists, but he has explicitly said that the set theoretic phase of model theory should be giving away faster than it is to a more geometrical phase. It’s is not about foundations. It is not about whether or not to use Zermelo-Fraenkel set theory.

**Corfield**: It’s hard to think of any other philosopher of mathematics than Maddy who would be useful to test MacIntyre’s theory.

**McLarty**: Well, the structuralists like to say they are talking about trends in recent mathematics. Mathematicians today have these structural methods. Not perhaps the ones expounded here.

**Corfield**: There is one limitation in this point of view. They say that mathematicians treat entities up to isomorphism, but there are times when they don’t do that.

**McLarty**: You can say it is all up to isomorphism depending on which category the isomorphism is in. Nobody looks at the complex numbers up to field automorphism, but you often look at the complex numbers up to real algebra automorphism. You take the real line as fixed and you look at conjugation as the only automorphism they have as a real algebra. And periodically you look at them as an oriented field, and do distinguish *i* from –*i*. When you’re doing complex analysis you distinguish holomorphic from antiholomprphic maps. You do not allow conjugation as an automorphism.

**Corfield**: So it would be quite easy for us to show a partiality of the point of view of these structuralists. They are fairly explicit when they talk about the need to capture mathematical practice.

**McLarty**: Yes but when they say “up to isomorphism”, they forgot to say which isomorphism. By isomorphism do you mean a holomorphic isomorphism, do you mean a real algebra isomorphism, do you mean a field isomorphism? Different parts of mathematics in fact look at these different forms of isomophism.

**Corfield**: Then there’s the further thought that at times you shouldn’t treat something up to isimorphism at all, such as when you treat a category up to equivalence. So it wouldn’t be very hard to show their partiality, but then how can this map onto the frustrations that they themselves are feeling?

**McLarty**: Well the question is why do we want a structural account? There’s no hope of judging our structural account unless we know why we want it. If we want a structural account because Benacerraf asks for one, then we just have to read Benacerraf’s paper to understand how good a given candidate for it is. But if we want a structural account because, as Resnik often says, this is how mathematicians do mathematics, then you have another standard for judging it. Let’s go and see. And here those mathematicians that do this, this is a tradition. And maybe that way of looking at it will help us connect a given candidate theory to the mathematicians.

**Corfield**: Do they perceive a problem in what they are doing?

**McLarty**: Well they certainly perceive research problems in structuralism.

**Corfield**: Can we get at the problems they perceive themselves to have and explain those problems?

**McLarty**: Resnik does want an account that matches fairly closely to how mathematicians do things.

**Corfield**: But if he carries on being oblivious to the fact that he is not matching what mathematicians do?

**McLarty**: Well, I wouldn’t say he’s oblivious. The question is how much accuracy is feasible and worth striving for. And here is one place where the craft tradition may help. We are not asking for a philosophical theory that reproduces what mathematicians do, because we can go to the library and find what mathematicians did. We are asking for one that is adequate to it in some way. So what we are asking is not what Lang wrote in his algebra, but what is traditional in that. What is important to the community in that. Not just the specifics. Can we match what is important to the community in that? And here I am trying to use MacIntyre to explicate how we should be evaluating these theories.

**Corfield**: Right. From our perspective it’s clear that the structuralist account is not adequate. I was just wondering how much they belong to their own tradition. Can we think about how they perceive themselves?

**McLarty**: Structuralism is a pretty small to count as a tradition.

**Corfield**: Shapiro, Resnik, Hellman. And they have rather different accounts, haven’t they?

**McLarty**: Yes, but they don’t want to consider themselves as an isolated tradition. They want to be part of some larger philosophic tradition

**Corfield**: Which certainly appeals back to Benacerraf. But even larger than that do you think?

**McLarty**: Benacerraf certainly saw himself as a part of a larger tradition. He was part of analytic philosophy. Now I don’t think that a lot of people today identify with analytic philosophy as a movement, although a lot are clearly descended from analytic philosophy, and have a lot of the same goals, and respects and disrespects.

And maybe this concept of tradition will help us, because it is a constant problem I run into, for example with that MacLane article – The Last Mathematician out of Hilbert’s Göttingen: Saunders Mac Lane as as a philosopher of mathematics – the referees come back and say “Now, this apparently talks about structuralist theories of mathematics. But it is not clear what goals this article has for structuralist theories of mathematics”. So you need to be clear about what structuralists are trying to do.

I contrast Mac Lane to structuralist philosophers of mathematics today. And the referee asks what are the goals of this structuralism. And one way to articulate these goals is to say everyone wants somehow to be referring to mathematical practice. What is mathematical practice? We could try to articulate it in MacIntyre’s way and say “Ok, here’s how to approach these goals”. Because the goal clearly can’t be to reproduce the statements of what mathematicians say. But it should somehow capture something about the statements they make, it should not have to do with the statements they make.

**Corfield**: Absolutely. Maybe even our concept of a philosophy of mathematics is a bit strange in a way. I’m intrigued to know when the term was first used. It is a strange idea that that is all you do. Presumably to the extent that the notion of tradition plays itself out in different fields, it will realise itself in different ways in different fields, but we may still learn something by the way it is applied to mathematics that will actually change the way we think about tradition. And this may be useful and helpful when we take it to other fields.

**McLarty**: I believe the mission statement for the philosophy department at Case Western Reserve in 1970 included the idea that philosophy should be philosophy *of* various sciences. Philosophers should be in contact with those things. Which from another perspective tends to look narrow and technocist. But the explicit intention of that time was to be involved. And yet it risks, as you said, becoming narrow.

**Corfield**: It seems that the philosophy of physics has gone that way, where the philosophers are more the physicist than the physicists. When there are surely things they can be learning as philosophers about physics which they should bring back to the mother ship of philosophy. I was stuck by one philosopher of physics who was telling me that he was talking to a metaphysician, and was shocked that this metaphysician had been talking about notions of space, “Of course all notions of space have a notion of distance attached to them”. Why isn’t part of what they should be doing to bring conceptions that they learn as philosophers of physics and challenge metaphysicians. Metaphysicians play this pseudo-scientific game, talking about space-time slices to wonder whether Tibbles the cat is completely there at any moment, or just a slice of him. It is not a serious engagement with what physics could provide to metaphysics.

**McLarty**: And this is how Howard Stein has argued in a lot of papers that we ought to understand that scientists of our time *are* the metaphysicians. The leading metaphysician of Newton’s time was Newton. And don’t worry that some other metaphysician didn’t agree with him. You might study them, but you must understand that the leading metaphysician of this time was Newton.

**Corfield**: This is Collingwood’s perspective. Metaphysics as the study of changes in the fundamental presuppositions of fields.

**McLarty**: Because if philosophy is ‘philosophy of …’ that doesn’t mean that a particular person is a ‘philosopher of …’. I would be interesting to see a history of ‘philosophy of …’

**Corfield**: But it really has become like that. You are labelled as being a philosopher of mathematics, as though that’s all you are. So it’s surprising to some people when I talk about MacIntyre and Bernard Williams. Things have become so horribly segregated.

**McLarty**: In France you can be *epistemologue*, which is like our philosopher of science. But the term is used precisely to indicate that this isn’t a specialized philosophy of science. This is epistemology.

**Corfield**: Yes, when you look at the Wikipedia articles on epistemology and *epistemologie*, they are completely different! It’s wonderful.

**McLarty**: So if MacIntyre becomes a way of healing this.

**Corfield**: These meetings with titles – Towards a new philosophy of mathematics – I don’t get any sense that they imagine their study of mathematics could transform philosophy. And yet if you look back through the history of philosophy, changes in the conceptions of mathematics have made such profound changes to the course of philosophy. It seems as though people can’t imagine that this can happen again.

Had we better get back to narrative?

**McLarty**: Well, what is the relation of community to narrative? If it turns out it’s a close relation then we had been talking about narrative.

**Corfield**: If the way the community constitutes itself is via a form of narrative, storytelling constitutes the community then clearly there’s a relation. And the retelling of stories. It’s not just a single telling of the story, it is also the way the telling of the story changes.

**McLarty**: So how far can you understand community that way? It needn’t be that the community constitutes itself by telling stories. It could be that we can understand community in the term that we understand narrative. Not so much that they were telling stories but that we would understand their constitution in the terms we understand narrative.

**Corfield**: Part of the story that we will tell of that community will involve the stories that they tell. There is always that dimension, isn’t there? It is the story of changing stories in some sense. One question for me is how historians delineate pieces of mathematics. How do you isolate a stretch of time as a whole contiguous entity? What is a suitable piece of mathematics to write a history about?

**McLarty**: Alexander was right on that point today. I mean he claims to find pieces, these images of mathematics. But he doesn’t see those as constituting communities of mathematicians particularly.

**Corfield**: No, they are just resources, aren’t they? Externally provided resources in order for you to conceive yourself.

**McLarty**: Yes, or for other people to conceive you. He certainly wants to say that the mathematics produced by these loners, these misunderstood geniuses, is a different mathematics than what was produced by the explorers.

**Corfield**: Is there a causal relationship going over there? This understanding *causes* a new type of mathematics?

**McLarty**: I don’t know which direction the cause goes but they certainly cohere, these disaffected loners produce a pure mathematics uncontaminated by the world. And mathematics that’s not drawn for physics. Maybe it can answer some questions but it’s not drawn from it.

**Corfield**: And there is some kind of correlation going on but it is not clear whether there’s a causal direction of influence between the image and the mathematics. But, typically, what are the topics historians of mathematics focus on, to the extent that they moved away from large scale history. There was a shift towards institutions, wasn’t there? For example, to look at Göttingen at a certain time, or to look at Bourbaki.

**McLarty**: I would guess that professional historians of mathematics would tend to aim for the biographical. Certainly there’s been a lot of good biography.

**Corfield**: So you can focus on an individual, you can focus on a community of some sort. Leo Corry has recently been working on algebra from 1890 to 1930 by looking at the *Jahrbuch*, and seeing the way the classification of branches has changed. And the *Structures * book.

**McLarty**: Which stretches back to Dedekind and Weber. He’s right there’s an appalling or interesting ambiguity here. We hear about structure, but it turns out that structure has meant a whole lot of different things to different people. A structure to Birkhoff is a lattice. The structure of a group is its lattice of subgroups.

**Corfield**: We are talking about the notion of community and how they identify themselves. Has that already gone beyond the level of community to take us from Dedekind to Bourbaki? Although Bourbaki must conceive of themselves in some sense as being the inheritors of what was done by people like Dedekind.

**McLarty**: Well famously as failed inheritors to what was done by the French. It’s constitutive of Bourbaki’s identity that French mathematics was destroyed by a failure to protect the promising scientists in World War I. They did not inherit the kind of tradition that they should have inherited. So they have to start from scratch. They go to Germany to learn, and then they are going to write the Elements to reconstitute French mathematics. Their initial goal was a text book that would change French mathematics instruction, but very quickly by the time it became the Elements of mathematics it was to change the world conception.

**Corfield**: Amir asked me about conceiving of all of mathematics at any time as part of a tradition. It’s not an easy thing to do to carve up a large range of activity and say this tradition is going on here and that tradition there. It is far more nebulous than that, isn’t it?

**McLarty**: But the question is are these valuable terms to do it in, and how close is it to the understanding of narrative. And I think yes these are valuable terms and it seems likely enough that you are going to understand a tradition this way. It’s not that you want to say that they used a narrative. The structure of a tradition is the structure of a narrative. It has characters for a start. A tradition certainly has characters. And they have roles in it.

**Corfield**: And they make pronouncements which are part of the narrative that we tell now about them as tradition.

**McLarty**: So in a way maybe we can take this as a definition of a tradition. A tradition is not just anything that you can write a chronicle of. You can write a chronicle of anything. A tradition is something that you can write a narrative of.

**Corfield**: But then when Leo writes his narrative that is contained in the book that you mentioned before, there’s a narrative going from Dedekind up to Bourbaki? And yet we don’t want to call it one tradition.

**McLarty**: Well we don’t have to. We are not committed to saying his book is one narrative.

**Corfield**: That would be the way out, to suggest it’s a sequence of narratives. Although, one can see a continuity between them.

**McLarty**: Yes, you might want to say on a larger scale yes, but not on a smaller scale. One candidate for what makes a narrative: one narrative has one beginning, one middle and one end. Whereas his account of conceptions of structures has a bunch of beginnings each of which has a middle, and most of which have an end. Some peter out without quite ending, and one hasn’t ended yet.

**Corfield**: I am not sure MacIntyre gives much of an account of what he takes a narrative to be. What other recourses we can appeal to there? There are the resources of the narratologists.

**McLarty**: How does MacIntyre decides what is *a* tradition?

**Corfield**: That is a good question. You only see him do it from moral philosophy. They are the only histories he ever gives you. And he can certainly say there are many Aristotelianisms.

**McLarty**: But from what I know from his work it wouldn’t be much of a stretch to say the reason he says there are many Aristotelianisms is because he can give an account of each one. A narrative of each one. And this fits into a larger narrative of Aristotelianism. But he doesn’t say by what criteria he individuated tradition. And we might want to use narratological criteria.

**Corfield**: As far as I am aware there are no technical resources to pick out a tradition. So, a beginning, a middle and an end?

**McLarty**: It’s what little I know of narratology. And I learned it from Aristotle in *The Poetics*. I believe it’s a tragedy that has a beginning, a middle and an end. Aristotle’s work on comedy is famously missing.

**Corfield**: Oh, so they ought to have been resources MacIntyre could have been using then. So it would be interesting to think about the history of theories of narrative. Traditions in the theories of narrative.

**McLarty**: I was wondering who are the characters. If we are going to talk about Bourbaki’s conception of structure, certainly some people are characters in that and some are not. Riemann is not really a character in Bourbaki’s conception of structure. He is a figure of conception of structure for a lot of people.

**Corfield**: As some sort of ancestor or origin figure?

**McLarty**: The example people go most to is his approach to complex analysis. And he wants to say, well to put it very coarsely, we are not going to do complex analysis by looking at series and functions. We are going to do it by looking at these surfaces and relations between surfaces.

**Corfield**: So a tradition has a beginning, a middle and an end, but often the beginning refers back to some ancestor. But one needn’t count that figure as the beginning of the tradition. It’s just some figure in the past.

**McLarty**: But every historical narrative can be put in some larger historical narrative. But we can tell *the* story of Bourbaki that is a story of Bourbaki. And that story doesn’t include Riemann.

**Corfield**: Although it’ll include their accounts of Riemann.

**McLarty**: Well, not prominently because they don’t give prominant accounts of Riemann. Whereas for a larger conception of structure, a lot of historians are now interested in saying modern mathematics descends from Riemann. In this ‘modern’ is a lot like ‘structural’. And Corry’s narrative of Bourbaki doesn’t really end. He gets this from Pierre Cartier and I’m very sympathetic to it.

It runs up against the failure of their structure theory and the fact that there was a better alternative right there that they didn’t take. Or better in some ways.

**Corfield**: Why doesn’t that mark an end?

**McLarty**: Oh, it peters out instead of ending. There is no outcome. There’s only dissatisfaction.

**Corfield**: Does the end have to be a more decisive kind of outcome? Cartier talks about the perception that there is no need for Bourbaki at the moment.

**McLarty**: Cartier wants to say Bourbaki died in the fifties. The continued existence of the Bourbaki seminar is a lovely thing, don’t fail to go to it when you are in Paris. But it is not Bourbaki.

**Corfield**: How do we construe that in terms of community? There is his story then of Bourbaki as a community, and it’s a plausible story. But there are more contemporary people who would want to perceive themselves as Bourbaki.

**McLarty**: Well if a narrative recounts an approach to a goal, then I think this is a good way to understand Cartier and Bourbaki. The goal, which they stopped pursuing in the fifties, was to present one wholly unified account of mathematics founded on their structure theory. Their structure theory failed and they couldn’t use it.

**Corfield**: You can have a continuity where there is a slight change in goals. This is a radical giving up of a goal.

**McLarty**: This is not slight!

**Corfield**: So shouldn’t we agree with Cartier?

**McLarty**: I like to look at their book on homological algebra which took something like twenty five years to write because the original conception in 1956 was already out of date. So it took them 25 years to finish because it was not a Bourbaki project. Their book on integration led the field. Their book of homological algebra was out of date when they conceived it, and worse by the time it appeared.

**Corfield**: There has been a theme through the conference about whether you can call mathematical concepts *characters*, an idea I find a bit dodgy myself. I am much happier thinking of people as characters. People with goals and responsibilities. That would fit a lot more neatly with a MacIntyrean account.

**McLarty**: Yes.

**Corfield**: So we agree that we can happily reject notions or visions or concepts or whatever as being characters?

**McLarty**: Yes. I have been leaning on goals here, but as you said the methods changed and they stopped having the Bourbaki congresses of the kind they had. They stopped shouting at each other. Cartier points out they stopped having Jean Dieudonné write all the final drafts. On a lot of levels, they just stopped being Bourbaki. He talks about the unity of consciousness there. Up until Dieudonné retired, you could turn to him and ask “What was our result on such and such?” . He would go to the shelf, pull off the book and go the exact page and show you. Once he retired nobody could do that.

**Corfield**: So Bourbaki presents a rather neat case, doesn’t it? So do we find other similar communities or things are a bit more nebulous elsewhere. There’s nothing quite like Bourbaki for having a strict sense of what it wants to achieve and an idea of how to go about doing things. Are we going to find that elsewhere?

**McLarty**: The Noether School. The founding of modern algebra.

**Corfield**: I can imagine us finding a lot of good examples, but does every mathematician have to belong to a tradition or community? Is there not something good about being self-consciously organised as a community and a tradition Is it not better, healthier, more rational to organize yourself in that way as some kind of school? Are conceptions of encyclopaedic rationality infecting, if you like, certain people, certain mathematicians so they are acting in a rather individualistic way? And something is lost by their not belonging to some form of fairly delineated community. In the Bourbaki case. Who benefited most? The members. Imagine having all these incredible mathematicians around to talk to in an intense way. What a wonderful thing. Or the Göttingen School. MacIntyre’s point is that that kind of social organisation is a good thing to try to achieve. One can say that bits of mathematical activity don’t really take place in this kind of framework.

**McLarty**: Did Grothendieck’s seminar have *esprit de corps* or did they have cultishness? Doesn’t this description depend on whether you approve of it or not.

You know an interesting case – Erdös. Erdös does want to claim that he is in a Hungarian tradition. But Erdös is a visibly isolated individual.

**Corfield**: But why is Erdös the one chosen to have the number with? It proves the point that he is isolated in a way. You use him as the reference point to see how far away are you from him. He is the metric of how far away you are from someone else.

**McLarty**: He does stand out as going against the trend. He does particular problems. He does problems.

**Corfield**: You know Tim Gower’s article on the Two Cultures of Mathematics? He puts together on one side algebraic geometry, algebraic number theory, Langlands, Grothendieck, etc. And on the other side, combinatorial, graph theoretic, Ramsey theory kind of work. There is almost as apology going on within it. To people outside it seem as though the problems are trivial. There is a justification for what they are doing in that behind these apparently trivial problems, there is something unified. It is not a body of theory, and he calls the Grothendiecks and Langlands *theory builders*. But his side is deriving a body of technique. He knows that to make any advances in Ramsey theory, that some extraordinary idea will be necessary. It won’t be expressible in that big theoretic way within a grand framework. It won’t justify itself through its realization in that particular problem. It will then be useful in some quite subtle way in an apparently quite distant field, possibly. Something like arithmetic progressions amongst the primes.

**McLarty**: And when I looked at it I liked that he says that he’s talking about different kinds of mathematicians, not different periods of mathematics or something. What I don’t like is people who want to say mathematics used to be problem oriented then it got theory oriented. Because nobody can identify which period was which. Is the Langlands program a pursuit of a problem, functoriality, or of a theory?

**Corfield**: But this work is quite divorced. He’s not using any like category theory, for example. But there is a body of understanding that is applicable in a range of situations. We had a discussion about it on our blog. We were speculating as to what it would be like to do something that would be relevant to both ways of reasoning and we really couldn’t make much of a bridge. If you were to draw a graph theoretic network, representing the mathematical activity of the present, you find some parts almost cut off from others. It is not hard to go all the way from Alan Connes to the Riemann hypothesis to Langlands. With various steps and hopping about, you can cover a huge area.

**McLarty**: If you took one of the most famous tiny equations that we wanted to understand better, the Fermat equation, and ask, how did we answer that question you would find yourself in the Langlands program. That is not a bunch of hopping about.

**Corfield**: Yes, I don’t think that you can get from Fermat’s Last Theorem to what Gowers is doing. They are disconnected. Maybe that is too extreme. But with the lines of communications as such, if there are any, they are quite thin.

**McLarty**: If you want to solve problems in combinatorics, say the numbers of solutions to polynomials in the two element field, but that is also looking at algebraic extensions. Let’s solve the Weil conjectures. You’re on the doorstep of the creation of the Topos theory. On the doorstep of. You are engaged in the creation of derived functor cohomology.

**Corfield**: Can you get from that to what Gowers is doing?

**McLarty**: I don’t know Gowers’ work. Ok but how do we understand the problem orientation. Do we say that it is only a problem orientation if it doesn’t contact theory?

**Corfield**: Well I don’t think it was a sensible terminological distinction. I think it was a bad choice. But he could have said the two bodies of theory, they don’t make much contact with each other.

**McLarty**: But traditionally there is this concept of problem-oriented mathematics that should be about little identifiable problems you want to solve. Surely the Fermat’s last theorem is one. The Weil conjectures were one.

**Corfield**: I agree with you it’s not a sensible way to make a distinction. But we were interested in Erdös and his successors, and actually they do have a sense of tradition, of which Erdös is an important part. You can tell a story that takes us up to Gowers now and we can construe it as a tradition. And even if its original conception in as going against the tide, that’s just an initial founding act.

The MacIntyrean question, though, was whether it is better to be part of a tradition. Erdös almost constitutes himself as someone outside the tradition. And in the act he forms a tradition.

**McLarty**: Well he insists that there has always been a Hungarian problem oriented tradition.

**Corfield**: Ok, so he has already identified himself in that sense.

**McLarty**: Not so much von Neumann, although he also comes out of Hungary. But he can legitimately say that von Neumann does a very different kind of mathematics than the central members of the Hilbert School. And this is because he is Hungarian. I would be willing to give an account of von Neumann as a Göttingen mathematician who was not shaped in Göttingen but in Hungary.

**Corfield**: So there’s a blend going on? He does a whole host of amazing things, which you know get developed by a whole range of people, but von Neumann doesn’t seem to generate a school. Poincaré, we mentioned before, doesn’t create a school around himself. So this is the point I was trying to drive at, is there some failure in that? Has he failed in some sense?

**McLarty**: Well he hasn’t failed to locate himself in a narrative. He has this huge body of work that is really about locating himself in a narrative.

**Corfield**: Right, but he hasn’t developed the people that will carry on his tradition. So in some sense he has failed at some level.

**McLarty**: Yes, but it would have been a much more drastic failure if Solomon Lefschetz hadn’t in fact continued him without ever actually meeting the man.

**Corfield**: But that was by luck rather than by design.

**McLarty**: Well he designed the merits of his ideas.

**Corfield**: Ok, but he didn’t do all he could have done to assure the survival of his outlook.

**McLarty**: No, he seems to have done nothing to help students. We can say that that is a weakness in Poincaré. When you look at Poincaré and Hilbert you might ask which is the greater mathematician. Hard to know. Which had the greater impact on their profession? No comparison. Because Hilbert could work with people and he did.

**Corfield**: So that’s something you could bring to contemporary mathematicians. Go organize yourself in schools. It is an interesting insight that one could bring as a philosopher to the subject. In Göttingen, the survey writing of Hilbert and Klein was intended self-consciously for school building. Whereas the genealogists might worry about schools because they are going to be instruments of power…

**McLarty**: That famous Göttingen *nostrification*, where they would take credit for what other people had proved because they had restated them.

**Corfield**: So, more generally, to what extent can we use this notion of the importance of the virtues for the health of some mathematical community. Perhaps Poincaré has lacked some virtue in a sense.

**McLarty**: Poincaré was aware of this, at least according to Hermite. He knew the work he sent out was far from done. But he thought other people would be better at finishing and he would be better at having more new ideas.

**Corfield**: Which brings us to the case of Thurston, and his realisation, according to his story anyway, that this was not the way to proceed and all he would end up doing would be to leave a dead area behind him. There comes a point where he has to stop, go back and pick up people to bring along behind him. There is something Aristotelian or MacIntyrean about his reply to Jaffe and Quinn –“we must give credit to other types of mathematical activity”. And Rota makes a similar point that not enough credit is given for good exposition, and Hilbert’s *Zahlbericht*, his number theory, is something that he is very much remembered for. It shows you the importance of that kind of activity.

**McLarty**: But of course in the *Zahlbericht* he comes up with substantial theorems as well.

**Corfield**: And what do we mean by exposition? Can we wonder about the percentage of how much is reformulation of what has been done and how much of it is new material?

**McLarty**: And if we want to pursue that then we could use ideas of narratology because Hilbert’s position is clear that: “I was told to explain current results in number theory. And I did that. But the explanation of those results was a group of substantially new theorems. I didn’t just go further. What I did further was to explain.”

So now if we want to ask: Is this a fair judgment? Was he really explaining or was he breaking new ground? Surely we’ll want to use tools of narratology to distinguish explanation from further idea. Is he telling the story of those results or is he giving new results?

**Corfield**: But how are the tools of narrarotology going to help us with that distinction? How can one decide?

**McLarty**: One narratological attempt I’ll offer here: Is he assigning those theorems roles in one story or is he proving new theorems? Well surely he is proving new theorems, but is the force of that to assign these roles or is it just new stuff? And I think he did say, “Yes, I am showing you the roles these things actually have in a better understanding than anybody did have. People didn’t see these roles, but these were the roles they had. And I have made that explicit.”

**Corfield**: I was talking to Barry a lot yesterday about explanation. Nobody that I have ever read has suggested that narratology has got anything to say about that. But that is an interesting thought. I mean there is a vast philosophy of science literature, isn’t there, on explanation which has absolutely nothing to do with that kind of historical story-telling.

**McLarty**: And there are two kinds of explanation. There is explaining particular facts. but there is also explaining a body of knowledge *as* a body of knowledge, and that is closer to narrative.

**Corfield**: Right, the classic distinction in philosophy of science is between the unification and a kind of causal mechanism, which are both forms of story-telling in a way. Like the example of Salmon’s, when you are in an aeroplane and you are taking off down the runway, and you are holding a helium balloon, which way is it going to tilt as you accelerate? Well, it tilts forward, and there are two ways to see this. One way is to invoke general relativity: accelerating is the same as being in a gravitational field, and if you were lying on your back you would expect the balloon to go upwards. Whereas a causal explanation the other would want to talk about air molecules and pressure differentials and so on. So they are forms of story-telling.

**McLarty**: But Hilbert does not explain the various theories of number theory at that time in the sense of telling you why they are true. He is explaining them in the sense of putting them in a context. It is just a different sense of explanation?

**Corfield**: But are the philosophy of science categories useful for us? Is there some sort of unification going on there?

**McLarty**: Yes, there certainly is.

**Corfield**: But you don’t think it is the whole story.

**McLarty**: Yes. It is the right unification. It is an explanatory unification. Because it shows you the roles of these theorems compared to each other.

**Corfield**: Is that something to do with the ordering of concepts? The proper organisation of the field? And you are thinking that narratology could help us there by understanding what that would mean – The proper organisation?

**McLarty**: Yes, proper in a very rich sense. When the German mathematical union would assign these things to people they assigned lots of them. What they were expecting was a clear summary of which are the important results. But they weren’t expecting what Hilbert produced, which was to reconstitute the whole field. And yet it is the reconstitution of that same field, I want to argue. Well this is to say that the characters of this reconstitution are the characters that were in that field.

**Corfield**: What do you mean by characters?

**McLarty**: The things that have roles.

**Corfield**: Because I thought before we were hoping to restrict characters to people.

**McLarty**: No, you were!

**Corfield**: Oh, I thought you happily agreed with me when we mentioned the characters in the Bourbaki narrative we meant the people.

**McLarty**: I agree with you about that. But that is not the only way we could use it.

**Corfield**: Although there are narratologists who were not so happy with the thought of mathematical entities as being characters.

**McLarty**: Well I am not sure I am thrilled with it either, but I’m using it here. Well they talk about characters as data types. The characters in computer role playing game are data types in a fairly straightforward sense.

**Corfield**: One of the narratologists said that certain members of the data type count as characters but not the data type itself.

**McLarty**: But when you buy a SIMS game it comes with data types that can be made into characters if you want to say that.

**Corfield**: They are templates you put particular features on, set various values on the template.

**McLarty**: If you say a wizard is a character in Dungeons and Dragons, then well a wizard is a different data type than a warrior. Wizards come with different attributes. All wizards have different parameters for the same attributes. Warriors have different parameter values for a differnt set of attributes.

**Corfield**: Sure, but there are differences between the wizards. They have some differences.

**McLarty**: When people talk about Bollywood movies there are characters. There is the father, the mother, the young man, the young woman. The character in that sense is rather like a data type.

**Corfield**: Well I think Chris would have wanted to say the particular instantiation in a particular film can be a character. They wouldn’t want to say the role is a character.

**McLarty**: What I want to say is that the role of the father in a Bollywood movie is a lot like a data type. Laurence Olivier’s performances of Hamlet is not much like a data type.

**Corfield**: Ok, I think they wanted another word for this. So what do we have in the *Zahlbericht.* Things like ideals appear as the main players?

**McLarty**: No, I am thinking of theorems. He is going to organise whole kinds of result.

**Corfield**: As some sort of template? So that is more like wizard or father in a Bollywood movie.

**McLarty**: Well, like the quadratic reciprocity theorem. There are lots of reciprocity theorems. The *Zahlbericht * made them instances of ‘reciprocity theorem’.

**Corfield**: So, more like Barry’s template. But the question was does narratology help us? Probably we can avoid its terms. We could say Hilbert is template forming. There are things that appear to be different, and he’s putting them under the same rubric. They’re instantiations of the same template.

**McLarty**: I’m happier saying that narratology is going to explain what we mean by tradition.

**Corfield**: Well there is a lot of work for us to do there!

**McLarty**: Which is good. How else would it constitute a tradition? What do you have to say about heritage versus history.

**Corfield**: So Grattan-Guinness’s distinction. The historian really is doing something very different. They’re not at any moment going to gaze into the future to see how something will be understood.

**McLarty**: Which is exactly the opposite of what Barry said. He said that when historians of mathematics write about mathematics they have a position, they have a stake.

**Corfield**: That’s not the way historians view themselves.

**McLarty**: No. Their pretence is that they have no position.

**Corfield**: Did you mean ‘pretence’ as not true or just as claim?

**McLarty**: I meant it in the French way. Maybe with some suggestion of the other, as it has in French. We say in English the Pretender to the throne. Very few Pretenders to the throne are just about to be crowned when they’re called that. There’s a hint that there’s some gap between you and the throne. It doesn’t mean you are not legitimately the heir.

**Corfield**: What worries me is that it is possible that the historians with their strict sense of history are going to deny themselves access to something about the truth of mathematics. There is a growing wariness of any thought that one needs to take into account what the mathematics of any moment will have become. That future perfect tense. How can we do justice to a particular mathematician, say Poincaré, thinking at a particular moment without taking into account what his thinking will have become later on? Can one really have grasped his thought without taking into account what it will become in time. And I think you are going to miss something of the truth of what he was thinking unless you do.

**McLarty**: In my experience, there’s even just a practical point. Poincaré in his *Analysis Situs *proves things about the homology of surfaces which I could not have learned from him if I had not read later accounts of the homology of surfaces. No one did learn it from Poincaré until decades of folks had hammered on it.

**Corfield**: Right, if you ever want to talk about some of the people you see as the most brilliant mathematicians, how unlikely is it going to be that you can do this just using current public language, without even his private language in which he’s talking to himself in some sense? Why should the resources of that period’s public language be enough to allow you to understand what’s going on?

**McLarty**: Nobody else at the time was capable of thinking like Euler, why should I be able to?

**Corfield**: So how do historians cope with this problem? Are they really relying on later understanding? Or do they avoid working on what the leading matheamticians were thinking?

**McLarty**: They tend to say they’re using the concepts of the time. As if anybody but Euler had the prerequisites. Grattan-Guinness works by staying pretty close to the text.

**Corfield**: Does he miss something then in his histories? Does he capture the ‘truth’ of mathematical thought?

**McLarty**: I wouldn’t try and read that history without reading a modern account. Or else I would be missing something. I would not try to understand Euler on what we now call calculus of variations without brushing up on my calculus of variations.

**Corfield**: Are they denying themselves access to something of the truth of mathematics? That’s the question.

**McLarty**: I think so.

**Corfield**: So we should be worried about his distinction. There could be collaborations between mathematicians and historians, rather than a ‘two state solution’ where one is doing heritage and the other history. There are two forms of anachronism. Mathematicians’ accounts can feel pretty anachronistic. Arnold reading back topology in Newton?

**McLarty**: I found some of that quite persuasive.

**Corfield**: Ok, but there are some examples of mathematicians’ anachronism where things have gone too far, and historians are right to criticise. But is there a kind of anachronism the other way, where by denying yourself the use of things that emerged into the public language later than a certain date, you misunderstand a mathematician?

**McLarty**: It’s like trying to read Plato as a typical man of his times. As Stanley Rosen says we don’t do this. Why would it be Plato we read?

**Corfield**: But does that then explain why the historians wanted to move away from the individual ‘genius’? That is not their role to tell us the history of the thinking of the genius. And instead look to much broader considerations. The social context of what is going on in say French politics.

**McLarty**: Well, there have been different phases. There is obviously a lovely account, I am persuaded by it, that an important event in the whole history of French mathematics is when Napoleon sets up the Ecole Polytechnic, or a version of it. And puts people in there and sets them work on their own stuff. That the conception of the Ecole Polytechnic produced for example Lagrange’s work on the foundations of calculus. So I think there is a social explanation to be given there. But we also have to look at this in terms of periods because right now biography is the main mode of the history of mathematics. Although there is a question of whether these are historians’ histories. These are histories published by Birkhäuser, often written by mathematicians.

**Corfield**: What or you actually doing in your paper when you sort out a more adequate history of Gordon? What are you doing by that?

**McLarty**: I am trying to share his values, and Hilbert’s to a lesser extent.

**Corfield**: But have you become a historian?

**McLarty**: People say this.

**Corfield**: Is it that you as a philosopher are drawn to do this kind of history? As philosophical work?

**McLarty**: Yes. If I want to know the nature of mathematics, I want to know the nature of *existing* mathematics.

**Corfield**: So is writing the proper history of mathematics necessarily a philosophical venture?

**McLarty**: Absolutely. To put it in narrative terms: you cannot have a narrative continuity of a direction towards a goal, without having a conception of the goal. Conceiving goals is philosophy, as for example Saunders MacLane would have it. Saunders has no interest in mathematics versus philosophy. Mathematics is a love of wisdom for Saunders. And the largest questions about the direction of mathematics, do set yourself the task of trying to tell what they are.

**Corfield**: How many people think like that? It seems a quite rare way of thinking.

**McLarty**: I think Jeremy Gray believes that one source of important history is philosophy. That it directs you to certain problems. Goal setting.

**Corfield**: He certainly sees some mathematicians as philosophers. Do you consider him a philosopher?

**McLarty**: I consider his work important to the philosophy of mathematics. When people say I’m a historian. Well my pay cheque comes from a philosophy department.

**Corfield**: I don’t know if you have noticed that history and philosophy of science departments are splitting. We’ve left behind the days when each thought the other necessary for their work. But the divide has always been clearer in mathematics.

**McLarty**: Why is it so much harder in mathematics than in other sciences? Is it right to call them other sciences? Instead of the sciences.

**Corfield**: Is it not linked to the founding act of analytical philosophy?

**McLarty**: That’s a compelling way to look at it. Precisely, our conception of history of mathematics is linked to a philosophy of mathematics. And it’s a hostile linkage between the philosophers of math and the historians of mathematics. But the reason historians of math don’t look like historians of physics and chemistry is because of a one-time philosophy of mathematics.

David Rowe. He tries to do history of math that will look like history of physics. And he does some nice stuff. And Leo Corry. But an awful lot of the people who show up at the history of math meetings are responding to a one time philosophy of mathematics.

**Corfield**: My supervisor Donald Gillies was in Harvard for a year and he found affinities with the historians of science. He met up with people like Peter Galison and Anne Harrington and he’d say “Those are fascinating histories you’re writing, what can we gain philosophically from what you are doing”. And they would deny that it did have a philosophical importance. From Donald’s perspective that couldn’t be true.

In Galison’s *Image and Logic*, from my point of view there is a clear expression of a philosophical viewpoint on science. It is a vision that goes against Kuhn, and his carving up of science into periods interrupted by revolutions. There is an interweaving of threads of different subcommunities of science, which by breaking at different times makes science stronger. To my mind this is a philosophical kind of thought.

**McLarty**: Any philosopher of science who thinks that *How Experiments End *isn’t philosophy of science, it is because they are not even interested in refuting SSK (the sociology of scientific knowledge). They think SSK is so obviously not what they do that they are not even interested in arguments against it. I wouldn’t be interested in a philosophy of science that couldn’t include what Galison does. But I don’t engage much with philosophers of science. I engage with mathematicians and historians of mathematics. And I engage with philosophers of mathematics.

**Corfield**: There’s still a huge job of work to be done pulling it all together.

The post Corfield interviewed by McLarty appeared first on Thales + Friends.

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]]>**Mazur**: In your essay you give a panoptic view of the field of narratology, and of the various models that are used, which is extremely clarifying. You offer a view of the goals and meta-goals of narratology and you seem to be taking a very mathematical approach to your field.

Herman: I think I was trying to envision what concept might allow for maximum overlap (or at least an optimal means for comparison) between the domains of mathematics and narrative theory. Models and modeling processes struck me as being ideally situated at the intersection of these two domains. Hence I started to work backward, and review what kind of research on models had been done in fields that are at least allied with mathematics. Not having enough background in mathematics to know how mathematicians actually use models, I had to rely on the accounts of models outlined by philosophers of mathematics, theorists of applied mathematics, and so on. In composing the paper, I drew on that research as well as my own “native” knowledge of the models that circulate within the field of narrative inquiry.

**Mazur**: You refer to Max Black’s taxonomy of models.

**Herman**: In connection with Black’s taxonomy the notion of meta-goal that you mentioned before is quite pertinent. Or, to use an idea I develop in my paper, Black’s taxonomy has significant implications for metanarratology–that is, the study of what kind of theory the theory of narrative purports to be, or might aspire to be. If you look at models in the field and then try to place them somewhere within Black’s taxonomy, it seems to me that many of them fall in the category of analogue models. That is, they’re structural approximations of some sort: they try to mirror the

structure of stories, the web of relationships among elements of narrative. In working on my paper, I began to consider what it would take for narrative analysts to move from such analogue models to theoretical models, as defined in Black’s taxonomy. For Black, theoretical models, beyond functioning as structural approximations, have to explain or predict in some sense–in the way the computer models used by meteorologists predict

the weather, with more or less success. My impression was that in mathematics–and correct me if I’m wrong–there are such things as theoretical models, models that explain and predict and don’t just mirror. Accordingly, to try to figure out stepping stones that might lead narrative theorists from analogue models to theoretical models, I reviewed some implicit criteria that practitioners in the field seem to be using to evaluate existing

models. In exploring why scholars of story tend to favor some models over others, my larger goal is to work toward an account of why (and how) narrative theorists make such choices, using this sort of metanarratological inquiry to lay groundwork for new modeling practices in the field itself.

To be sure, there are already areas within the field that, leveraging empirical research, seek to explain and not just approximate the structures of narrative. Cognitive psychologists such as Richard Gerrig, for example, use empirical methods to investigate correlations between specific textual features and readers’ responses to those features, as measured by reading times. This kind of work might provide a pathway from analogue to

theoretical models. But in other parts of the field, the main focus of research does seem to be more about attempting to replicate through various kinds of models what a narrative is, what it looks like.

**Mazur**: In writing and thinking about narrative, why formalize at all?

**Herman**: I think that the formalizing enterprise or mission has been part of the field from its inception. Narratology, at a fundamental level, really is about narrative structures and the attempt to capture them formally. And here is where an understanding of formalism and formal models would help me–if I had a better sense of the range of uses of models in mathematics and in other, neighboring fields.

**Mazur**: Of course in mathematics one often develops a language to deal with some kind of formal structure. But usually one doesn’t do that unless one hopes that the structure will pave its own way–or pay its own way.

**Herman**: Otherwise one would just stick with the lower-level language, if I understand correctly?

**Mazur**: Yes. In mathematics I think there is a slight negative view of “museums.” Any model you build, if you simply put it in a case after it is built, is not paying its own way. Models should be used somehow. And your models are surely being used!

**Herman**: Yes. I think there’s something about narrative that requires distillation—for example, distillation from natural language texts to arrive at some sense of their underlying structure as stories. You can capture aspects of narrative through that sort of structural analysis that you couldn’t otherwise see. Analogously, if you parse sentences using linguistic models, you can see constituents and relationships among constituents

that would remain invisible without the help of those models. Indeed, the power of linguistic models to reveal such structural relationships is a main reason why linguistics came to be viewed as a “pilot-science” by the early narratologists; it inspired those theorists to assume that by developing models of narrative structure, you can capture aspects of stories that aren’t necessarily evident from their surface forms. Take, for example, Genette’s model of temporality and his account of the temporal relationships between the story plane (or what happened) and the discourse

plane (or how what happened is presented). Genette formulates exact ways of discussing order, duration and frequency, elaborating a systematic taxonomy. With respect to order, his model maps out what counts as chronological narration and what count as departures from it, such as flashforwards and flashbacks. Duration, in the model, can be defined as the ratio between how long it takes for an event to unfold on the story plane and how much text is devoted to its recounting on the discourse plane, with pauses, scenes, summaries, and ellipses corresponding to different narrative speeds. And frequency is the ratio between how many times something happens and how many times it’s recounted. Genette’s work is thus an example of a model that successfully captures and organizes a system of narrative possibilities–a system that you couldn’t easily envision without the model. You can look at individual narratives and see how they instantiate parts of that system.

Furthermore, by giving due attention to the features captured by the model I think you can avoid certain problems that enter into other traditions of narrative analysis. Are you familiar with William Labov and his work on conversational storytelling from a sociolinguistic pespective? Labov is particularly interested in what he terms narrative evaluation, his claim being that every storyteller has to underscore the point of his or her story, signal its reason for being told in a particular context of telling, to ward off the dreaded question “So what?” If you’re telling a story and somebody

interposes this withering rejoinder (to use Labov’s phrase), then you know you’ve failed. By evaluating the events being presented in the narrative, you can highlight the narrative’s point and thus avoid the “So what?” question. In turn, one of the strategies that Labov uses to describe evaluation is to posit what he characterizes as a standard or default syntax for narrative, whose basic unit is the “narrative clause.” For Labov, narrative clauses are such that they cannot be transposed without changing the underlying semantic interpretation of the story; furthermore, he argues that these default narrative clauses are very simple, marked chiefly (in English-language storytelling) by past tense verbs in the indicative mood. The sequence of narrative clauses featuring these verbs iconically matches the order of the events they report, and the “baseline” clauses themselves are devoid of modal auxiliaries, negatives and hypotheticals, and other complexities of structure.

But are such complexities of structure always evaluative in their function or force? In contrast with Labov’s model, Genette’s model suggests that any of the structural elements encompassed by the narrative system may or may not be deployed in a marked or event-evaluating way–from “iterative narration” via constructions involving the modal auxiliary would (we would go to the ocean often that summer) to time-bending flashbacks and flashforwards. Conversely, Genette’s model captures a resource for evaluation not registered in Labov’s account: namely, the way fluctuations in narrative speed can be used to mark off salient actions and events from background circumstances and happenings, as when a narrative shifts from a summarizing to a scenic rate of presentation.

**Mazur**: Is Labov’s account an empirical summary, though? Possibly Labov is focusing on the story-tellers he’s studied, rather than the general range of story-writers.

**Herman**: Yes that’s true. That’s one way to justify the model Labov has produced. He models precisely what he found in the specific corpus of narratives he looked at. In contrast, Genette’s work focuses on literary narratives, where other constraints on communication are in place. Written narrative allows for more complex structures because readers can flip back through the text, reread the same sentence multiple times, look ahead, etc. So there’s a way in which, you’re right, the communicative situation might explain the contrast between Labov’s model and Genette’s.

But actually there have been follow-up studies by researchers also doing empirical work on face-to-face narration. They argue that Labov was looking at one subtype of narratives produced in interactional environments where an interviewer prompts somebody to tell a story and then stands back as that person tells his or her narrative in a relatively monologic way. But there are many, many other types of interactional environments involving co-narration, intense competition for the floor, he-said-she-said gossip, and so forth, and so other analysts, seeking to diversify the corpus

of stories on which frameworks for inquiry might be based, have developed models that attempt to capture the full range of storytelling situations. There’s a tremendous book–it’s the best book on storytelling in face-to-face interaction that I’ve ever come across–called Living Narrative, by Elinor Ochs and Lisa Capps (Harvard University Press, 2001). Ochs and Capps develop the notion of narrative dimensions, including linearity, tellership, tellability, embeddedness and moral stance. Any storytelling act will be located at different points along these five dimensions, with Ochs and Capps thus providing a highly nuanced picture of the different systematic possibilities available to people using narrative in face-to-face discourse. So theirs would be another model that we could talk about, alongside Genette’s and Labov’s.

**Mazur**: There is, it seems, a link between some mathematical developments and what goes under the heading of structuralism in anthropology, and–I would imagine–in structuralist aspects of literary theory. Can you tell me more about that?

**Herman**: You know, this link has certainly been prominent in the history of the field. And actually I would like to hear more about Bourbaki from you because I only really started to read about the group while doing research for my paper, though I’d known vaguely about Roman Jakobson’s interactions with some of the group’s members. In general, structuralism was an approach to literary and cultural analysis, especially prominent

in the 1960s and 1970s, that used linguistics as a pilot-science (in the sense mentioned previously) to study diverse forms of cultural expression as rule-governed signifying practices or “languages” in their own right–with narratology being an outgrowth of this general approach. Before the French structuralists, however, the Russian Formalists had already begun to develop important tools for narrative analysis back in the early part of

the twentieth century. Commentators like Viktor Shklovskii, Boris Tomashevskii, and Vladimir Propp were affiliated with that group. These scholars developed for example the fundamental distinction between story (fabula) and discourse (sjuzhet), which made its way to France partly because of Tzvetan Todorov, whose knowledge of Slavic languages helped bridge the Russian Formalist and Francophone structuralist traditions. The high point of structuralism was in the mid to late sixties, and the term narratology itself was coined by Todorov in his 1969 book Grammar of the Decameron, which presented a structuralist analysis of Bocaccio’s text. Like other domains of structuralist theory, structuralist narratology was based on a Saussurian understanding of language, which divides language into langue or system and parole or message (or utterance). Further, the structuralist narratologists, following Saussure, wanted to background individual narrative messages (or texts) in favor of the larger narrative

system, or narrative langue. For Saussure, the message or individual utterance is too contingent, too dependent on factors that cannot be accounted for in linguistic terms. The structuralist narratologists likewise sought to focus on the system that makes possible the production and understanding of individual narratives. They thus argued that narratology should not be viewed as a handmaiden for interpretation; its goal is not to interpret

individual narrative texts but rather reveal their underlying structure–just as the linguist does with sentences.

This broadly Saussurean approach led in turn to an uncoupling of structural analysis from literary interpretation, although the narratologists did recognize that attention to these structural possibilities could enable interpretation. That is, if you could set out the full system of narrational possibilities it might allow you to say something interesting about why you have the interpretation that you do of, say, Proust’s In Search of Lost Time. With Proust, a key issue is his tendency to use iterative narration (a type of frequency that involves telling once what happened more than once), so in this case the narratological categories do in fact allow for a more precise analysis of the temporal structures of Proust’s novel.

In the decades since they were first formulated, however, structuralist approaches to narrative have come under attack from a number of directions. One line of criticism has been that narrative theorists really do need to engage with issues of interpretation–on pain of having to face the “So what?” question in their own right. Another critique concerns the language theory that the structuralists derived from Saussure’s ideas, which aren’t necessarily the best for capturing how language works in situated communicative contexts. When it comes to studying aspects of language in use, other theoretical frameworks are arguably more powerful—frameworks such as pragmatics and discourse analysis, which focus on how using certain language designs in certain kinds of contexts licenses inferences about utterance meanings, as discussed in the work of Grice, for example. These traditions of linguistic inquiry are particularly important for understanding extended stretches of text or talk–that is, discourse. So the structuralists

were in some cases working with less-than-ideal linguistic tools when they investigated how narratives function as structures. In some of my own work I’ve tried to retain the basic structuralist insight that language theory provides invaluable resources for narrative study, while drawing on different, more recent linguistic concepts and methods in an effort to flesh out that original insight.

**Mazur**: Are the formal models of narratology analogous to the formal models of music in musicology? Using formal models of music enables you to talk about and to decide what key you’re in. That then manages to structure your experience of the music for it determines what the dominant is, what the subdominant is; it tells you how certain harmonic progressions will end up, or at least suggests such endings, giving you expectations and anticipations. This kind of judgment helps the listener to hear the intentions of the composer and helps the composer to organize the piece, helps the

performer to perform it so you’ve got the composer, the listener and the performer, all in a kind of loop.

Herman: That’s a great point and I’ve thought about some analogues in narrative theory. Generic codes would be clear parallel here, since genres too set up expectations and anticipations, allowing interpreters to channel and delimit their inferential activities as they engage with a particular text. Or take musical compositions involving both horizontal and vertical dimensions, or melody and harmony. Analogously, some of the models of plot feature not only the linear unfolding of events over time, but also interrupted or truncated action lines that are caused by conflicts among characters–conflicts that cause some of the characters to refrain from performing actions or attempting to pursue their goals. So then you have a range of ghostly, unrealized action lines paralleling or shadowing the main plot line–which would mirror compositional structures involving melodic as well as harmonic relationships. Another parallel, in this same connection, might be afforded by multiplot narratives that follow several plot-lines, involving different groups of characters–as in the Victorian novel in the British tradition. As for modulations of key, perhaps one could talk about adaptions that shift narrative registers, as when a story about serious events is renarrated humorously? I think the disanalogy here would be that–with the exception of work on storytelling in face-to-face interaction and some of the emergent work on computational narratology–narrative theory has largely focused on the reception side of things rather than on story generation or production. True, in the tradition of sociolinguistic analysis, researchers working in the area known as Conversation Analysis have developed a method for analyzing ways of producing discourse. Conversation Analysts assume that people signal their understandings of situations by how they talk; hence these scholars explore how interlocutors create social situations in part by producing particular kinds of utterances–utterances that index a certain understanding of what is going on and thereby help bring about, in a self-fulfilling way, the type of situation that corresponds to the interlocutors’ own sense of what’s happening. This research has examined how storytelling works and, more specifically, how building narratively organized sequences in discourse requires–and involves the socially coordinated display of–certain attitudinal stances and strategic maneuvers. So this approach does get at the design element of narrative.

But in literary narratology the focus has been more on processes of reception. That said, however, there is a strand of research on literary narratives that is rooted in rhetoric, which of course studies how communicative situations involve language producers as well as language interpreters. This tradition was pioneered by Wayne Booth, in his classic 1961 study The Rhetoric of Fiction.

**Mazur**: Isn’t there a moral thrust to Booth’s work?

**Herman**: Arguably, there is. Two literary critics, Wimsatt and Beardsley, wrote an essay in 1946 called “The intentional Fallacy,” in which they objected to methods of interpretation that invoked the intentions of the author. Partly in response to this line of argument, I would suggest, Wayne Booth developed the concept of the implied author. One of Booth’s concerns was how readers might be affected by texts that didn’t overtly censure narrators or characters with suspect values. If you define the implied author as a kind of value system and explore whether a narrator or a character is favored or disfavored in the terms afforded by that system, then it’s a natural step to extend the focus of concern to how readers are impacted by this process of negotiating the value system (or systems) bound up with a given text.

For what it’s worth, in my contributions to a forthcoming co-authored book, titled *Practicing Narrative Theory: Four Perspectives in Conversation* (co-authored by David Herman, James Phelan, Peter J. Rabinowitz, Brian Richardson, and Robyn Warhol, and in press at the Ohio State University Press), I’ve developed a critique of approaches based on the concept of the implied author. (My co-authors, I should stress, defend and deploy the idea of the implied author.) I’ve also proposed there a way to explore issues of value that, drawing on the concept of norms, contrasts with Booth’s.

**Mazur**: Modern American authors are sometimes not only not implied but are rather quite explicit and try to preserve the literary imagination from too four-square an insertion of ethical strictures; for example, Philip Roth once said: “in my imagination I betray everybody.” Value judgments regarding narrators must become particularly complex when there are layers of different narrators involved in the telling. Genette, for example,

discusses the different narrator-strands in the Odyssey.

**Herman**: Yes, that’s right, and in fact theorists have developed a variety of models to account for this phenomenon. One such model is basically a vocabulary of levels deriving from Genette’s work. In this model you have a sort of matrix scenario, or baseline narrative reality, and then embedded scenarios–so that as you descend within the frame structure you arrive at a more and more distant remove from the matrix frame, so to speak. This is where the term diegetic comes into play. The diegesis or the diegetic level is the “primary” level or frame evoked by a narrative text, and a hypodiegetic level can be created if within the main narrative level a character starts to narrate. So in Chaucer’s Canterbury Tales, for example, there is the baseline reality of the pilgrimage in the context of which the characters tell stories to one another; those characters are thus intradiegetic narrators who tell hypodiegetic narratives. This analytic machinery is an attempt to capture how texts can have different narrator-strands, as you put it, and it provides a basis for exploring how different kinds of value judgments might be more or less pertinent depending on which strand you’re focusing on.

**Mazur**: Is it possible that narratological theory could be used as a schema for the language in which we tell anecdotes about our daily life?

**Herman**: There have indeed been attempts to use narratology for this purpose–for example, in one of the first systematic contributions to this field, a 1966 essay by Roland Barthes titled “Introduction to the Structural Analysis of Narratives.” Barthes argued that there are two kinds of fundamental plot-units: kernel events (or “nuclei”) and the satellites (or “catalyzers”) that expand upon them. In the terms afforded by Barthes’ model, one can say that to paraphrase a story you would at least have to capture the kernel events–whereas how many of the satellites you bring along with them is negotiable and will vary according to the needs of the paraphrase. On the production side of things one could deploy this same model to talk about how a person narratively configures his or her past experiences. In telling a story about some formative event, I know that I need to include at least some kernel events. Further, an implicit principle of relevance then allows me to determine how many and which satellites to include. And the metric of relevance will change with the context–depending on your audience, what kind of narrative you want to craft, and so forth. Is it for a family scrapbook? Is it for an e-mail message?

This is just one (small) example of how ideas from narrative theory might throw light on the structures and dynamics of everyday storytelling. Again, I’d recommend Ochs and Capps’ Living Narrative for its brilliant discussion of a whole range of relevant issues.

**Dellaportas**: Can I ask you something related? In a model focusing on music, the temporal dimension is key. Thus, when you construct a model of music you would expect a certain combinatorial argument there. I’ve seen a lot of studies in literature that count how many times certain things happen etc. Likewise you would expect narratological models to take into account this temporal dimension. For example, after highly dramatic moments you would expect something calmer. Do narratological theories take this sort of temporal rhythm into account?

**Herman**: Some of these issues have been explored under the umbrella category of plot—that is, the study of plot structure and the overall trajectory of a narrative over time. Thus the German theorist Gustav Freytag developed the so-called Freytag’s triangle to account for the structure of plots; the triangle suggests a movement from the introduction to the climax and thence to the denouement. Subsequently, Meir Sternberg developed a different account of exposition and temporal ordering, which suggests that the exposition of background material can be differently distributed in a given narrative. It can be concentrated in the beginning or else dispersed in various places throughout the text. Sternberg also draws a distinction between exposition and scene and argues that each work establishes its own “scenic norm,” which is a pace of narration that suggests you are no longer in the domain of exposition but instead in a scene-like structure which is important and therefore catches your attention. So Sternberg is interested in the play between exposition as background, which narratives typically provide in a relatively rapid way, and scenes, which the text presents in a comparatively slower and more detailed manner. Work of this kind does try to take temporal rhythm into account.

**Mazur**: I’m guessing that narratology is not an attempt to model our internal thinking. Is this right?

**Herman**: Some of the recent work in narrative theory does try to connect up three domains: (1) humans’ underlying cognitive abilities; (2) linguistic and more broadly semiotic reflexes of those abilities; and (3) aspects of narrative structure that bear on both (1) and (2). Some of the research on narrative perspective (or, to use the technical term, “focalization”) that I discuss in my paper seeks to explore links among these three

domains. Further, another tradition of inquiry in the field draws from the interdiscipline of narrative psychology, a rich, integrative approach that takes inspiration from some of Jerome Bruner’s foundational work. This approach combines ideas from the philosophy of mind, social psychology, and discourse analysis and tries to move away from a notion of the mind as internal and instead tries to embed the mind, or mental processes, in

discourse–and in narrative as key type of discourse. From this perspective, the mind is less something located inside the head than a structure spread out among participants in narrative, the story itself, and the setting in which the narrative occurs. Hence we could be talking about something cognitive that would not however be internal–or at least not wholly internal. In current work I’m exploring some of the implications of this approach when it comes to studying the nexus of narrative and mind.

**Mazur**: Is there a piece of literature that makes you feel “if I didn’t have this equipment, the narratological models, I would be less of a reader of that piece”?

**Herman**: I’ve certainly had moments in which I felt that I had to pull every narratological trick out of the hat to try to figure out why I was affected by a text in the way I was. Here I think that one again works in a backward manner. I mean that you get struck by something in a text and then you try to figure out why that aspect of the text had such an impact on you. You ask yourself: what aspect of the narrative’s structure has affected me in this way, and how can I best describe it? This is where narratological models become helpful—and fruitful. For example, many postmodern narratives involve a structure that Genette identified as “metalepsis.” Metaleptic narratives are marked by a conflation of diegetic levels. At issue is a sort of M. C. Escher-like phenomenon, where a hand comes out of the page and draws the other hand, so that you’re not sure what’s the actual world and what’s the imagined or create world. That would be a metaleptic pattern: the embedded layer swallows the embedding layer endlessly in a kind of looping structure. There are stories with this same structure, such as Julio Cortázar’s “A Continuity of Parks,” where a character reads a story in which he, that same character, is about to be murdered. In this context, knowing about the theoretical frameworks allows you to appreciate for

one thing the subtle complexity of the writer’s design. It also allows you to assign a structural description to the text that provides an account or basis for your reaction. You can say, “that’s why I found it so strange!” You can do this intuitively, of course, but with

this vocabulary you can develop a fuller account of the text’s defamiliarizing effect, and also think comparatively about how that effect plays out in other, more or less closely analogous texts.

**Mazur**: We’re TV viewers and film watchers and I don’t know much about this but I do know that the manner in which scenes are cut changes drastically from decade to decade. There is nowadays a quicker tempo; a lot more is assumed now because we’re very practiced viewers of film, so we can manage when scenes quickly cut into other scenes. And yet there are certain moments, certain cuts that, apparently, are never done. There are still restrictions but there’s an evolution of the class of allowed cuts; I wouldn’t call it progress, but I want to call it a trajectory of change based on viewers’ cinematic experiences. Do you see that in the representation of time in literature and do you see a way of integrating a dynamic component into narratological theory? If you had this dynamic component, you might be able to predict, like the linguist can vis-à-vis changes in the vowel-system of a language, future changes in the narrative system when it comes to temporal structures.

**Herman**: That’s a great question. In fact, James Morrison, in his contribution to a volume titled Teaching Narrative Theory, writes about the shorter scene durations in modern films. Shots used to be something like an average of nine seconds long in classic Hollywood films, like Orson Welles, but now they are something like three seconds long.

**Mazur**: There’s got to be a literary equivalent to this and surely narratological theory is going to be an absolutely crucial tool for understanding these dynamics.

**Herman**: There have been efforts to build in a dynamic component of this kind—not so much in the area of temporality, but rather in the area of narration, and more specifically in the presentation of characters’ consciousness. The Austrian narrative theorist F. K. Stranzel developed an approach of this kind. He identified three main narrative situations: the authorial narrative situation, which involves third-person narration and a distanced view of events; the first-person narrative situation; and the figural narrative situation, which features third-person narration but a filtering of the action through a character’s consciousness—as you get with Virginia Woolf, Henry James, James Joyce, or Franz Kafka. And Stanzel’s argument is that although the figural possibility was always in play more or less throughout literary history, it became concentrated only at the end of the 19th century into the early 20th century. Indeed, Stanzel’s approach has led to the proposal for a diachronic narratology, which would look at how narrative structures might have changed over time—how different structures or techniques were used to produce a given effect, and how the same technique may have acquired different functions over time. I’m not sure whether this approach has predictive value, but at the very least it has historical scope.

I had the privilege of editing a recently published volume that explores directions for inquiry in this subfield of narrative research–that is, diachronic narratology. The volume is titled The Emergence of Mind: Representations of Consciousness in Narrative Discourse in English (University of Nebraska Press, 2011). It features contributions by specialists in different literary periods who use specific case studies to explore changing trends in the representation of fictional minds in English-language narratives. All in all, the volume covers the period stretching from around 700 to the present.

**Mazur**: It might be particularly useful to study the evolution of narrative structures in the work of a single author.

**Herman**: Joyce would be a good example.

**Mazur**: Do you use narratological models to study Joyce? And if so, what do you think about Frank O’Connor’s remark that Joyce’s “The Dead” is something like a novel, in contrast with Ulysses, which he thinks of as a short story?

**Herman**: Yes, I’ve looked at a number of Joyce’s texts from a narratological perspective–they are so rich and multidimensional! Regarding O’Connor’s comment, my understanding is that Joyce did originally intend to write a short story called “Ulysses,” but that his subject refused to be limited or constrained in that way. But beyond these specific considerations, I would reply to O’Connor by asking a more general question: how do you define a short story (in principle)? What constitutes the dividing line between short story and novella, for example? I don’t know if there’s an exact divide, but it’s an interesting question. Is it size? Is it complexity?

In any case, there are lots of interesting narratological questions to explore when it comes to Joyce, and different strategies for posing those questions. In my chapter on “The Dead” for The Cambridge Companion to Narrative, for example, I draw on several traditions of research to examine how Joyce goes about representing his characters’ minds.

**Mazur**: Do you discuss the maid? What’s the maid’s name?

**Herman**: Lily.

**Mazur**: Her speech is about the dead, and about how men nowadays are all palaver and what they can get out of you. Do you make out the resonance of the word cadaver in palaver?

**Herman**: That’s a great point! Indeed, the opening scene with Lily exemplifies the structure which has been called free indirect discourse, which happens when a character’s voice (or mind style) inflects a narrator’s voice. Another interesting aspect of Dubliners is that it is a short story collection but the stories are linked. Some of the characters, situations, and events seem to cross over the boundaries between individual stories,

which is an interesting narrative structure in its own right. In “Two Gallants,” one character’s last name is spelled “Corley” but it’s pronounced “Whorely,” and sure enough he deceives a housemaid (like Lily) into giving him her whole month’s salary.

This incident resonates with Lily’s comment, in “The Dead,” about how men try to take advantage of women. There’s a finely wrought language pattern here, one typically associated with poetry rather than prose.

**Mazur**: David, the sweep of your analysis covers panoptically the grand structure of narrative as well as the syllable-by-syllable music of imaginative literature. I think you wrote a wonderful essay. Thank you!

**Herman**: Thank you, Barry, for your excellent questions!

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]]>Alexander:…

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]]>**Alexander**: The first question that I have is about how you arrived at the main subject of your essay, the question of non-Euclidean mathematics, and specifically the use of narrative in it. How does this fit in with what is your main field of interests and your general research program?

**Plotnitsky**: The key epistemological problematic of the paper, that of nonclassical epistemology, is a subject which I have extensively explored in my earlier work. There are, however, significant new aspects of this problematic in this essay, especially those related to the question of narrative. Most of this earlier work was related to the 20th-century philosophy and the epistemology of quantum theory. I’ve also done some work on mathematics along these lines, and I developed a concept of non-Euclidean mathematics earlier as well, although in a more preliminary way. It is, however, in working on non-Euclidean mathematics that I came realized the significance of the relationships between nonclassical epistemology, defined by the irreducibly inaccessible nature of the ultimate objects considered by a given nonclassical theory (be it mathematical, physical, or philosophical), and the multiplicity, which appears equally irreducible, of different fields in working with these types of objects. These relationships appear more pronounced in mathematics than in quantum physics, and they have emerged earlier in the history of mathematics, sometime in the early-nineteenth century, although the epistemology in question was not perceived until the twentieth century. A really new dimension of this paper for me is its narrative dimension, which was not addressed in my previous work on physics or mathematics. I have written on narrative in the context of postmodern philosophy and literature, where the relationships between epistemological nonclassicality and discursive multiplicity are found as well.

**
Alexander**: Before we get to the narrative issue, tell us a little more about this multiplicity of fields, how this fits into this notion of non-Euclidean mathematics.

**Plotnitsky**: This is not an easy question to answer. While it is not surprising to me intuitively, I’m not altogether sure in analytical terms why nonclassicality (the irreducibly inaccessible nature of certain objects) and theoretical or discursive multiplicity combine in mathematics, or elsewhere. Both, however, jointly characterize non-Euclidean mathematics, as defined in my paper. Historically, this type of relationships appeared in number theory sometime around 1800, when, more generally, what I call non-Euclidean mathematics appears to have emerged, although as I said, the radical (nonclassical) epistemological implications of this type of mathematics were not initially perceived by mathematicians. Indeed they still appear to be less perceived by mathematicians or concern them less, as against the way physicists react to this type of epistemological situations, because quantum theory and, to some degree, relativity brought them into sharper focus and made them a subject of an intense debate in physics. The epistemological questions, including those with nonclassical implications, have been significant in mathematical logic and set-theoretical foundations of mathematics, especially in the wake of Gödel’s theorem and related findings. In any event, around the time of Gauss, in his own work and that of others, mathematical fields of arithmetic (number theory), algebra (specifically theory of polynomial equations), geometry, and analysis, or their particular subfields, were deployed jointly and interactively in approaching certain problems that would seem to belong to a particular single field or subfield. The mathematics of complex numbers, my main subject in the paper, had a special significance in this context. But there were other developments contributing to this, as it might be called, new mathematical situation, the rise non-Euclidean geometry, for example. Mathematicians such as Gauss or, especially, Riemann (a student of Gauss) were able to use and connect seemingly heterogeneous and distant mathematical areas and concepts. Indeed, part of Riemann’s way of doing mathematics was to work with and to relate certain properties of mathematical objects without necessarily fully specifying these objects themselves. One could, on the other hand, believe in the existence of such objects for various reasons, specifically because of their capacity to have effects upon other objects, including those that we can expressly describe. In the case of complex numbers, for example, however controversial the claim of their existence might have been at a certain point, it was difficult and even impossible to account for certain mathematical “effects” without introducing them.

**Alexander**: The image that I gather from you is that there are certain junctures in the development of mathematics at which new concepts do not fit into previously developed moulds, resulting in the introduction of entities that are, as you said, “unthinkable.” Nevertheless, historically these “unthinkables” become the focus of a whole range of new practices that cluster around them.

**Plotnitsky**: Yes, that is correct. Now, one could argue that at some later point such objects could be properly defined or constructed, thus resolving the situation on more classical lines. Sometimes this indeed happens, as, for example, in the case of irrational numbers, which were arithmetically ungraspable to the ancient Greeks, but eventually became (more) tangible mathematical objects. On the other hand, it is also possible that some objects of this type would remain inconceivable indefinitely, as is the case, thus far, with quantum objects in physics (at least in certain interpretations). Consider infinite-dimensional spaces. Can we possibly have a rigorous mathematical concept of such objects, especially as a space? I am not sure we can. Nevertheless, we can still work with such spaces in mathematics or in physics, which is one of my main points: nonclassical epistemology does not disable mathematical practice, and in certain circumstances, it might enable it.

**Alexander**: The issue here is how much you want to require of mathematical objects to be perceived in that sense. Physics is naturally about the world, and if our theoretical constructs don’t fit the world then we face a problem. This is different in the case of mathematics in which mathematicians are not as conscious of such problem. They think: Well, I’ve defined an object, at least formally and I didn’t find any contradictions, then that’s really all I need.

**Plotnitsky**: Sometimes it’s indeed like that. It might have something to do with the Platonist view of mathematics. (I am not saying Plato’s view, because I believe Plato himself held a different view.) I think that there could be a belief on the part of some mathematicians that, once you define enough properties consistently, such an object probably exists and, for some, even definitively exists as thus defined in a certain Platonic realm. But it is not necessary to have such a point of view of mathematical objects, and there are alternative views (intuitionist, constructivist, etc.). Indeed one of my main points is that our inability to define or even conceive of such objects doesn’t stop mathematics. Of course, encountering the impossibility of even conceiving the objects with which one is concerned, and building one’s theory in spite of it or, especially, as based on it is not easy and requires a kind of “negative capability,” as Keats called it.

**Alexander**: I keep thinking about examples from areas that I’ve worked on, such as the development of the early forms of the calculus the 17th century, which involved deliberate immersion in known paradoxes. For example Evangelista Torricelli, who was Galileo’s student, was constantly trying out paradoxes of the infinite, demonstrating how subversive they were to his work on infinitesimals. And yet, he continued developing his infinitesimal method without ever defining what the objects he was dealing with actually were. Is this equivalent to the moments in the history of physics and mathematics that you talk about?

**Plotnitsky**: I am not sure about the epistemological equivalence, but I would agree that the situations are epistemologically similar; they might be just about equivalent in terms of the “negative capabilities” required to confront them. To make a radical claim, we really know next to nothing about most mathematical objects. In a certain sense, mathematics may have been nonclassical all along, even if without realizing it. Frege once said that it is a scandal that we don’t really know what numbers are. I certainly do not think that we have an access to complex number qua complex numbers via their geometrical representation as points in the Gauss-Argand plan. This is the most controversial claim of my paper, since the commonly accepted view now is that the Gauss-Argand plane properly represents complex numbers geometrically. This was not always the case. As I note in my paper, Gauss himself was ambivalent about this issue, and others, Cauchy, for example, were even more skeptical. During the subsequent history of mathematics, the Gauss-Argand plane became so ingrained in the mathematicians’ thinking that they came to believe, nearly universally, that it is a geometrical representation of complex numbers, while it may be (I argue it is) just a diagram that helps us to work with them rather than geometrically represents them.

**Alexander**: So in your view this isomorphism between the complex number field and the Gauss-Argand plane is fundamentally different from, say, mapping the real numbers on the real line. Or is that also a diagram?

**Plotnitsky**: This is not easy to answer by yes or no. For, as I say in my paper, and as appears from Alain Berthoz’s work discussed in Bernard Teissier’s paper, that there are complexities in the case of real numbers as well. So, we may ultimately not have a rigorous concept of real numbers either, or any numbers, although a geometrical representation of, say, natural numbers (by discrete sets of points on the line) is more accessible to our intuition. But we certainly have mathematics of real numbers, or of complex numbers, a very rich and beautiful mathematics. In this sense, the situation becomes similar to quantum mechanics, where, however, we, again, deal with certain undefinable, inconceivable entities materially existing in nature and detectable with experiments. On the other hand, mathematics, too, may be more experimental than we think. As Federica La Nave’s paper shows, Bombelli was thinking along these lines, and his mathematics was kind of experimental. He accepted the reality of negative roots, the roots of negative numbers, because he could do things with them, without knowing what they are.

**Alexander**: There is a tradition in the 17th century of what can be called experimental mathematics. John Wallis for example is very influenced by Baconian empiricism, and his idea of a proof is: here’s a statement; I try it once, I try it twice, I try it three times, it works. OK, proven. He literally experiments. Along the same lines Bernard de Fontenelle wrote that our knowledge of the calculus is experimental.

**Plotnitsky**: In this case, however, unlike that of physics, we also create things that we are testing, or more so than in physics. For, I take a more Aristotelian (rather than Platonist) view, according to which mathematics and its objects are not pre-existing somewhere but are the products of who we are as human animals.

**Alexander**: That leads me to an issue that you’ve mentioned several times in your paper, which is that “Euclidean thinking may reflect the essential workings of our biological and specifically neurological machinery born with our evolutionary emergence as human animals and enabling our survival”. This suggests that Euclidean thinking is somehow “natural” to us. But Euclidian proof seems very unnatural to me. The very notion that you could prove something was completely remarkable to the ancients, and it only arose in a single historical context, that of ancient Greece. In fact, it’s very difficult for people in general to learn to think in terms of proof.

**Plotnitsky**: This is to some degree true. I would, however, be inclined to qualify the case. Euclidean geometry may seem unnatural because of its (especially by current mathematical standards) relatively abstract nature. On the other hand, it is reasonably natural, as the word geometry suggests: measuring the earth. Euclidean geometry or, in part correlatively, classical physics, as Bohr and Heisenberg note, represents the refinement of the way we generally think in everyday life, in which the proximity between both is rather transparent. Elsewhere in mathematics the distance between both becomes much greater and could eventually lead to a near unbridgeable separation, a divorce, between mathematical and everyday thinking. That is one of your own research subjects. Essentially, I think that whatever we can form a conception of and especially visualize is classical. On the other hand, quantum objects and behavior, or the ultimate nature of certain mathematical objects, is not something we can really conceive of in our thought. Thus, nature and our mathematical, or philosophical, thought (which is still the product of nature) bring us to the point where our thinking constructs the possibility and indeed necessity of “objects” that we cannot think, cannot conceive of. This is another crucial point of my paper, beginning with the question of the diagonal of the square, with the fact that the Greeks arrived at this type of rigorous mathematical construction of the irrational from and through the rational. That was, at the time, a kind of “scandal,” at least, as I say in the paper, in some modern “reconstitutions” of the case (and there, I admit, alternative views of the situation). Mathematics was also a model, the model of rational thinking. Geometry helped to avoid this particular scandal, albeit in a kind of proto-nonclassical way, while arithmetic was restricted to its proper domain of dealing with proportions. That this subtle proto-nonclassicality and hence the scandalous nature of geometry were not perceived at the time helped both ancient Greek mathematics and culture, or a certain Socratic Greek culture. The preceding, tragic, Greek culture, as Nietzsche calls it, was more open to and even defined by such conflicts, since the conflicting entities—being and becoming, the Apollonian and the Dionyssian, and so forth—were accepted and even celebrated in their coexistence, and hence were not seen as scandalous.

**Alexander**: Let me recapitulate so we can get back to the question of classical mathematics as against non-classical, non-Euclidean mathematics. It seems that the crisis of the irrational was the first scandal, and that it was settled, at least provisionally, by splitting geometry and algebra apart. But in the 16th century the trouble emerges again, with Bombelli and others: there is the scandal of imaginary numbers as well as an ongoing scandal of infinity, which becomes the central object of mathematics in the 17th century. This pattern seems to question whether there is just one big break in the development of mathematics, that between the Euclidean and the non-Euclidean, or whether perhaps such breaks are endemic to the development of mathematics.

**Plotnitsky**: It’s a good question. As our earlier discussion suggests, the history in question is even more multiple and multifaceted than this picture suggest. There are both “scandalous” breaks and more continuous developments, involving various degrees of both key features of non-Euclideanism: nonclassical epistemology and the interaction between different fields involved. The combination of both is crucial to my conception of non-Euclideanism, and, the second aspect of the phenomena emerges with an eruptive force around 1800, with the rise of many new subfields and branches of mathematics, and of new forms of interaction between them. On the other hand, it took a while to realize the radical, nonclassical epistemological features of this mathematics and some of the mathematical objects where these features are more pronounced have indeed appeared later (so there is a history in this respect as well). Of course, the multiplicity in question has its history as well, a long and rich history. As Barry Mazur astutely observed in commenting on my paper, the scandal of the diagonal reflects, including in epistemologically proto-nonclassical terms, the relationships between arithmetic (and by implication algebra) and geometry, and friction or obstructions in trying to bringing them together. The harmonious bringing them together is a kind of hope, an eternal and eternally unfulfilled hope, of mathematics. While it’s not quite clear to me (not being a historian of mathematics) to what degree the ancient Greeks were thinking in these terms, they must have been thinking about the connections between both. This thinking is manifest in the Pythagoreans’ discovery of the irrationals, or, to begin with, in the idea of measurement, which defines geometry. With Descartes and his project of analytic geometry that dream reemerged with a new force and appeared possible to realize, now with the help of algebra. But then a much richer architecture of multiplicity (and with more radical epistemological implications) has emerged in the nineteenth century.

**Alexander**: Right, you can think of the case of analytic geometry in these terms — geometry and algebra are two fields, and they are combined into a single field. What I think you are referring to in the 19th century is more radical — truly different fields that somehow converge to inform each other on a particular question. This seems to be a later phenomenon, and Barry Mazur’s paper demonstrates it very clearly.

**Plotnitsky**: Yes. Thus, while, as Federica La Nave’s paper, again, beautifully demonstrates, Bombelli and others already showed earlier how you could do things with imaginary roots by combining algebra and geometry, the explosive power of the mathematics of complex numbers and its impact elsewhere in mathematics and physics only emerged in the nineteenth century. So much was gained as a result in number theory, algebra, analysis, algebraic geometry, and so forth.

**Alexander**: Do you see, then, a fundamental difference between the use that Bombelli makes of imaginary numbers, and the field of complex analysis as it was developed 200 years later?

**Plotnitsky**: Yes, very much so. Bombelli’s approach is important, conceptually and historically, but the difference is essential, all the same. I don’t think that one could really speak of Bombelli’s mathematics as non-Euclidean in the present sense, even though one can locate certain elements (the link between algebra and geometry, for example) of non-Euclideanism in it.

**Alexander**: It seems to me that there is a qualitative difference here. Bombelli was simply saying that we can solve this equation by acknowledging ”imaginary” roots. But the field of complex analysis emerges at the period that you talk about at the nexus of your story.

**Plotnitsky**: That is right. It is not only a matter of gains for mathematics (and one can also speaks of losses, insofar as certain things previously useful were no longer possible), but also of the emergence of a different type of mathematics. Other developments were also crucial, for example and in particular, non-Euclidean geometry.

**Alexander**: Right, Non-Euclidean geometry is one of the cases in which, as you say in your paper, the problem becomes the solution. The problem that you cannot prove Euclid’s 5th postulate becomes the starting point for a new mathematics and a new physics. I find this historically fascinating because it brings out the aspect of multiplicity that emerges at this moment. To me this is very strong evidence for the break that you write about between non-Euclidean and Euclidean thought in this period. As Barry Mazur notes, the issues of friction between geometry and algebra have a long history, but nevertheless it seems that something critical happens at this point.

**Plotnitsky**: Well, first of all, one also had analysis, a field that is algebraic in the sense of formal manipulation of its symbols but that deals with things, such as continuity, that are not part of algebra. So, by that time, there are three main areas of mathematics rather than only two: algebra, geometry, and analysis. And, again, one also had complex numbers. These “resources” allowed mathematicians to give very rich structures to the mathematical objects involved, to enrich old ones and to create new ones.

**Alexander**: So it seems that this process was very gradual. Geometry is of course very ancient as is arithmetic, modern algebra dates to the late 16th and early 17th century, the calculus to around 1700, and then general analysis to the 18th century. One can conclude that it is a slow process in which each crisis generates a new object, a new field of discovery.

**Plotnitsky**: I would say, along the lines I suggested earlier, that the process is long, because, while at certain junctures things may accelerate fast and punctuations may occur near to each other, it takes time to cohesively integrate new ideas even within the original field of their emergence and, all the more so, on broader scales. I would also argue, however, that this integration is often productive and sometimes necessary, it is in general partial, especially on broader scales. The history of mathematics continues to bring its diverse concepts and fields together, but, it appears, without the possibility of an ultimate unification (partial or provisional unifications are possible). Such an ultimate unification may not be possible even within a more limited configurations or, in the language of Barry Mazur’s paper, templates, such as those created by programs like that of Kronecker or, later, that of Langlands. For, as these programs historically develop, things not only come together but also break apart, creating new separate trajectories, sometimes new programs. It is a great interplay of continuities and discontinuities, interconnections and disconnections, necessities and chances, and so forth, which becomes richer and more complex as we move to greater configurations of knowledge within and beyond mathematics. It’s like an archipelago, where you can build bridges between islands. In fact, I don’t even like the metaphor of bridges, I prefer that of ferries that move between different islands and sometimes run aground or arrive at unexpected destinations. Hegel already thought in terms of similarly multi-component and multi-interconnected architecture of knowledge and its history. But he still appears to have believed that it is possible, in principle, to bring it into a form of unity through the dialectical synthesis of difference and contradictions. I think, however, that even for mathematics, let alone for knowledge in general, that does not appear possible, and non-Euclidean mathematics is a manifestation of both this impossibility and of the productive use of interconnections without an ultimate synthesis.

**Alexander**: Could you say something about the work that narrative does in your theory, and what it adds to your methodology. Is it essential to it?

**Plotnitsky**: It may be more essential methodologically than I thought before working on this paper, although I have always recognized the essential role of narrative in the epistemological problematic in question. Narrative is obviously essential for this paper, both conceptually (it is about the narrative) and methodologically. For example, although it is not strictly a historical paper, I was interested in certain historical situations that have played themselves out in cases where the construction of certain narratives was crucial, because this is how culture and politics work. In your own paper, the figure of the mathematical martyr became part of a certain narrative operative in culture. The scandal of the irrational is also part of a certain cultural narrative. I stress “part,” because such situations are not reducible to narrative, even though their narrative component is irreducible. One can also flip the coin. The story a person goes and drowns himself, or herself, after discovering the irrationals may be an allegory of the shipwreck of the Pythagorean arithmetic; but it may have also been true. Gauss had famously concealed his discovery of non-Euclidean geometry for year for fear (justified) of being laughed at by philistines. It should be noted, however, that it is disputable and even doubtful that Gauss had ever fully established the existence of non-Euclidean geometry. Although I’m ambivalent of many of Socrates’ ideas, he was executed for them after all. Nietzsche even says that the Greeks had good reasons to prosecute, even if not execute, Socrates because he threatened to replace the tragic culture of the Greeks with a new culture, the despotic logical culture, with an androidal culture, if I may say so.

**Alexander**: True, his political propositions do not seem appealing at all… Thinking along your lines about narrative, and the role it plays in those schemes, it seems that once you start to think about this not just abstractly but temporally, about crises that occur historically at particular times with particular agents and certain stories, it becomes a very powerful (as well as convenient) way of talking about these transitions.

**Plotnitsky**: It plays a major role. It is not only a matter of it being convenient or powerful, although this is also true. Even though other ways of thought remain important and irreducible, there may not be another way to think about temporality and history without narrative.

**Alexander**: You bring in the narrative in the crisis moments that we talked about. But I wonder what happens to narrative in between those crises. Would you say that it plays a role then as well?

**Plotnitsky**: I would , again, advocate a more multiple and dynamic picture, along the lines suggested above. Many histories happen at the same time, and the “same” event may be a crisis, a radical break, in one history and, in Kuhn’s terms, normal practice, part of a continuous trajectory, in another, either contemporaneously or in retrospect. These types of events may, however, have a great transformative power.

**Alexander**: In thinking of narrative in this context I can see that it has an enormous transformative power. It seems to me that narratives by their nature are polemical, and that they would naturally come to the fore at those moments of crisis. Their function in-between these points is to me an open question.

**Plotnitsky**: That is certainly true because I think narrative is a crucial part of the rhetoric of culture. Let me clarify something, however. I think that almost everyone, even the most rigid Platonists, would subscribe to the theory that mathematics is an exploration of the unknown. But nonclassical epistemology entails a different narrative model, in which, at a certain point, you can no longer continue this type of exploration and move into an unknown territory and gradually make more and more of it known through exploration. Nonclassically, the unknowable is always part of knowledge, which also entails a different type of narratives of knowledge. Now, to return to history, via the question that Peter Galison asked me concerning a possible concept or model of history grounded in nonclassical epistemology, coupled to the irreducible discursive and, especially in this context, specifically narrative multiplicity. Would you agree, as a historian, that it is not an easy model of history to develop or practice?

**Alexander**: Definitely.

**Plotnitsky**: Indeed, even if applied to events that involve encounters with nonclassicality, the models and narrative of history used in my paper or that we have invoked thus far are more or less classical models, coupled as they may be to discursive and narrative multiplicities. That is, however, different from constructing a non-classical model of history and a corresponding narrative, which is hard.

**Alexander**: It is very hard; on the other hand, it is also very clear today that classical narratives of history are sorely deficient. You can tell all the grand narratives you want, and you might capture various things about, say, technology, the march of progress, or politics, or the neo-conservative march of democracy or whatever else you like. You can tell those stories, but whichever narrative you tell, it more than falls short. It is almost an act of wilful desperation for a historian to try to tell a single “classical” narrative.

**Plotnitsky**: That’s quite true. Let me, however, reiterate the distinction between the narrative of history that involves encounters with nonclassicality, in which narrative itself could still be classical, and the role of nonclassical epistemology in the history of mathematics and science, or elsewhere. Dealing with nonclassicality does not in itself imply a change in our thinking concerning the nature of history or our practice of history.

**Alexander**: I perfectly agree.

**Plotnitsky**: However, non-classical narratives of historical events, even apart from any role of non-classical epistemology in this event themselves (say, those of classical physics) is, in principle, possible, and perhaps necessary.

**Alexander**: Yes, that’s what I’m trying to get at. What would such non-classical narratives look like?

**Plotnitsky**: I cannot think of historical studies where this type of history is practiced (perhaps because of the disciplinary constraints), although one might find certain non-classical elements in such studies. We might need to look at literature for examples of enactments of non-classical historical and narrative models—Joyce, Beckett, Woolf, or, closer to history, Faulkner. I do think that Faulkner gives us a non-classical picture of American history. I am not unhappy that I am not a historian and that I don’t need to write non-classical historical accounts, because they would be difficult to construct and, I suspects, even more difficult to publish.

**Alexander**: Very difficult. The examples that you give are of self-contained literary narratives, but to apply these to a historical narrative is very hard. You probably end up as with mathematics – with a multiplicity of fields and a multiplicity of narratives. And that is in fact what you see in actual historical practice – a multiplicity of narratives. You find so many overlapping, and competing stories, so many different ones, and none of them can make a claim to being hegemonic.

**Plotnitsky**: It’s difficult for psychological reasons, it’s difficult for rhetorical and institutional reasons. It is one thing for a writer like Faulkner or Beckett to write a novel of this type; it’s another thing to write a history of mathematics and science that operates in a radical register of discourse.

**Alexander**: But that’s the thing; what separates historical narrative from literary narrative is the claim of the factual truth of actual events, which literary works don’t require. You can never give up this claim as long as you’re doing history.

**Plotnitsky**: You’re quite right. How do you describe a fact or event nonclassically? That could, I think, be done in principle, for example, on a literary model, say, the way certain events, especially conversations, are described in Faulkner. Whether it could be done in practice of your discipline is another question, because a specific narrative of certain events, often key events, such as the discovery of the irrationals would be like a novel, more like Faulkner’s Absalom! Absalom!, Joyce’s Ulysses or Finnegan’s Wake, or Beckett’s The Unnameable, a most fitting title here.

**Alexander**: Yes, it will not be a historian who will write that history. Historians are willing to qualify their truth claims to some extent, to say that they tell stories about the past, that they use imagination, empathy, etc. But ultimately, the only basis, the only reason for the existence of the field is that in some way it’s about “what really happened.” Nevertheless, things have thoroughly changed. Not so long ago it was generally assumed that underlying all of history was a single unified narrative; you might be telling part of it, somebody else might be telling a different part of it, but all ultimately join together into a unified narrative that will tell us what the past really was. So a researcher looking for a topic would search for a topic “that hasn’t been done.” The point was to cover a little bit more of this great unified landscape of history. That is no longer the case. Historians do not pick topics in that way. The current assumption is that there are so many different possible questions, so many different possible stories, so many possible heroes and villains that you could talk about. And this multiplicity is now integrated into the structure of history. Without any acknowledged crisis, the underlying assumptions of the field have been transformed. This strikes me as profoundly related to the non-classic move.

**Plotnitsky**: To what degree did the history of mathematics itself follow this shift? It seems to me, for example, that the heterogeneity or non-Euclideanism of mathematics has not as yet found its way into the multiplicity of narratives about mathematics. It also seems to me that mathematics itself has continued to have and portray a more homogeneous Platonist image of itself than sciences have, especially biology and information sciences and technologies, which recently replaced physics as dominant sciences in culture. It appears that the field of the history of mathematics—I don’t mean the particular historians, but the institution or the field as a whole—has not as yet portrayed this multiplicity, which is always historical (an intersection of events and trajectories) of modern mathematics. If we had adopted a more Aristotelian view of mathematics, which would be related to the world, the history of mathematics could also benefit from it. On the other hand, this divorce of mathematics from reality, from everyday phenomena was also extraordinarily effective and productive.

**Alexander**: Extraordinarily productive, no question… But here’s the paradox of mathematics. I think you make a wonderful case about this multiplicity of mathematics and those overlapping fields and making it almost a prototypical non-Euclidean, non-classical field. On the other hand, mathematics has a cultural role which can be described as “the guardian of modernity,” of the modern hierarchy of knowledge. For example, you are right when you say that information science or technology is the dominant form of knowledge, at the moment. But mathematics has a role here; it may not be at the centre of things, but it guarantees their validity. It guarantees that information technology is rational, that it works and so on. Because of this function there is a widespread sense that mathematics must be kept unified and pristine, insulated from the world.

**Plotnitsky**: The event, to use a narrative category, like this—the conference, the book, the interviews—is significant from this point of view because it tries to bring the world to mathematics, and mathematics to the worlds, through narrative. It might help a greater dissemination of mathematics in culture and, as such, may have an effect upon the history of mathematics. It seems to me that our discussion now arrived to what defines the event of which it is a part: the role of the narrative in mathematics at various levels, from the cultural narrative of the kind you analyse to the kind of epistemologically defined narratives that are subject of my paper to the quest narratives analysed in Apostolos Doxiadis’ and Michael Harris’ papers, to the narratives of journeys of mathematical ideas themselves considered by Barry Mazur, so forth. Or, from the other pole (although there are more than two poles) of this event, that of narrative theory, there is an introduction of certain, let us say, quasi-mathematical models into the theory. This bringing narrative into our understanding of mathematics could, I think, also be productive in reaching the field of history of mathematics and, again, in enriching the relationships between mathematics and culture in general, in part because, while much of modern mathematics is divorced from the world, most of the world in which most of us live, the narrative is far from being so divorced.

**Alexander**: I agree. Mathematical narratives are a kind of “Jacob’s ladder”…

**Plotnitsky**: I think, to return to my earlier metaphor, it’s a ferry that can bring us to the island of mathematics or run aground close enough, so we swim across. I prefer horizontal metaphors.

**Alexander**: I understand why.

**Plotnitsky**: It brings us close enough to see and, at least to some degree, understand, the architecture of mathematics, which, as we discussed here, is more like a city than a building because of its interactive heterogeneity, multiplicity. This, I now realize, is itself a metaphor and a narrative that connects mathematics and the (urban) world, at least, formally, kind of mathematically. But in any event, I think the narrative can help us come closer to mathematics.

**Alexander**: I would definitely like to do that…

**Plotnitsky**: But I think that’s what you are doing here, what we are doing here. And historical narratives, such as those told in your paper, are part of that process, as are also fictional narrative, such as that of Apostolos Doxiadis’ novel. They are, let us say, narrative aspects of the grand phenomenon of mathematics. The narrative could serve to open, to use a metaphor of Gilles Deleuze, new lines of flight, in both directions, between mathematics and the world.

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]]>Harris:…

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]]>**Harris**: I like your way of viewing the history of mathematics in terms of guiding stories. Whether or not they correspond to reality, they are certainly a product of their times; at the same time they provide a context for people either to fit in or not.

**Alexander**: It’s not so much a matter of everybody fitting those models so much as they become models. They become archetypes of what mathematics is or, in some cases, what a mathematician is. Obviously, Galois is a very extreme case of any biography which is why he’s so famous, but he becomes the model of what a mathematician really is. That’s what I’m looking at.

**Harris**: Had I known I was going to interview you, I would have made an effort to get a copy of your book. You are writing another one now. Is it going to be on these three periods?

**Alexander**: Yeah. My first book was specifically about the emergence of infinitesimal methods and what I call, exploration mathematics. It was a very specific historical study of a specific period and the emergence of a narrative of exploration, encouraging brave men to go out and possess distant lands. I then follow this narrative from its origins all the way to the actual mathematical techniques used. I find that the story emerges in the 16^{th} Century in association with the voyages and it draws upon two different Medieval narratives. First the crusading narrative says ‘Let’s go back to the Holy Land and possess it,’ except that instead of going back to the source, this “crusade“ leads to the ends of the Earth. Second, it draws on the knight-errant romance, which dies contain this theme of going to the ends of the Earth but, unlike the crusade narrative, it has no element of possession. The knights-errant go to distant lands, but leave no mark, discovering only their own worth. It takes of both of these narratives to come up with the 16^{th} century notion of exploration.

I start there and I see how explorers themselves used this story and how it shaped the geography of the land. I discuss how mathematicians were involved in the voyages, and focus specifically on Thomas Harriot, who was both explorer and mathematician. Overall it’s a very detailed historical study of one particular period.

**Harris**: Before you go on, I realize I should ask you about your training, how you got into history of mathematics, how you came to have the idea of approaching history in this way, and how you put together your “narrative of the narrative” from source material.

**Alexander**: I was an undergraduate in Jerusalem, where I studied history and mathematics. Later on I combined my two interests and started looking at mathematics historically, using some of the approaches I‘d learned in history. I also saw this as something of an intellectual challenge because mathematics is the most abstract discipline. I wanted to see if I could make a convincing argument about how those seeming opposites relate to each other, influence each other.

**Harris**: It sounds to me as if you’re saying that it’s a challenge to do a materialist history of mathematics. It’s very interesting that you do that by way of narratives.

**Alexander**: One influence in my work that would be fairly obvious would be Foucault. Another one would be the anthropologist Clifford Geertz, who shows how people telling stories about themselves shape their own world.

**Harris**: And neither one of them was a historian.

**Alexander**: Foucault was, in a way, a historian. When you look at *The Order of Things *there are a lot of details one could argue with, but at the end of the day you really feel like he told you something very deep and very true about the order of knowledge in the 18^{th} and 19^{th} Centuries.

Historians are trained in looking at a lot of sources and reading them properly and accurately but what they are not always so good at is making sense of it. They have an ingrained suspicion of generalizations or anything that smells like it’s going to turn into philosophy.

**Harris**: Moving to the next project…

**Alexander**: The next project is more of a step back. The emphasis here is less on any particular period, but rather a gesture towards a general storied history of mathematics. I offer some broad strokes of what a history of mathematics, based on stories, would look like. As you heard in my talk, I suggest several different periods in the evolution of mathematics. I have the three periods that are most advanced right now: the 17^{th} Century, which is the age of indivisbles; the 18^{th} Century, in which mathematicians were viewed as “natural men” in touch with the inner harmonies of the universe; and finally those beautiful martyrdom stories of the 19^{th} century, which appear in the 1830s practically, it seems to me, out of nowhere. They have no precedent among mathematical biographies.

**Harris**: The stories of the period of the Age of Discovery are well known and people learn about them still. In the 18^{th} Century, the stories are rather different; it’s the Enlightenment and the French Revolution…

**Alexander**: They are different stories. I do have some examples of stories of exploration in the 18^{th} Century and it’s very interesting how different they are from the stories of the 17^{th} Century. Maupertuis leads an expedition to Lapland in Finland and the point is to determine the shape of the Earth; to measure the degree of latitude. (He wanted to) compare the degree of latitude at a higher latitude to one closer to the equator. Another expedition went to the Andes of Peru to measure a latitude there. What I find interesting about that is that it’s a totally different notion of what exploration is. The mathematicians in the 17^{th} Century were interested in the exotic other world, penetrating it through forests, rivers and mountains, and seizing its gold. When you go into the 18^{th} Century, their focus is not to discover new facts about Lapland or Peru. What they want to do is show that wherever you go, you’re still subject to one rational system. That is, they want to show things are everywhere the same, whereas in the 17^{th} century explorers wanted to show how different they were. It’s very much about spreading this rational grid around the word and making it homogenous rather than penetrate it to detect hidden secrets.

**Harris**: Let’s talk about some alternative narratives. In the 17^{th} Century, another popular narrative centers around Fermat and Pascal. France is a point of stability in the 17^{th} Century when there were wars everywhere else. It’s the period of stabilization of the centralist monarchy. Do Fermat and Pascal fit in with the exploration theme?

**Alexander**: My sense is that the tradition in France is very different, starting with Descartes. When Descartes was young, he experimented with indivisibles and infinitesimals and tied to mix various calculations with those. Later on, as he developed his more systematic philosophy he ruled against it. Infinitesimals for him were not sufficiently clear and distinct. They’re paradoxical, problematic, obscure, and he ruled them out of bounds of acceptable mathematics. Instead, he developed analytic geometry, which seems to me to express an alternative tradition to the calculus. Infinitesimal methods are based on a materialist intuition — they are a mathematics abstracted from what we feel that matter is like. It’s “mathematics from below,” — you start with materialist intuition and you abstract from there. That’s something that Descartes was very unhappy with. Mathematics was supposed to be a model of rationality. It was the anchor you could rely on in the face of the uncertainty of our intuitions. If mathematics itself is an abstraction from sensible matter, what is left? So Descartes created a method in which you start with mathematics and then define the world accordingly. That is, you take a mathematical expression, an algebraical expression, and then you draw what it is like in the world. So instead of mathematics being an abstraction of the world, he much preferred the world to be an expression of rational mathematics. For Descartes, the world is rational because mathematics is rational and the world expresses mathematical principles.

I’m not saying that all of French mathematicians were Cartesians, they were not, but these kinds of concerns about mathematics being independent of the world were much stronger in France than they were in England or Italy. There was a really wonderful exchange between Fermat and John Wallis in England. Wallis published it because he published everything. They understood each other because they both wrote good Latin but apart from that, they understood nothing about what the other was doing. On the one hand, there was Wallis doing experimental mathematics. He had this idea that to prove something (you needed to say) ‘Try one case. Try two cases. Try three cases. You see a pattern here. Proof.’ He would have no problem dividing by zeroes and multiplying by infinity. It was all very wild, his kind of mathematics, but he reached the results so, that’s all he needed. Mathematics was an experiment, an exploration. ‘Go there and see what we come up with.’ That was perfectly fine for him. Fermat of course was appalled. This was not mathematics. He showed him his number theory and all the speculations leading to Fermat’s theorem, and Wallis says ‘Well, you can do that but, what’s the point?’ I find this symptomatic of the cultural divide between the kind of mathematics that was being created in France and the contemporary mathematics in England. My sense is that the Galilean school in Italy was conceptually close to the English school.

**Harris**: Were there exploration poems and dedications in other kinds of literature as well? Was the imagery more concentrated in mathematics?

**Alexander**: Those kinds of poems and that kind of imagery was very common in other fields of science during this period. ‘Natural philosophy is exploration’ was a very dominant theme of the scientific revolution. In some ways, I find that the mathematicians felt like they were being left out. in the 16^{th} century the bounds of the old world were shattered and there was a sense that a whole new and unknown world was out there. The reformers of knowledge, like Bacon and Galileo, promoters of Empiricism and Experimentalism say ‘well, let’s go out and do the same for the natural world. There are great continents of knowledge to be discovered elsewhere as well. We are going to do exactly what those explorers do. We will go there and look, and try and see what works.’ Exploration was a wonderful model for experimentation and empiricism. You don’t just stay home and speculate. You actually have to go and explore and see what you can find, with no preconceptions. Mathematics was left out because, according to the Euclidean ideal, all geometrical truths are in principle implicit in the assumptions. There are no surprises. Geometry was considered a known and completed field that left no room for exploration.

**Harris**: There were the classical problems that Federica is talking about. The trisection of the angle…

**Alexander**: And there was a renewed interest in trying to resolve them in this century and this sense of mathematics making new discoveries and not just repeating its old results, as it largely did for a long time previously. These new mathematicians looked at what was going on in other fields of knowledge and said ‘We can do that too, we can explore mathematics. We don’t have to stay trapped in those old abstractions. We can go and explore and find new things. We can go and map out what a geometrical surface is actually like and what a line is like and what a cylinder is like. We can explore and discover new surprising things about them that no one ever knew before. And, we’re willing to take risks that were considered illegitimate, because exploration is always risky.’ To some extent, you can look at it as mathematicians joining the band wagon of the new empiricism.

**Harris**: The method of indivisibles creates the internal structure of geometric figures. But it is seen as a process of discovering hidden treasures. Triangles had to be imagined pre-existing in order to bear the weight of exploration metaphors. You can’t explore a triangle-

**Alexander**: -unless you constructed it first. That is a problem. That is why it seems so strange that you can think about Harriot drawing a material picture of what the continuum is like after imagining the continuum. It’s very strange. I agree with you. But they saw mathematics as practically material. They thought that when they were doing these thought experiments they were actually discovering what the structure of the continuum actually is; what the structure of a triangle is. Not that they put it there. Not that they imagined it a certain way, but that they are looking into it and discovering it. They, including Harriot, used paradox as a tool of discovery because it helped them explore what the actual structure was. For example, Galileo in the *Two New Sciences*, used a device called the “wheel of Aristotle”. There are two wheels around a common center, and when the outside wheel makes a complete turn the inside wheel also makes one complete turn. But somehow, it makes a line that is longer than its circumference! How can this happen? Galileo concluded that there are gaps between the points that make up a line, and that the gaps are bigger on the inner line than they are on the outer one. So, through the paradox, you learn something about what a line actually is.

**Harris**: Newton was a narrative all by himself, I guess. People tell stories about the apple. I suppose they started telling that story during his lifetime. It doesn’t fit in with either the 17^{th} or 18^{th} Century. He’s such a singular figure that maybe he doesn’t have to fit in a pattern.

**Alexander**: I’d like to fit him in. But when I talk about the 18^{th} Century, I talk very little about England and Italy which were central in the 17^{th} century. The dominant tradition is certainly French.

**Harris**: That was true of mathematics in general. It was not a particularly creative period in England.

**Alexander**: The standard story, that may or may not apply, is that the English mathematicians were so devoted to the glory of Newton that they were slow in accepting continental methods. I’m certainly aware that when I talk about dominant stories, although I emphasize time period, they clearly have a geographical boundary to them. What I hope to do is to have some sort of scheme of what this kind of history will look like. I don’t mind qualifying it geographically or tying-in other narratives.

**Harris**: The first obvious point about the 18^{th} Century is that your chapters are almost exclusively about France. Even Euler is seen from the standpoint of Condorcet. How important was the image of the natural man outside France, for example in Catherine the Great’s St. Petersburg? Was Euler appreciated as a natural man?

**Alexander**: Good question. The Russians were certainly Francophiles, and the Russian aristocracy all spoke French, But I can’t actually say that he was viewed in that way. By all accounts he was a very down-to-earth, practical man; not playful. Just a solid Swiss-German bourgeois citizen. I would imagine he would not think of himself in any sense as a romantic, natural man. But for those who, like Condorcet, wished to view him on this manner, he would in some ways fit the mould — especially in his suspicion of the sophisticates and all of their refined manners. The main guy for the story of the “natural” mathematician is clearly d’Alembert, who left a very long record of non-mathematical writings. In his “Preliminary Discourse” to the *Encyclopedie* he also wrote a concise description of what he thought mathematics was about, and 18^{th} Century mathematicians really followed it (his view) very closely. I would say he gives a very good description of a contemporary understanding of what mathematics is: an ultimate abstraction of nature that never leaves nature. Algebra is the final frontier of mathematical abstractions, but still remains anchored in the actual world. Leaving the world would be a disaster that would deprive mathematics of content.

**Harris**: My impression is that the calculus of variations is the typical field of mathematics in the 18^{th} Century: Euler, d’Alembert, Lagrange.… Is the Principle of Least Action part of the narrative of the natural man or of the narrative of the principle of nature?

**Alexander**: I would say the latter. The assumption is that of the mathematical universe. The reigning assumption in the Enlightenment, among those who supported mathematics, is that mathematics is about the world because the world is essentially mathematical. For Mauperuis, the Principle of Least Action is the fundamental underlying principle of what the world is like.

There is a lot of opposition to mathematics as well. Buffon and Diderot were both very interested in mathematics early on, but concluded that it was too abstract. They thought mathematics leads one so far from the real world and that one can never get back. Diderot has this beautiful image of mathematicians standing on their high mountains; when they try to look down, all they see is mist below, which prevents them from finding out anything about the real world.

In his *‘Letter on the blind’*, Diderot talks about the blind mathematician Nicholas Saunderson, a Cambridge professor who was blind from birth and who was a wonderful mathematician nevertheless. Diderot uses him to makes the argument that the greatest mathematician is essentially blind because he doesn’t need the world, he builds his own structures. Saunderson is a brilliant mathematician, he even teaches optics in Cambridge, but he knows nothing of what the world is like because he’s never seen it. He doesn’t even know what seeing is because he’s been blind from birth. For Diderot mathematics is about building castles in the air, in isolation from the world, and is therefore useless for our understanding of the world. In the “Preliminary Discourse” d’Alembert agrees that mathematics should be about the world, and responds to Diderot by saying ‘no no, the world is mathematical, and mathematics is always about the world.’ He also adopts Diderot’s image on the mathematician standing on ‘the high mountain’ and looking down. Below is the world, and if look down and gradually you can see the harmonies that govern it. Those harmonies are visible to a mathematician precisely because he is so high up on the mountain.

**Harris**: I have one more question about this period. Were other scientists praised for being “natural men”? I‘m thinking of Benjamin Franklin, for example, who was celebrated in France.

**Alexander**: Perfect! I hadn’t thought about that but it fits very well with the story I’m telling. That science was about the world and scientists were the ones who had a natural understanding of the world. The mathematical twist to it was the claim that mathematics also belonged in that company because mathematics is also about the world.

**Harris**: When I was first interested in mathematics, I was exposed to Bell’s *Men of Mathematics*. Of course, I didn’t know that this had been going on for so many years. Nobody else I knew was interested in mathematics or these stories but romantic stories of Abel and Galois seemed appropriate. Did they present these characters just because we were teenagers? Because they appealed to us? Why did they not talk about the other mathematicians?

**Alexander**: I think those were the stories that appealed to everyone. These stories have an interesting genealogy, beginning with the deaths of Abel and Galois to their canonization by Bell. Oddly enough, Galois was celebrated as a hero when he died not for his mathematics but because he was a hero of the Revolution. He had all those radical friends who railed against how he was an oppressed genius, how his mathematics was ignored by establishment mathematicians. His close friend wrote an obituary for him shortly after he died and he called him ‘a martyr to his genius.’

**Harris**: There was an Abel centenary in Paris a few years ago. I believe they told the story of how he came to Paris and wanted to meet with mathematicians. They didn’t have time for him but four members of the Academy nevertheless wrote for him.

**Alexander**: He was a rising star. It’s true. He did not have a lot of success in Paris but he met interesting people. He was friends with the scientist-politician Francois-Vincent Raspail, who years later told in a speech in the Chamber of Deputies how this poor mathematician was ignored in Paris and had to literally walk back home to Norway.

**Harris**: It’s significant that they didn’t mention that part. They did mention that he published his articles.

**Alexander**: Legendre wrote admiring letters to him in his final years. He could have had a position in Berlin years before except that he wanted to stay in Norway. Crelle never understood why on Earth he would want to go back to that snow-bound no-man’s land to Christiania (modern Oslo). Christiania was very small and very poor, but people there tried to help Abel as much as they could. Other professors added to his salary from their own pockets because they recognized he was a very special talent. At that point, they could not get him a permanent chair, but the man was just 26. And then Crelle was working very hard the whole time to get him a position in Berlin and he succeeded – but too late for Abel. So Abel was not a tragic figure in his own eyes at all. He dies just when so many opportunities are opening up for him and that is perhaps a personal tragedy. But he was not alienated or marginalized. Nevertheless, the minute he died, the story of the persecuted genius immediately emerged. People were pointing fingers both in Paris and in Norway about “who killed Abel”. And ‘Why didn’t he get a permanent position here?’ It was kind of sad because the people who were being accused were his friends who helped him. The people making the accusations were the people who’d never heard about him and never cared, but in his death had found a hero.

**Harris**: Why was the romantic stereotype so effective all of a sudden in the 1930s when Bell wrote his book and at least through the 1960s when I was exposed to it?

**Alexander**: In France, the Galois story was already canonized by that point. It was a founding myth of official French science. That happened in the late 1800s when Galois became this misunderstood hero. The inspector general of the Ecole Normale himself wrote the most authoritative biography of Galois to this day. This was a great irony because Galois didn’t want to go to the Ecole Normale. He was kicked out of the Ecole Normale. He wrote horrible things about the Ecole Normale. Two generations later, he becomes the emblem of the greatness of the Ecole Normale and an icon of French science. During World War I the Belgian George Sarton, founder of academic history of science came to the United States. He knew the story from the Continent, and he started publishing in the US and he spread the story of Galois. Bell adopted it from him, and then it really became widely disseminated. So the popularity of the story actually dates from long before the 1930s and, from my experience, it continues to this day.

**Harris**: It still works. It’s kind of a paradox because the romantic stereotype is still effective and yet very few mathematicians see themselves as misunderstood revolutionaries. Some do but they are usually crackpots, outside the official structure. Mathematics is a highly structured activity. A mathematician may be misunderstood by some people in the establishment, but is rarely misunderstood by everybody. Then there is also the myth of the suffering romantic artist. The dynamics in mathematics are not the same, but it’s pretty clear from your description that the romantic artist and the romantic mathematician arise at the same time.

**Alexander**: I think they do. Chopin. Byron. The tragic story is not original at all. It’s fairly standard in a mathematical version. The interesting thing is that it aligns mathematics with activities like music, art, poetry and so on and not with physical science which is what the previous generation thought about. The Ecole Polytechnique tradition saw mathematics as an extension of physics and the study of the natural world.

You really don’t find that kind of tragic imagery among scientists. Einstein is viewed as an eccentric, but not a tragic one. The imagery of the scientist, from d’Alembert to Einstein, is that of the eternal child who is still curious about simple things and looks at the world with big eyes. Einstein is a natural man, not corrupted by social refinements. That’s why you have those pictures of him making funny faces. He’s just a kid playing around, expressing his natural wonder at the world. It’s a very different imagery than the ones you find associated with the tragic mathematicians.

**Harris**: Having established that the romantic myth retains its attraction and provides motivation for people to enter the field in the form of role models, I would consider the theme of self-consciousness. It certainly fits with the broader cultural themes attached to names like Marx, Darwin, Nietzche, Freud, and with the mathematical theme of the foundations crisis. Cultural history is a little bit of a risk. You see what you want to see. In psychoanalysis, if the heroes are tragic figures, it may be because they are mentally ill. You have Cantor and Nash, Gödel is borderline, Turing and Wiles are presented as obsessive. These are the images that sell popular books.

**Alexander**: Herbert Mehrtens wrote a book about it which, unfortunately, is in German. It’s specifically about modernist mathematics and talking about the early 20^{th} Century. He had an argument along the lines you were saying, seeing the foundational crisis in mathematics in the context of all the cultural fault-lines of the early 20^{th} Century. When you say that those 20^{th} Century mathematicians were not exactly tragic but were rather insane, you see a possible connection there between self-consciousness and…

**Harris**: and Freudianism.

**Alexander**: (That’s) something to look at. I looked at them as repetitions, as versions of the Galois story but I am willing to consider other options. I’m convinced that there is a strong element of continuity there. And that article I mentioned earlier today in the *New Yorker* about Perelman and Yau. I was very much impressed with that story which could be very unjust to Yau. It basically seemed to replay the old drama of Galois and Cauchy or Abel and Cauchy, the young genius who gets swept by the wayside, etc. Except that Perelman did get the Fields Medal!

**Harris**: The story has very much to do with traditional Russian notions of personal purity. He turned his back on the world. He was not trying to improve the world. He was not trying to make a stand.

**Alexander**: He’s not peculiar in any other way. Perelman in the article seems perfectly happy. He chose this kind of life. He lives with his mother at that institute in Moscow and he seems perfectly satisfied in that life. There’s certainly nothing tragic about that. That’s certainly a difference. He cares about the truth. He doesn’t care about worldly rewards. And then, on the other hand, you have his nemesis who cares very much about worldly rewards, credit, power and so on. In basic ways, it does follow the old paradigmatic story.

**Harris**: This is a story written for an American audience but it concerns two figures who are not really present in the 19th Century mathematics- a Soviet-trained mathematician and a Chinese mathematician. There’s a limit to which these stories can really be exported. I have no idea if there are any figures comparable to any of the ones you mentioned in the Chinese literary tradition. I don’t know what stories are motivating contemporary Chinese mathematicians.

**Alexander**: I don’t know what they are either but I suspect they are very different. This story in the *New Yorker* is told from a Western perspective and uses Western tropes.

**Harris**: You write: “*The correct results, which for Lagrange were both the purpose of Analysis and the ultimate guarantee of its viability, were for Abel merely a puzzling aberration*”. Where did Abel get this idea and how do you compare it to the shift from natural history to analysis? How did this actually get realized in the case of Abel?

**Alexander**: Where did he get this notion? I’m sure he was influenced by Cauchy. Even while he was in Paris, in his letters, he complains about his treatment but said that he is the only one who truly understands what mathematics is. The only one who was a true mathematician there.

**Harris**: So how did he know that?

**Alexander**: He read everything. Cauchy was publishing like mad.

**Harris**: How did he know when he read Cauchy that he was the one who really understood everything?

**Alexander**: I think part of the mystery here is that you have a break with tradition, and this means that you cannot see a direct continuity with what went before. Where did Abel get the idea that 18^{th} Century mathematics was dull mathematics? I don’t know. It’s an interesting question but it’s not the question I am ultimately interested in. Overall, you can see a general break, a clear change of directions in the understanding of what mathematics is about in a large group of mathematicians in France and then in Germany; along with this change comes a clear break in the stories that are being told about mathematics and mathematicians. Ultimately, more important for me than causality is the isomorphism between these stories: it shows that this kind of mathematics is intimately related to these kinds of stories. That is really what I focused at; telling a story that is not necessarily continuous but rather has breaks in it, that the breaks occur both in the stories and in mathematics.

**Harris**: Something big happened around 1800 that shook up Europe quite a lot. We can certainly see how the romantic hero arises in that situation but what’s surprising is how this romantic hero gets associated with the notion of mathematical rigor.

**Alexander**: In a book called *In Bluebeard’s Castle *George Steiner talks about this period as the generation that is after Napoleon. They were very bored after their fathers’ great generation. This produced a whole generation of alienation. He talks about the poets and how they try to look beyond this mundane, boring, depressing world. They tried to look for purity and purpose away from this world, in an alternative universe of beauty, poetry, and perfection. And I think there is something of that movement in mathematics as well. True mathematics resides in a beautiful alternate reality that is no longer part of this world. It is over there, ruled only by its pure self-referential standards, not by the limitations of our corrupt reality. So there is certainly a parallel there between poetry and literature and mathematics. The emphasis on mathematical rigor is a result of the conception of mathematics as self-referential, that mathematics is not part of the world. In the 18^{th} century mathematics was legitimized by the fact that it correctly described the world, but in the 19^{th} century it has to be true only to itself, not to the world around us. How do we know if a mathematical statement is true to itself? Only by following its own inner standards rigorously, because there is nothing else to fall back on. In other words rigor becomes really important because that is the only standard that you have – perfect, internal coherence. Whereas, as long as you think mathematics is abstracted from the world, rigor might be nice, but it’s not essential. The big movement is taking mathematics from our world to its own world, and the emphasis on rigor follows from that.

**Harris**: This morning I made the hypothesis that the alienating nature of mathematics education explains the persistence of the image of insane mathematicians, marginal figures in popular culture; the idea that people who are not mathematicians have an image of mathematicians as unstable figures.

**Alexander**: But you could turn the argument on its head and say that the reason why mathematics is taught in this way is because a certain image of mathematics prevails. The way mathematics is taught in college is very, very difficult and unnecessarily so. Students are basically asked to forget everything they know or think they know about math, and build an alternative new world. Students come in with a certain idea of mathematics. They know that a derivative is related to the slope of a graph. An integral is the area under a curve, and so on. When they come into college, they have to start everything from scratch. Forget about slopes and areas, instead, let e be greater than 0 and go from there. Bright students can follow a professor’s proof step by step, but have no idea why he’s doing it because they are given no context – they are operating as if in a vacuum.

I think that comes from that particular understanding of mathematics which has its origins in that period that we’re talking about. Mathematics exists I an alternative world operating according to its own rules. A student’s materialist intuitions are therefore useless and he/she should start from scratch. Going back to your question, perhaps it works both ways. This flawed education causes people to view mathematicians and strange and otherworldly, but also the view itself produces the flawed education.

**Harris**: So, you’re thinking that the people who teach mathematics have internalized a view of themselves as separate from the world.

**Alexander**: Of their field, maybe not personally of themselves.

**Harris**: Well, yes, but their intellectual life is not connected to the world, it is not subject to the vicissitudes of material life.

**Alexander**: That’s mathematics.

**Harris**: That certainly is an attraction.

Another thing I mentioned as a possible periodization is the rise of conjecture and internationalization as a source of self-consciousness in mathematics. When Hilbert stated his 23 problems in 1900, it was consciously an international research program in the setting of the International Congress of Mathematicians. The act connects in several ways with self-consciousness more generally: you analyze the field and try to determine where it’s going to go on the basis of the incorporated self-consciousness with respect to foundational questions.

**Alexander**: So you are saying the very fact of setting up research, the self-reflection of the field is indicative of a larger move towards self-reflection in perhaps other cultural areas?

**Harris**: Orientation to the future operated in mathematics in the other periods of exploration. In that sense, it is consistent with the tradition as well.

**Alexander**: Those great problems remind one a little bit of the famous ancient problems, like the trisection of the angle or the squaring of the circle. It’s an interesting question seeing how those people actually talked about those questions in antiquity and how they saw them. Was there a research program to resolve the great remaining riddles? I don’t think there was the same sense of progress in mathematics in antiquity, of this open-ended research program. They basically felt that they knew what they were doing but there were few of those problematic islands which were tough nuts to crack.

**Harris**: There was a very self-conscious program for modern music.

**Alexander**: I agree, that’s the place to look – music, architecture, art. I’m really interested in Peter Galison’s analogies between modernist architecture and the Bourbaki project. It makes a lot of intuitive sense.

**Harris**: I had some other images regarding 20^{th} Century mathematics. It’s harder to compare them to what you call “tropes” in popular culture.” I already asked what “popular culture” meant in the 16^{th} Century, but you more or less explained that by saying that in travel narratives, for instance, people had a shared culture that they could read.

**Alexander**: Yes, they could. In the 16^{th }century those travel stories were enormously popular. It seems like every explorer, from admiral to ship’s boy published a book of their adventures, their discoveries, and all the hazards and the hardship they faced. This was very widespread in that period.

**Harris**: Bourbaki provides a new image. I don’t know whether Hilbert and the Bourbaki really should be separated as systematizers. They fit with the Puritanical impulse that you have already identified in the 19^{th} Century, but are by no means tragic figures. At the opposite end, people like von Neumann and Ulam are in the middle of a completely different sort of adventure. Ulam has written an, I guess, not–so-popular book called *The Adventures of a Mathematician.*

**Alexander**: And then there is Turing, who is a tragic figure though. I recently saw another description of Turing’s story very much in that vein.

**Harris**: We’ve still got a few more topics but I haven’t got to the 21^{st} Century yet. There is one survival of Puritanism I identified. I don’t know to what extent it goes back to the early 19^{th} century but there is strong and irrational disapproval of giving up any research time to pursue other activities. This is something of which I was already aware as a graduate student. Students were gossiping about famous mathematicians who had accepted administrative positions — obviously because they were no longer able to do research. The only reason you would do anything other than mathematics is that you’ve lost it. There’s the myth that you can’t do mathematics after you’re 30. I don’t know when that got started.

**Alexander**: It started right about the time of Galois and Abel. Euler published new work well into his 70s, as did Lagrange and Legendre. The myth of youth, I think, is related precisely to those young martyrs, whose youth was a sign of their purity. They’re young. They’re pure. They’re not contaminated by the world, and as a result they are the ones who can see into the true world of mathematics. When you grow older, by implication, you become too corrupted by the world, by power, by interests, by children, by family, by all those things that pull you down. You’re no longer pure enough to access the beauty of mathematics. My sense is that this emphasis on youth is related to purity; the purity of mathematics and the purity of mathematicians.

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]]>**Galison**: I thought maybe we’d start by giving a little background. I don’t actually know how you came to this complex of questions. Perhaps you could say a little bit about the kind of mathematics that you have been most interested in and then I’ll ask about how you got interested in the more philosophical aspects. We’ll move into those questions and then we’ll look at the intersection of math and narrative.

One of the things which strike me in reading what you’ve written and in talking to you is that there’s something unexpected to me about your trajectory, so let me say what it is.

You have a strong interest and an involvement with Bourbaki and ideas of rigor in the Bourbaki tradition. I’m a complete outsider, but from the point of view of a theoretical Physicist they stand for a certain kind of rigor and for agnosticism about philosophical interpretation and meaning, a deliberate bracketing of issues like: what is the referential structure of these symbols? On the other hand many of your interests seem actually quite sympathetic to Poincaré with his muscular movement of shapes and this Helmholtzian theme of understanding geometry through the manipulation of rigid objects and so on. It is striking as a combination to have these two interests, so I wanted to pursue that a little bit with you.

**Teissier** : At the time when I began to work for Bourbaki, I already had these preoccupations, but I didn’t see things the way I see them now. I was not close to Poincaré in my reflexions. It was rather a very strong unease about all these things which I wanted to make sense. Working with Bourbaki had two aspects which I liked, one is that it was an ascesis. Feeling that need for meaning as I did, I thought that since I was given the opportunity I should really look at what these people, who all were older than me by definition and excellent mathematicians, were doing, and ask myself: what do they get out of this approach of going to the higher structures? why do those guys seem to be perfectly at ease with all this while I am ill at ease?

At the time I was already educated in structuralism. When I was in Classe Préparatoire I used to skip classes to go to the Collège de France to listen to Levi-Strauss, and that was another strong influence. So I was attracted by this ascesis of doing things in a way which was not natural for me and thus trying to understand what they got out of it. To be in the midst of it and to collaborate is the only way to understand how it works.

There was also another aspect: Bourbaki, as I viewed it and still do, is as a kind of public service, trying to make things as clear and accessible as possible to working mathematicians. I liked that. I didn’t teach so I thought I had a social obligation to do something for the community. Working for Bourbaki fulfilled that need in a way which I found interesting.

And I wasn’t disappointed, I really got a lot of understanding of this structuralist approach. It is true however that Bourbaki was not at all interested in meaning in the same way that I am.

**Galison**: Do you think of Bourbaki in its totality, in the volumes as they develop, as being a kind of a narrative about mathematics, a story of a kind?

**Teissier**: Yes, it claims to be that and I think to some extent it is that. It is a story with many digressions. But it is true that it claims to be a kind of story of origin. The work of Bourbaki begins with set theory and everything that goes with it. And then he said OK from there on let’s see if we can move up: algebra is like this, topology is like that, so it is a kind of story of origin. It doesn’t tell its name of course but as a matter of fact it is that. But it is not a story of the origin of meaning. If you assume that mathematics flow from set theory from to the ends of a mathematical tree, then it is a description of that flow. At the same time it is logically organized, with proofs, so it is also a foundation of truth of the statements that it contains. And finally it tries to bring to light the “deeper reasons” why the results are true, the structural reasons.

**Galison**: Here is one of the reasons that I asked whether Bourbaki is a kind of a narrative about mathematics. It seems to me very interesting that you have a story that de-emphasizes time, it is a story that in some sense is outside of time, which at first may appear completely paradoxical, but the sequence is a kind of logical one although it is not exactly logical, it is not Russell and Whitehead, it is a story where the flow of time is replaced by a conceptual flow. And did they think about set theory as the only way they could have started or did they think of it as a good way to start?

**Teissier**: That’s an interesting question, I don’t know. I do believe that at the time they began they were convinced it was the only way to do it.

**Galison**: So the idea of a kind of foundational story?

**Teissier**: Yes. But you see it is the only book that gives a gradient: it says that it goes from the general to the particular. If you think about it it is amazing because it occurs only 30 years after the birth of set theory; in fact it is not even 30 years. Suddenly a group of people decides that this is the point of origin of all mathematics. I think it is a major intellectual statement to say we’re going to take set theory as a point of origin and from there everything flows from the general to the particular. I think somewhere Levi Strauss says that genius is remarking on a simple fact and following its consequences to the end. In a way it is a good description of the Bourbaki enterprise. You could say that taking set theory as the foundation was a rather natural step at the time, in view of Russell-Whitehead, the Hilbertian program and all that, but to decide to follow the consequences to the end, that’s something absolutely new.

I think working with Bourbaki stimulated my desire to find foundations of meaning in the “human nature”, as one used to say.

**Galison**: By the move in the other direction.

**Teissier**: Precisely. However the manner in which I became interested in cognition to the point of giving it the central role in the foundations of meaning is not at all a deep philosophical evolution. It happened like this:

I was working on a problem related to catastrophe theory. I wanted, in a precise situation, to eliminate critical points of real analytic functions. Then I chanced upon a paper on the mathematical modeling of vision where people used the heat equation to eliminate the critical points of “grey functions”. This was an important idea in what is called multiscale analysis, introduced by David Marr and other people in the modelisation of vision in the sixties or the seventies. Perhaps you’re familiar with this. You take a grey function g(x,y) encoding a black-and-white picture and consider its two dimensional graph which comes with critical points: maxima, minima, and points of index one like mountain passes. And then you apply the heat question to g(x,y), thinking of it as describing a distribution of heat in a plane region at an origin of time. And what happens is that under the evolution determined by the heat equation the function gets smoothed, the critical points minima and maxima, points of index one are after some time eliminated. In any case that’s the idea. For functions of one variable it works perfectly, for functions of two variables like g(x,y), actually it fails but in a very interesting way: evolution according to the heat equation can create new critical points instead of eliminating them.

**Galison**: that’s interesting.

**Teissier**: This is exactly the phenomenon which I stumbled upon; it showed that the technique which I was thinking of using could not work but on the other hand it got me interested in the theory of vision. Then I started to read and got interested in why people were thinking of applying the heat equation, but with the time replaced by a “scale” parameter. Somehow it explained that we could see at the same time the (small scale) details of a face and also just the oval of the face. In principle when you apply the heat equation all that remains for large parameters is this big blob, the oval of the face. So I started to really try to understand what’s going on and that’s how at some point I got introduced to low level vision. Learning what people did for vision led me to apply the same kind of idea to meaning.

**Galison**: When was this?

**Teissier**: I’m trying to remember; this was in the late 80’s, about twenty years ago. I was working on it very slowly since it is not my main activity of course. But the outcome of it was that I started talking to people who worked on vision, particularly Alain Berthoz at the Collège de France In Paris, a very interesting person, he was very open. About ten years ago he invited me to speak about space in his seminar and at that time I was really angry with Kant because I couldn’t understand so I started by thinking I couldn’t accept, because for me it is just a term, that space is an a priori of perception. So I started to think about it exactly in the way I’ve been talking about today. That’s more or less the path that lead me there; it started with a problem in mathematics which by chance was connected to a problem with also occupied the people in vision, it moved me to try to understand why and how they were interested in it, and finally I thought I could see the trace of meaning, so I learned about vision and it is only about five years ago that suddenly I realized that the real problem of meaning was: can I give a meaning to the real line?

Fortunately by that time I had acquired a certain knowledge, a certain experience from talking with these people and I knew about the definition of the visual line and I knew from the Berthoz seminar about the vestibular line and some of the experiment that had been done. And I thought: ah! but the thing is that they are isomorphic from the view point of the structure and what nature, or evolution, did is to identify them for us, and that’s why we can think in several ways about the “same” real line and that’s why it is a mathematical object! That was the itinerary.

**Galison**: So Berthoz himself had not been interested in the real, in the mathematical side of it, he was more at the physical line?

**Teissier**: Berthoz, I don’t exactly know how to say it, he’s interested in many things but his real work is really neurophysiology. This isomorphism is called the Poincaré – Berthoz isomorphism in homage to both because Berthoz tried to draw the consequences of Poincaré’s idea that the position of a point in space is coded by the muscular tensions corresponding to the finger being positioned at that point. He made experiments inspired by that. For example you might believe that the gesture you make is not unique but in fact there are experiments to prove it is, that the natural gesture you make is severely constrained and in fact it is unique. Of course you can force variations, but the normal gesture will always be the same. He has many experiments in that direction.

**Galison**: That’s interesting

**Teissier**: He made literally dozens, maybe hundreds of experiments of that nature, on how you perceive symmetry, one of these experiments that struck me early on, the unconscious perceptual system knows that the sum of the angles of a triangle is equal to p. Put someone in a dark room, turn a certain angle, tell them to walk a certain distance and then turn back to its point of origin and people do it rather well….so as an inertial system of navigation the vestibular system is pretty good, pretty efficient. He’s not really interested in the meaning of mathematics. Just interested enough that he likes to know how a mathematician thinks about space from this view point and that’s how we have all these interesting discussions, but I don’t think he’s deeply interested in the meaning of mathematics. He’s deeply interested in understanding the relationship between vision, the motor system, the vestibular system, the integration of all that, how it works, how the neurophysiology of it works but at the same time as I said he’s a very open minded person: at his lectures and seminars there are dancers, actors, because people who master some kind of real expertise in the way the body moves in space interest him. One of the things that I said when we first met, which he liked very much, is that finally I would like to show that explaining a proof is like a dance in some abstract space and that idea struck him. I don’t know if I would say exactly the same thing today, but at the time I had this notion, it is dynamic, it is constrained in a given space, a dance in a way is a narrative. So he liked that idea very much and talked to me several times about it and it encouraged me to think about it. To explain the space where we dance is to explain the meaning of the mathematics we are doing, to some extent. It was very helpful to me to talk to people like him.

**Galison**: Is there any relationship with the tradition of Piaget and his students

**Teissier**: Well, certainly Berthoz is quite well aware of the work of Piaget and his students, some of Piaget’s students come to the seminar. However, there is a strong criticism of the Piaget view of the learning of space and the learning of mathematics and I think it is clear from what I said, I’m m not at all convinced that the sequence that Piaget described is valid.

**Galison**: His “staging.”

**Teissier**: I tend to think it is much more complicated than that, you don’t learn topology then metric. I think infants have some notion of metric, which is developing, some notion of topology, also developing. They interact for a long time, creating the sensory-motor complex, which has metric and topological aspects. In this topic it is extremely difficult to design experiments which do not incorporate our preconceptions. Piaget’s vision I think is not really a myth of origin, it is really a kind of a scenario and I don’t think it is very believable.

**Galison**: But somehow the goal of taking simple operations and trying to enquire into them experimentally in this way is interesting even if he gets a lot of the sequence wrong, the hierarchy.

**Teissier**: Absolutely, I think in that he was probably a pioneer, making this kind of experiment with children. First of all it was very daring at the time although now people do that all the time with other preoccupations, but it is true that all those who think about this, including Berthoz, including myself, have a great admiration for Piaget.

**Galison**: Einstein once had a famous exchange with Piaget and Einstein encouraged Piaget to look at the question of simultaneity and to see whether if was innate, as people often claimed in criticizing Einstein. Piaget did those experiments that show that actually simultaneity is a complicated notion and that in children, it is all entangled with the speed for instance, and if you have two trains and one goes fast and one goes slow then you turn off the power to both of them at the same time, the children think that the one that’s gone farther stopped later, so it doesn’t seem to be strikingly a priori that way.

**Teissier**: No I think that’s typically a notion which is not an elementary construction of our perceptive system; it is something elaborate. And perhaps that’s why people had so much trouble with the EPR (Einstein-Podolsky-Rosen) paradox and perhaps that’s also why ultimately they don’t have so much trouble, simultaneity doesn’t really talk to our basic intuition of the world, perhaps because it is important only in very very constrained circumstances. It is an abstract notion for us, we have to become a physicist I suppose to have an intuitive idea of what it is.

**Galison**: It has often interested me that, in a kind of folk way we often think of things as innate that are rather complex constructions, simultaneity would be one.

**Teissier**: I think it is a really a major challenge to try to understand what our perceptual system actually says.

**Galison**: Yes

**Teissier**: I’m pretty much convinced about what I said about the real line, that’s very likely.

**Galison**: Yes

**Teissier**: But there are other things like boundaries which are more difficult. Are they really elementary constructions? Are they really low level thinking? I believe that simultaneity is not. To make a list of these low level thoughts is a real challenge.

**Galison**: So Berthoz had, early on, read and been interested in Poincaré and when you encountered Berthoz had you also been thinking about Poincaré or did you come to that through him?

**Teissier**: Of course I knew without any great precision the ideas of Poincaré. At that time it certainly had not occurred to me to try to follow them through because the literary style of Poincaré made it very difficult for me to think of this as a basis for building a solution to my preoccupations. So while of course while, like all mathematicians, I am profound admirer of all the things I understand of his work, I was not an admirer of him as a philosopher. I think it was really Berthoz who convinced me that there was something deep because he made those experiments. I was convinced because he showed the neural connections, he conducted the experiments that connected vision with matter. Poincaré’s is a nice idea for sure but as I said, for some reason I’m very much allergic to the explanation of words by other words. Perhaps that’s due to my original love for the Greek language but anyway wherever I see that in a text I tend to dismiss the text, which is a mistake. But certainly I had that reaction with Poincaré.

**Galison**: Too many words, not enough logic or experiment.

**Teissier**: Yes, not enough experiment, not enough meat, too many words like “ beauty”. But I feel I should have been more humble, and respectful as well. When somebody like Poincaré writes certainly you should take it seriously even if at first it seems it is a bit *facile. *Perhaps what he writes about beauty will turn out to be also a very deep comment at some point, and that would be a typical example of what I think is happening. People like Poincaré and Weyl, they really had this idea that all this has to be somewhere in our perception of the world. That’s really, as you said in the beginning, the tradition to which we belong as opposed to the formalistic tradition. And all these people thought that, and they were right to think that but they didn’t have the tools, they didn’t have neurophysiology, they didn’t have the experiments. Now we are beginning to have them, and I think all their ideas are going to be completely transfigured and probably they will become the great visionaries of the Philosophy of Science in the next 20 years. I really believe that. And I fondly hope that the formalistic view of philosophy of science will be reduced to its natural place, which is not at all empty but much more limited than what it has grown to be.

**Galison**: So in the end you have the view that the meaning-providing vestibular and muscular perception, and in a general sense the low level thinking is quite compatible with the Bourbakian aesthetic more structuralist account they are not the same project but they don’t clash, they don’t contradict one another.

**Teissier**: That’s true. They don’t clash, absolutely not. I lack the vocabulary to explain. Bourbaki’s work is for a fairly large part what I would classify as language creation. He tries to find the best possible way to state, to prove the basic facts. Language creation is a fantastic operation. The other project is rather about meaning, as I say it is about creating myths of origin. Of course the two things are not disjoint. You must ultimately give meaning to the words that you’ve used or invented when you create language. But the two endeavors are clearly distinct. They both answer needs of the human mind, basic pulsions. We need to create language, we need to create these abstract structures but we also need to have meaning. If you look at history, before the non-Euclidian Revolution the meaning of mathematics was not a problem because it was in nature: the principle of Dirichlet was “true” because it is a natural fact. At the scale of scientific thought this was yesterday. Then there was a spell where we had to move away from the roots of mathematics in nature because they had proved to be unreliable; globally we lived without roots for a while, built a reinforced concrete platform on which to base mathematics, and rebuilt them on this by the axiomatic method. The impressive success and richness of this construction caused a certain type of philosophy of mathematics to develop. It was also boosted by the incredibly strong urge we have to find causes and that is, in my language, a low-level urge. But we can’t live without roots for very long. Now the time has come to make roots again and we have the tools. I think this will occupy us for the next few decades.

**Galison**: Tools like neurophysiology.

**Teissier**: Yes. Also I think we’ve learned in this experience with the relationship of mathematics with the real world that one should aproach these things with great humility and not claim to understand the roots immediately. I think it is a very complex process. I fondly hope abstract mathematics will continue to develop pretty much in the way it does and I have no dissatisfaction with that at all. My only dissatisfaction is that I have so many problems understanding the objects that I really need some methods to do it. Probably some people are more gifted and they don’t need to attach meaning to the mathematical objects to be able to be happy with them.

**Galison**: Do you think that’s so?

**Teissier**: Yes, I can think of some people who don’t have that need at all, or at least do not seem to. I was once in a committee promoting one of them, say X, and I asked an extremely respected mathematician, Fields medalist, for a letter. He wrote a beautiful letter about X and his results, and at one point in the letter he says “he’s seen what us simple mortals have failed to see”, or something like that. This X is an extreme case of somebody who I think is truly a mathematical genius and sees things immediately and doesn’t seem to need any meaning. I was there once when some people were explaining why they could not prove some theorem and X was in the audience and he proved the theorem then and there for strictly structural reasons… he said because of what you said this arrow has to be unique and this proves the theorem. He has a sort of structural view of mathematics which is extremely interesting. I think he defended his PhD when he was 16,17.

**Galison**: Is he still active?

**Teissier**: He’s still active. So I believe there are people like that and it is good. They do not need any kind of meaning, but I do not think they suffice to create all mathematics or determine all interesting directions or research.

**Galison**: So when you have, in thinking about this meaning, one direction would be pursuing in the laboratory either with neurophysiology or with experimental psychology with children for instance, to try and understand the ways these are constructed. Another question might be: once you begin to think about this low level thinking that gives meaning, you might want to change Pedagogy in some way. Is that something you’ve thought about?

**Teissier**: Yes, I have thought about that, I think it is really important.

**Galison**: It is presumably still early days in the sense we only have a detailed account of low level thinking, but what kind of Pedagogical consequences would there be?

**Teissier**: Well I think you can think of that on several different levels. One of them is to illustrate, let’s say. It is perhaps the easiest way to go at it now. In the early years of mathematical teaching when you introduce a notion, explain to the children that a line is not just an object you define, a line is an object of an environment where you can see lines you can walk on straight lines, make them, use them just as Einstein, make them imagine they are sitting on a ray of light, walking in a straight line, making smaller and smaller steps and approaching a point and stopping, so that they can feel with their bodies a tiny part of the mathematics. But I think it is important because many of them are lost at that stage. There was a famous textbook definition of the real line for the *Classe de Quatrième*, where* *the kids are about what 13, and the definition that was given was exactly the definition of the line as a principal homogeneous space under the translation group. The definition took five lines and I don’t remember it by heart, but if you read it, it said it is a principal homgeneous space under the translation group. So give that to another teen and you hope that they will do something with it, but of course most of them don’t… they who have help or are extremely gifted will survive but most of them don’t. So I think teaching the teachers that many of these notions the notion of area for example is a very complex notion; it is not part of our elementary intuition of the world; our perceptual system doesn’t evaluate the area of a lake in the precise way it evaluates our distance to its shore. Actually the notion of area is a late invention in the history of mankind, probably sixth century BC or something like that in Europe. In Homer, on a few occasions he spoke of the size of the cities and it is determined by the perimeter. So one could explain to the children that there is a problem there and make them realize it is important because of fields and crops, just very simple things like that. I have talked to a kid who was learning about angles, this was in the mountains, and the kid had no idea that the angles of his compass were the same as the angles in the school book. Just nobody told him, how is he supposed to know? Of course he was probably not extremely gifted, not a genius, but to see that the circle he had to draw in his exercise book, marking 90 degrees, 45 degrees and so on, that was the same thing as the circle of his compass had just not occurred to him. So how do you want him to understand the usefulness of adding angles. So this kind of totally elementary thing which sort of makes us understand that we live in space and that our body rotates, etc., we have actually totally eliminated from teaching. Of course this has to stop at some point you can’t explain the third degree equation in simple terms like that so at some point you must say now we have seen problems, that’s what the Greeks did. They looked at X^{2} equals 2, X^{2 }equals 5, X^{2} equals 7, then some genius said: let’s look at all equations because then they saw the other problems where there are linear terms in X. So the problem is to solve all possible equations. This I think the kids can understand.

**Galison**: Like we heard about from Bombelli from Federica.

**Teissier**: Yes exactly. So of course people will say this will take infinite time but I think it is much better to have kids of 16 who really understand what numbers are and really understand what angles are, what length is, what area is, they won’t become mathematicians but they will be at ease in life, than to have kids who really…

**Galison**: Panic

**Teissier**: Panic exactly, every time they hear the word. What’s the best solution?

**Galison**: One thing I thought I would ask is, once you begin to move from the low level thinking to more abstract and more complex combinations of things, you mentioned in the discussion today that you can go beyond the sort of primate constraints that…

**Teissier**: Yes

**Galison**: Are given to us, on neurophysiology, that’s interesting maybe you could say a little bit about that.

**Teissier**: I think it is partly this process of sedimentation

**Galison**: What do you mean by that?

**Teissier**: I mean that we start with this, say, proto-mathematical concepts as I said I would love to have even 5 examples to give you, but I can only give you one, maybe two if I add boundaries and maybe three if I add ordinals. But not many for the moment. And we start doing things with them. Mathematics in some sense talks about the regularity of the world, regularity with respect to time, homogeneity of space. Our friend Meister said I think at some point, that narration is a way to tame the chaos of the world or something to that effect. It seems to me that mathematics originally is a way to talk about or capture the regularities which are observed: the sky, homogeneity of space and things like that, talk about this. The line is a prime example, so we start to capture the regularity of the world, and then according to our low level pulsions when we start to do Euclidean geometry we tend to go all the way we tend to make it complete because I think that one of the basic pulsions of low level thinking to complete what is not complete. So you push what you are doing until you feel there is a kind of completeness to it. So lets say in some sense Euclidean geometry represents a more or less complete state of affairs starting with, say, configurations of lines in the plane. A that point, Euclidean geometry becomes part of our equipment, we think of it as a whole, more or less, as a view of the world which we didn’t have a as primate, as a primate we have of course much more elementary views, but once science has acquired that, its there we can use it to build something else. So we can do in different directions. I’m not saying we say ah! let’s go to non Euclidean geometry …but for example we can’t help to try to use that to work on curved surfaces, which helped to invent the Riemannian geometry, we are led naturally to Riemannian geometry. And for example I think its very striking they try to explain parallel transport is kind of cabled in our brains in some way which is not fully understood, but it is not an exaggeration to say that parallel transport is cabled. Now when Elie Cartan invents parallel transport I’m tempted to argue that he invents it *because* it is cabled in the brain, posibly for evolutionary reasons because its very important to be able to detect angles and parallelism is an extremal situation in the evaluation of angles.

Then, since this parallelism is cabled and becomes crucial in euclidean plane geometry, when we have a space that is no longer the plane of vision, we try to extend that concept to that new space. It may fail or not, in the case of parallel transport it works and if you look at the way Elie Cartan actually describes parallel transport, which is richer than what you find in differential geometry textbooks, it really sounds like: I would like to think of myself following a path and being able to carry this direction with me and all the consequences that it has. So then in turn, that becomes something on which you can apply low level thinking once you have realized that your, lets say primate in fact this case, its probably not primate, its probably man … Primate feeling of or concept or whatever of parallel transport, can be extended then you start thinking you can extend also other concepts of Euclidean Geometry to differentiable manifolds. For example you said OK let’s see what happens if I make a loop, then you discover ahh! something goes wrong. So then you invent curvature, or something of that nature. So one way to tell the myth of origin of curvature is this, you take parallel transport, make a loop because that’s an operation a primate can understand perfectly well and lo and behold …

**Galison**: There’s a defect

**Teissier**: There’s a defect and then because of your low level thinking, when you find an obstruction, analyze it and give it a name. First, of course, you try to measure it, you see if it is intrinsic, you see if it depends on the loop… All these are in my mind are low level methods of reasoning, first is it independent of the loop? Yes? then let’s give it a name because it has something intrinsic, it is an invariant of something, what is it an invariant of? then you discover ahaaa! it is the obstruction to mapping isometrically the surface to a plane, and that’s beautiful because it is not at all in the construction you made. You started with parallel transport, you looked at what happens like a child makes some experiments to see what happens, something happened and you named it, then you tried to understand it, and found that it has varied interpretations. So that’s when I said: this is a beautiful theory. I feel beauty because I have an element of surprise, because I feel, after I understood the theorem I know that I’ve used nothing but the Riemannian structure to begin with. Nothing abstract, just that. And I find that I have measured an obstruction to an operation which I find important of course. Then from there you can go on to connections because you try to elaborate a world in which you can play this game but with things more complicated than tangent vectors or whatever. So that’s the way I like to think about these things. If I had to make a course of differential geometry I would try to say not begin with definition, rigid answer, but say OK lets look at that and then in the end, you can find you can reinterpret everything with some meaning. This is something which you don’t find usually in the books of differential Geometry except maybe for Spivak’s or Hermann’s books

**Galison**: Yes

**Teissier**: For some reason they are not the books which the majority of our math. students look at. But you can sometimes find this approach in physics books.

**Galison**: Yes, the Wheeler book which I talked about in my paper, that’s very much what he does, it is very physical

**Teissier**: Actually when I was in charge of the Math. library at the Ecole Normale I bought a lot of physics books for the reason that things like connections, Chern classes, are explained much more clearly in physics books, according to my feelings, than in the math books.

**Galison**: I mean in Wheeler you go from parallel transport you define these things then you get to the Ricci curvature tensor, you go in that direction.

**Teissier**: Yes but for what reason don’t you ever find that in a classical differential geometry book?

**Galison**: Because you want to go from the general to the specific, I don’t know, I mean it is a different strategy of narration.

**Teissier**: Yes.

**Galison**: Do you have students who are interested in some of these things?

**Teissier**: Actually yes, I think I’ve managed to contaminate some. Fortunately they don’t view things exactly as I do, but I have one student in particular who likes to teach in that way. He does it almost naturally, I think he is a fantastic teacher. He gave a course this year on topology according to Poincaré in which he really explained topology as Poincaré would. His students are very fortunate. He doesn’t have this philosophical aspect of things but as I say spontaneously he’s someone who needs to understand mathematics down to the roots. I think that’s part of the reasons why he became my student. But of course his thesis is written in the classical style and all his papers are.

**Galison**: So you said something this morning which I thought was interesting, which was people tend to conflate the foundations of truth and the foundations of meaning. Tell me little what you mean by that.

**Teissier**: Well it is more or less what we’ve been discussing …we claim to have foundations of truth when we have a coherent system of axioms inside which we can build the objects of which we are talking. Because in principle then we can prove whenever we know the proof of a theorem, we can translate it into a logical system in which it will be valid. For example the real line has several axiomatic definitions, you can define it with Dedekind cuts, you can define it as the completion of the reals, you can define it with the theory of real-closed fields or something of that nature. That’s what I call the foundations of truth. Again you can see that as a story of origin, as the existence of a story of origin to be precise, because once we have stated an axiomatic system we still have to tell the story as a sequence of deductions, which no one will ever be able to do, but in principle it is feasible. And the foundation of meaning is what I described, to say why are we able to think mathematics on the real line? It is because we transport on that object, pairs of intuitions of our perceptual system, vestibular and visual, and if we do that we organize the real line in some way and the mathematical expression of this organization is what we know, an ordered field, but the reason why we really can think about it in a meaningful way, is what I said. The claim I make, which one can certainly discuss, is that both are necessary if we just follow meaning we go into terrible trouble when we are high up in the tree but if we just keep the logic, the truth foundations, we lose meaning at some point.

**Galison**: We stay high up in tree and we don’t touch the ground.

**Teissier**: Not only that , but we become unable to progress, because to progress you must have desire, to have desire you must have something meaningful. Again some people, perhaps the mathematician X whom I mentioned earlier, get meaning from very high up in the tree, from the structure of the environment, somehow. That is what is meaning for him, knowing the place of the object in some huge category or n-category. and that’s enough for him to work with. But for teaching, for perhaps less gifted people and I think also for ontological reasons. I don’t think you can stay at that level.

**Galison**: Ontological meaning…?

**Teissier**: Yes, I think it is necessary, it is in the nature of mathematics that somehow we must preserve the roots of meaning. That’s a philosophical position, I’m convinced that it is true, of course I cannot offer no proof but I’m convinced if we do not pay attention to the foundations of meaning we would run into trouble too.

**Galison**: What kind of trouble?

**Teissier**: Lose interest, waste time on meaningless problems, this happens. Make uninteresting mathematics, which will degenerate into something purely academic. I think there is not a real danger of this happening nowadays for most of mathematics. But in the distant future I think it might happen. Also we lose, incidentally, this is not a philosophical statement, we lose students because they are not interested if we give them those abstract definitions and tell them: now go home and come back with a theorem!

**Galison**: Do you think that even knowing that there are some people who can come up with very interesting mathematics without passing through the connections that are given by low level thinking and some kind of physical or visual or motor understanding. Do you think that understanding in some way requires both the abstract and the meaningful? It is a different question… I’m not asking the question whether it is possible for certain individuals to continue to prove and discover interesting things. I take it as a matter of fact that there are such people; let us just accept that. But do you think that in a broader sense understanding in some important way requires this?

**Teissier**: Yes that’s a very interesting question, yes I do think so.

**Galison**: I do too

**Teissier**: It is a very delicate question I think it has to do with the nature of language again. We need the structure of language to understand what we say, just as we need the meaning of words. This morning Meister was talking about semantics and I almost objected. I think that semantics is a loaded word, let’s use meaning. I think it is more vague, semantics is too Tarskian or whatever.. But yes I think that’s one of the important issues and that why I think we must strive to construct meaning. The syntactic, or to speak like Meisters, the structural aspect is important. If I didn’t think that probably I wouldn’t be a mathematician I would be doing something else. But it tends to hide meaning after a while and so I think both are necessary and their interaction is necessary, their interaction is something that we can never really understand. It is transcendental, we can’t make an analysis of it even if we believe that the quantum harmonic oscillator is at work in the neural connections as Penrose claims I don’t think it will give us an explanation of the relationship between, the way our mind structures things and the way it builds meaning. But what I like to think is that to some extent we can approximate that, using those low level pulsions and thinking of them as a kind of cellular automaton: a sequence in which you apply a low level pulsions to some basic intuition then it gives a result then you apply another low level pulsion, is it invariant or something like that, and then perhaps we can produce a new understanding of the way meaning propagates… it will never be perfect but I would be happy to have some kind of vague image of the way it propagates, some good examples. And then perhaps once we have that we can guess a little at the way structure emerges, like structure emerges in cellular automata. You could say this is my program for the next 200 years. That’s the way the thing which I’m really interested, to see if it really makes sense to try to do that. And I admit that I start from a very tiny basis, the vision of the real line, and there is an enormous amount of work to do. But fortunately I am part of a large movement: there are now many people working not only on the neurophysiology, but also trying to understand what happens in our brain. It is very diverse. I’m in a group, actually coordinator for a group which contains two mathematicians Bennequin and I and a number of neurophysiologists. It is called neurogeometry and are trying to see how we can use mathematical ideas to understand what goes on in the functional structure of the visual areas, there are many fascinating comments with each little experiment costs so much time and energy so it is moving slowly. It is not like mathematics: in a way the obstacles are there all the time. Sometimes mathematicians have good ideas and that’s it, but there, good idea or not you have to do the lab work and there is no going around it.

**Galison**: This is a speculative question, but one of the discussions that take place at the boundary between physics and mathematics is whether the intuitions afforded by what for mathematicians are poorly defined objects or not well defined objects like path integration and so on, are of use in formulating mathematical theorems in algebraic geometry for instance, or Morse theory or knot theory or mirror symmetry. When you think about what’s going on when that happens you have to ask what is it that permits a set of intuitions from collisions among these one dimensional physical objects, using methods that were developed to think about point particles, electrons, scattering, to suggest results in algebraic geometry? It is so unexpected that that should have this kind of heuristic role or suggestive intuitive role. What do you think is going on when that happens? And then my follow up question will be is that anything like what is happening at a more fundamental level, with low level thinking? But let’s start with the geometrical.

**Teissier**: I tried to think about that. For example, for a mathematician or a physicist, again part of our sedimentation will contain operations like “evaluate a function” maybe even “integrate a function”. For example Feynman diagrams is an idea which I immediately loved, I think essentially because of that, I viewed it the first time I saw it as a sort of low level construction. It is so natural in some sense: you look at all possibilities and then you realize they have some structure and order and then you use that. And you integrate over that and you make a kind of average and then you use a kind of extremal principle and out comes physics. So yes I think this is definitely an operation …I don’t want to give the impression that low level thinking will eventually encompass all interesting ideas that’s not at all my purpose. But when I see somebody facing a problem which concerns path from A to B, say: let me look at all paths except this is not a thought of a primate, but this is not very far above the thought of a primate. It is not very far above an elementary perception of the world. And now it turns out, for reasons which are coming from the environment that this has a structure so that’s the part where the physics kicks in somehow. I think that even in the most abstract realms of thought when you are talking about homogeneous spaces, compact Lie groups, things like that, this kind of reflex: how many ways to go from A to B? Oh, there are many ways to go from A to B but clearly there is one that looks nicer, this kind of thing is part of our being, not as mathematicians, but as primates

**Galison**: A physicist says: clever primate!

**Teissier**: It is a clever primate, its an educated primate, the primate of a physicist or of a mathematician is not the same as the primate of a literary person. It has not had the same education so to speak. No, when we learn to be a physicist or a mathematician, we learn to talk to our primate according to a certain code, we explain to it e.g. to see a geodesic when you throw a stone and the primate says OK, and then you teach him to carry this image in more abstarct spaces, but the stone’s trajectory is still there. But one advantage of the primate which we must not underestimate, he has these pulsions, maybe not all the pulsions of a human being but many of them, he has this perception of the world. But he has an enormous advantage over us… he’s relentless, he never stops, he’s infinitely patient he will try and try again. We get tired and say this is boring, but the primate never gets bored and I think that’s how eventually we get problems solved in our dreams, we somehow explain, that’s compatible with what Poincaré says, we explain the problem to our primate and the primate is good natured, and he works and he tried and he tries and we go to sleep, we go to the movies and at some point somehow the primate, says ahhh! Then he rings a bell to tell our conscious self that he has broken through, and we perceive it as a new idea. This can take the form of a ghost, I knew a mathematician when I was young, who eventually took his own life, he was persuaded that his mathematics was given to him by a goddess, a ghost. He was an excellent mathematician. His mathematics was brought to him by a goddess, and he is not a unique case. One should think about that, there must be a reason why some people have ghosts or goddesses who bring them mathematical statements and we have dreams, dreams of mathematics. Sometimes I have dreams to tell me: what you wrote today is wrong, but sometimes to tell me: this looks good, and sometimes it works. It is not accurate but as I said it is sometimes close to the good idea, and with a little effort you work it through. Poincaré describes it, many mathematicians share that experience, but nobody seems to be interested in finding out why something like that happens, probably because there was no place to begin. But now we have a tiny window into the unconscious. The neurophysiological window is probably a tiny window but we must look through it to see what we can see, so I think that’s exciting.

**Galison**: We’ve discussed about the relationship of the true to the meaningful and the foundation of the true and the foundation of meaningful. In the non mathematical sciences true is often used in another sense in an empirical sense rather than a logical sense or coherence. But do you think that the empirically true falls on the side of the true in the sense that we’ve been discussing it or more on the side of the meaningful? Or is it a *tertium quid*?

**Teissier**: That’s a subtle question. I think its probably of the third kind, your tertium quid. Empirically true is meaningful only in so far as the experiment is meaningful. I know there are people in the philosophy of physics, who say that every experiment is meaningful by definition otherwise we wouldn’t be making it. But that doesn’t teach us much.

**Galison**: No

**Teissier**: On the other hand, the empirically true is always relative to certain framework of experiment so it is not true in the sense that we like to think of truth as eternal. It is a kind of local truth.

**Galison**: Contextual.

**Teissier**: Yes, so I think it really falls in that third category: contextually true and not necessarily meaningful. But even in mathematics we can make examples of a special class of variety or whatever, look for something, find it and be tempted to believe that it is true in a much wider class, local experimental truth. But to say that the statement is true in a wider class is obviously a conjecture. But it is an interesting question because precisely you could say that if it is empirically true in the given context of the experiment and so on, can you conjecture that it is absolutely true? Is that something one would like to call a conjecture, I know in mathematics it would be a conjecture or question.

**Galison**: We’ve discussed a little bit this morning the possible evolutionary role in picking out some of these primate… you might say I’m putting words in your mouth, a *prima facie* argument for the evolutionary nature of some of these elements of low level thinking is that we share some with primates, and the first order explanation order of that is that we come from the same place. How important is the evolutionary aspect of this account of low level thinking to you? Is that a kind of optional next step in the argument or is it a necessary step to the argument?

**Teissier**: It is partly a necessary step because if I want low level thinking to play its role in the construction of meaning, then I think we have to share it partly with primates. Again my definition of primate is not the next chimpanzee, it is our ancestral experience of the world which we for a large part share with primates and certainly is unconscious and does not depend on language. So it is important that we share it with primates, because observation shows that they have behaviors that are rather close to us and we have to explain that. My basic view is that to a large extent our experience of the world as primates is the main reservoir of meaning for us.

Of course if you told me there is no continuity between low level thinking of man and low level thinking of chimpanzee it would not destroy my need to understand, but the idea of continuity is so much nicer! You may say it answers a low-level need in me. On the other hand that’s a question which I have asked my self repeatedly: low level thinking is probably not a collection of pulsions as I describe it, the need to generalize, the need to categorize, the need to analyze, etc.

I think somehow what we inherit from the primates, in addition to way to perceive the world, is the possibility to create low level thinking rather than the set of low level thoughts. So it is evolutionary in that sense. Perhaps if one could make experiments with babies, which one can certainly not do, one could create babies who would lack some of these pulsions. Perhaps by accident there are such people who lack or have a defective pulsion to organize or to analyze or to synthesize. What is really important for me is that I need a driving force for the creation of meaning to go through mathematics. And these are the driving force: desires. There are driving forces for other activities, sexual desire, desire to command, to be the first, which chimpanzee for example have very strongly. You probably know that chimpanzees are socially very brutal, Bonobos are much nicer, resolve conflicts by sexual activity while chimpanzee resort to violence. So we leave that aside because it doesn’t seem to be very closely related to mathematics but it exists. In so far as mathematics is concerned I think that this low level thinking is part of our equipment to understand the world. We don’t inherit our understanding of the world. We inherit the pulsions that ultimately allow us to do that, we inherit the pulsion to form words as babies but we don’t inherit language. We form words and then we listen to the answer and by a process of learning we find the way to speak. So I think it is very similar, we have these pulsions we try them on the world we try to organize this and that, perhaps that’s a kind of experiment I would love to see. If we can observe infants, trying to organize the world according to certain rules because there are many things which infants do which we consider as totally meaningless but perhaps if we look at it in a different view, as trying to check that’s something’s is invariant by translation, that objects still exist when we turn around, perhaps we could observe things which gives meaning to some behavior of the infants that have to meaning a priori… I think we inherit that, and as we inherit it it has to have some structure, which I’m totally unable to imagine, and these pulsions are just there, there permanently one another.

**Galison**: I mean one question would be, with any inherited trait there are multiple possibilities about how it comes about …you could have something that has a selective advantage but you could also have something like what Stephen Jay Gould calls a spandrill, a side effect. For example you would have a trait A which is advantageous for natural selection. It allows me to type, but its not because I survive better if I can type. And another is drift, which concerns certain things that are not selective but turns out to be advantageous. So different mechanisms might be involved, all of which still could be inherited.

**Teissier**: Yes, I agree. I think I said something about that, about the line, we somehow construct the capacity to perceive curves but probably we don’t inherit the capacity to perceive lines. Line is a side effect it is a kind of minimal curve so as we perceive curves, there is a space of curves, there is an energy of perception of each curve and there is a minimal one and that’s the line and that’s it. It is not because we inherit from our parents the capacity to see straight lines or because detecting straight lines has a direct evolutionary advantage. I don’t believe that for a minute. Yes you’re right but how can we begin to understand how this kind of thing works? Because we’re talking about unconscious pulsions, we must invent experiments, that’s one of the things I like to talk about with Berthoz, because he has a great talent for thinking clever experiments. We must invent experiments and first try to identify how these things work and then see if we can find them in the chimpanzees…There is a lot of work to be done, but that’s faster than neurophysiology

**Galison**: The experiments on the individual neuron, that for instance my colleague Hubel did, those beautiful experiments where only a convex object or only a small object moving across the field of vision against the background would fire it, but not the background itself… they’re stunning experiments.

**Teissier**: Yes when I spoke about the neurophysiologists’ work on parallel transport, it is a very special case of thousands of very clever experiments on the operations that neurons and their connections are able to do. What they do is absolutely beautiful as an experimental corpus. and probably a lot of it is meaningful for geometry. I mean what I said, concerns maybe one thousandth of what goes on when we view a simple scene: we detect contours, we detect texture, we detect moving things. The neurophysiologists say that 90% of what we perceive does not reach the conscious level.

Perhaps finally what mathematics and narrative have in common is that both are ways to create, using language, coherent pictures of the world, which in some sense <explain> how it works.

We must remember that the first <explanations> of the world in our culture, by mythology, were a sort of modelization of the physical world by characters, the Gods, who were rather human in their pulsions. And we must remember also that just like mathematical objects, and just like the characters in a novel, the Gods acquired a life of their own, well beyond their use as <explanations>. The creation of mathematics and the creation of narrative are both, I think, powered by pulsions that are very close.

**Galison**: This has been terrific, thank you for now and we will continue our discussions another time and place, in Paris or Harvard.

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]]>**Herman**: Could you talk a bit about the “photos-and-movies language” that you use in your paper for the symposium? How does this language help us think about the family resemblances between stories and proofs?

**Doxiadis**: I do a lot of lectures, mostly to students to make them understand ideas about narratives or about mathematics or about the mixture of both. In one lecture, on Andrew Wiles’ proof of Fermat’s Last Theorem, I wanted to have a language that could be used to describe the proof as containing two types of objects. I said it really contains photos and movies which are ways of connecting them. It was the first time I used this. And because I was talking about mathematics, I wanted to have a very soft language that could relate not dots and arrows and vertices and edges, but rather photos and movies. You freeze and you move. I used the example of freezing and fast-forwarding your DVD player.

**Herman**: Just to put that in brackets for a moment…Is there any sense of (this) clashing with the graph languages, which is an earlier incarnation of the project (which you delivered at the Mykonos symposium)? Do you feel like you’ve reconciled these languages?

**Doxiadis**: No. I didn’t think it was clashing at all because one of the problems of the previous language was that both the dots and the arrows involve action. That was a bit confusing because you had an elementary atomism with two kinds of objects and both were made of the same substance: action. That was a bit confusing.

**Herman**: So, the photo is?

**Doxiadis**: Action.

**Herman**: And the movie is?

**Doxiadis**: The movie as a rule is invisible. In comics, the movies as a rule are in the gutter. I want to comment on this in another way. Let me leave it aside for a moment. In prose, the photos would be the action sentences. The movies could be, again, invisible…Do you know this poem by Jacques Prevere? I don’t remember exactly how it goes, but it’s something that all students of French learn because it’s so very elementary in language. It’s a great poem. It goes something like:

“He came to the table.

He opened the cupboard.

He took out a cup.

He put in the coffee.

He put two pieces of sugar in the coffee.

He picked up the spoon.

He mixed the sugar.

He added some milk.

He drank a sip of coffee.

He put the cup down.

He got up.

He put on his raincoat.

He put on his hat.

He opened the door.

And he left.

And I sat down and cried.”

Almost verbatim. In slightly more elevated French. I may be missing some words. It’s interesting because there is no moving, except in the last sentence. Although there is a mere conjunction there its function is like that of the key element in Forster’s definition of plot vs. story: “The king died and the queen died…of grief.” It is the equivalent of the “of grief”. And that is a case where you just have your photos and…in fact, it’s very interesting because all the first photos are connected with temporal links. There’s no direct causality there until suddenly the narrator, the woman (whom we understand from the last line to be the narrator), signals that this is about the end of a love affair. It is this which is causal. This is the power of the poem. In this series of temporal links suddenly a causal one has entered. So, for many years, I was trying to find a parallel for all this – and you know, this is relevant to the photos-and-movies language. If you have two elementary entities, why would they be made of the same substance? Why not call them one entity? I was trying to find a functional definition of narrative that was good enough for my purpose of developing a model for comparing proofs and stories; not for all eternity; not to be the cornerstone of narratology.

I actually went about these lectures in two ways. First, I went the atomic way. Let’s talk about the elementary. Simple narratives. Simple narratives are descriptions of simple actions. Action sentences. I didn’t use that expression but it was about that. Or molecular actions. A panel of dialogue, a sort of mini-action. Then, I said, from that you build longer narratives. And that seriality is always there. You meet more complex narratives and I said that it’s a series of descriptions of actions, which are ordered in some meaningful way, which may have, if they are of a more advanced form, a beginning, a middle and an end, and a definition–well, I don’t want to use the word ‘focused’. Let’s not get into that. I have a very precise definition of that.

I just want to take one line from that definition which surprised me when I read it but I didn’t like it as a definition. I wanted it to be cleaner than that but I couldn’t get past the sentence which said “..ordered serially in time and often also causally….” “Serially in time” was not enough; clearly for a narrative, there is something more than that. This is why “causally” is needed. If we take the Prevere poem, however, sometimes things are not causally linked. They are just temporally sequenced. Sometimes you see that when you describe passages from scene to scene, you will advise a novice director or writer to cut the in-between. But sometimes, some of the real world life of action is necessary to ground your action in the real world. You know, there are some forms of narrative where you have an excessive amount of causal connection. And when you do that, it is artificial. The narrative form that relies excessively on causality is, for example, the Victorian melodrama or the Hollywood action blockbuster. *24 Hours*. Have you ever seen that series? I was bored by it because it was all so causally determined. The tiniest little thing was of momentous importance. Only events of tremendous importance occurred and everything was significant. It was a paranoid vision applied to narrative all the way down. They took Chekhov’s maxim that if you see a shotgun in the first half of a play, it has to be used by the third to its total extreme.

To me, that is the sort of an example that is intensely causal. Intensely temporal are narratives like the ones little children tell. ‘Tell us how you passed your Sunday.’ ‘Well, I got dressed, I ate breakfast…’ And any English teacher would say, ‘we don’t need all of these things because they are just temporal’. So, in my definition, I had it as “temporally ordered with frequent but not constant causal connections.” So, because of those two words I could not throw out, to my mathematical instincts this definition was not very elegant. How could I get around this ‘sometimes but not always causal connections’ element. That’s silly. But I couldn’t get away from that.

So, it helped to think of the functions of the gutter in comics, and how it mediates between what I call photos versus movies. So, we just have action. Photos are just action. And when I looked at that slide I showed to kids with Wiles’ proof, I saw that anything mathematically significant was in photos. If you look at that slide, you can understand all the movies because none of them are mathematically significant. Movies are in Greek. We first observe that. You understand that. Now, from the construction, we made__________, you don’t understand what the result it but you don’t need to. There is a result which is by… It follows formally. Forget the “formally.” It means, follows mathematically, logically. All of these are very simple things. And the reason you can understand them all is that they do not all involve mathematics. They are just connectors. So, when you start to think about what types of things these things are and whether they are temporal or not, you start to see that one means for advancing a story is “and then” and another is “and so”. You begin to see that there are two kinds of movies and two kinds of passages.

And then I realized that these relationships could be captured in the non-linear structure of the graph. If a movie is “and then”, the link has one arrow going in and then one going out. In terms of the story as narrated, the sjuzhet, it just has one photo leading to this photo. If you speak of causality there’s always the example from Forster, namely, “The king died and then the queen died of grief.”

Let me give another example here. It involves a non-narrated connection: “Will you go to Paul’s party?” “I have to work.” What is that?

**Herman**: That would involve Grice’s Maxim of Relation, I think. The idea is that your interlocutor would assume that there is a relevant connection–would assume that there is a reason for mentioning the need to work in this context.

**Doxiadis**: What do you call it?

**Herman**: Grice’s term is conversational implicature.

**Doxiadis**: So, if we look at the Forster example, the interesting thing about “The king died and then the queen died of grief” is that people assume it to have more causality that real life. That is, if you’re telling them something (like in a murder mystery) you don’t just combine clues, you combine author strategies. Yes, but would an author like Agathe Christie tell us these obvious clues, or is she trying to mislead us–to make us think that if she’s showing us all those clues that it can’t be him? Anyway, with the example of “Then the king died and then the queen died,” most readers already think that the two are already related; but let’s assume the (author) wants us to read that as a neutral case of temporal order.

Now, if you say “The king died and then the queen died of grief,” there is again a sense that the king died and then the queen died. Then if you focus on that link and want to see the full non-linear structure, you will see another movie coming there, you will see, the queen loved the king very much. That is the only thing that had made her die. Otherwise, it’s a non-sequitur kind of thing. It’s an absurd kind of story. In the context of the story, you have to know that she loves him. In that sense, I realized it was useful to talk about photos and movies—because it gave you a kind of linear structure for time, which was what I call I-junctions, and a V- or Y-junction, where you have at least two things coming in.

In turn, that same structure applies quite naturally to proofs. In proofs, you have times where you are just doing one thing: I-junctions. For example, take the statements: ABC is a right triangle. ABC is a triangle. Moving from one statement to the other is an I-junction. That is what Kant would call an analytic statement, if ever there was one. The most usual analytic statement is a statement which contains all of the information. The most usual example of an analytic statement in a philosophy book is ‘A red rose is red’. Now, in my example, you have the implication from ABC is a right triangle to ABC is a triangle, which may be eminently useful you see. And it’s not trivial. It’s perhaps trivial to construct an analytic sentence but in the progress of a proof, remembering that ABC is a right triangle does not stop it from being a triangle. It may be eminently useful to talk about concepts of triangleness, e.g., that the sums of the angles of a triangle add up to 180 degrees. But, going the other way, is not a simple relation. You cannot say that because ABC is a triangle, ABC is a right triangle. It’s not necessarily true.

And then, because I was trying to find similarities of proofs to stories at the atomic level, when I use the photos-and-movies language, I got to that and I said ‘that’s it.’ We have something that’s strong. We’ve taken our optical instruments and we’ve moved to using telescopes—from using our eyes, to using a lens like Sherlock Holmes, to using a primitive sort of microscope, to using a huge electron microscope…and then we got down to the elements. And there, we saw something. One of my beloved sentences, I think it’s used in *Logicomix*, I don’t remember, is when people found fault with Frege’s model of a logical language, he replied: “I have invented a microscope. You cannot hold it against a microscope that it is not a good instrument with which to look at the stars.” I have the sense that at the microscopic level, I’ve gone down all the way to connections, and there are two kinds of connections in stories and proof: one is linear and the other is non-linear.

**Herman**: And so, as you’re working out these connections, how does that effort bear on (or derive from) your practice as a creator of narratives? Are you drawing on narratological theory or rather on ideas that are part of your repertoire as an artist? Was it your practical, hands-on experience with creating narratives that led you to make these refinements?

**Doxiadis**: Yes, but, I must open up a parenthesis here and then perhaps we can talk a bit about narratological theory. One thing I have not spoken at all about is my theoretical background. I have spoken about mathematics. I have spoken about stories and my work as a creative writer. But, I haven’t spoken about theory of any kind. I have to say that already at the level of my Masters degree, I moved from pure mathematics to mathematics applied to cognition to the study of the nervous system.

**Herman**: Is that right?

**Doxiadis**: My Masters was a micro-level study of the nervous system. I became immensely interested in cognitive theory, what today you would call artificial intelligence, cognitive psychology, neurophysiology. And that stayed with me no matter what I was doing. So, I first read Propp when I was 20 or so and I was so impressed by that I said, “Oh my God, this guy is really onto something.” I wasn’t thinking very much about stories then but I knew about fairly tales (and had loved them since I was a kid) and this was about the time I was doing my Masters and trying to look at models that went beyond the nervous system. I was very interested in structure, and I was shopping around for ideas in all sorts of things to help me combine a mathematical tendency for models and structures with cognitive models. And I saw that in Propp, so that when I started to operate as a writer, I started to think about it again. I went back to Propp and via Propp I first hit upon the word narratology and I started to read very selectively and in a very utilitarian way things that were relative to what I was doing. I did not read narratology as a writer and the things narratology told me as a writer, I found either evidently true or needlessly complicated. There was the odd occasion where I said ‘I know that. I have been doing that for many years but no one has put a name to it’.

But problem with Propp is that the moment you construct a graph of more than three nodes, you are already in the billions of possibilities. What can you do with that? And then, when I got into even more complex models, I would always think, ‘That’s too complex.’

**Herman**: One of my personal interests is in the distribution and dissemination of narratological research in different countries. Is there a tradition in Greece of modern-day narratological research that’s built on the original foundations? And, did you draw on anybody locally?

**Doxiadis**: No. Look, we are on the Continent. There is a lot of knowledge and interest about the French theorists; Genette is very well known. Barthes, Derrida, people like that are very well known and read. Of the more analytical, sober nuts-and-bolts kind of narratology, to which you belong, I would say that there is practically zero.

What I myself most got from narratology was a climate and I think that is huge. The climate of a feeling of narrative can be analyzed rationally, using tools which are tools of reason and logic, Aristotelian methodology. In fact, I see narratologists as children of the Poetics. When I do research, when I write novels, I constantly read around them. When I write about the Greek Junta, I read endlessly about Modern Greek history. When I work on *Logicomix*, I read about the Vienna Circle, I read about Belle Epoque Paris, and so on.. As I was doing this work, I kept my mind alive by reading narratology.

**Herman**: Let me ask you some more specific questions about the proof/quest/narrative connection. I’m still having a little trouble understanding where you’re locating the parallelism or the analogy or the commonality between proof and quest. And I guess, therefore, the commonality betwteen proof and quest narrative. In your account, the parallelism sometimes seems as though it pertains to the experience of doing the proof, i.e., the experience of following the proof – especially for the knowledgeable mathematician who has familiarity with the context and can almost reawaken the original effort that went into the formation of the proof. Even in the formulation of the your key propositions early in your paper, you’ve located the quest structure in the quest for the proof. “The quest for a mathematical proof is precisely a quest” (5). This makes the quest the search for the proof, the proof as discovered rather than the proof as published. But at other points in your argument, you want to attribute to the proof as published a quest-like structure in its own right.

**Doxiadis**: A mathematical proof is a quest—and not just a quest for a mathematical proof. I could give you two propositions, and both are true and both are related. The first is that the quest for a mathematical proof is a quest. And the second is that a proof is a quest.

**Herman**: A quest for the mathematician?

**Doxiadis**: A quest for QED. A quest in graph language. One thing that my analysis does as it advances is to begin from a more natural language which is intuitive and has to do with stories of proof as they happen and the historical reality of mathematics, then proceed to the graph language, and from there, once the focus on movies of graph language is established, to start to using increasingly only that more technical language to explain things. A good example is in the fifth section of my paper, which discusses the birth of proof and gives verbal examples. I am only seeing those in the graph language of photos and movies. I use these statements to create a sense of a huge realm of possibility. Once I do that, a proof itself is a quest in “P-space” and it is a quest in P-space, because with this diagram I have defined the requirements to have a proof. To have a graph-setting and a graph-setting in the proof is P-space. It is to have an A, a searcher and in the proof the searcher is the person reading the proof, the person moving the graph space. It is an entity moving graph-space. In fact, it is the entity which has to move temporally: A to B to C. An initial position is what we know, what a mathematical proof would begin with, namely, axioms and previously known propositions. The destination is the QED, the theory to be proven.

So, although I use “quest” in the everyday sense, I use it in order to build a graph language and once I build it, I restrict all my discourse–I can restrict it when I want, to designate just a proof and not the human quest. It is a quest in the sense that you have a place on the graph to be reached, a place on the graph to begin from and a way to get there. You might object that an enhanced tree is not a quest. But insofar as I’m focusing the idea of the published proof, I might be correct–in the sense that it’s an account of a journey.

**Herman**: Which is actually the passage I was going to point to. You talk about the ant and you put us in the position of the ant, as would be the case with an internally focalized narrative. We are looking at the steps of the proof as seen through the eyes of the ant. But then you say, because you are a very obsessive ant, you write down the record of your journey which then would suggest that it is a quest after the fact—that it can only be conceptualized as a quest after you’ve taken the journey.

**Doxiadis**: Keep endless notes that you report this to your superiors, to the deities. They will keep only the reports that are interesting. In a sense it’s like that story about the monkeys typing away and eventually typing all of Shakespeare’s sonnets. These are searchers who do their searches in often very random ways. A proof is whenever they start from an interesting proposition. If they get from there to a highly graded proposition that was previously unknown, they write that down as a significant journey. So, you generalize the language of quests to journeys, and you define a journey as a trivial quest in which there are not too many problems.

**Herman**: What I am actually trying to do is search for disanalogies between these categories of proof and stories.

**Doxiadis**: There are huge disanalogies. You know I am not a lunatic! The most basic disanalogy, if we are not talking about the formal language but the language of proofs and stories as they occur in life, is that proofs and stories are imitations of actions, human or anthropomorphic, whereas proofs are progressions in the world of mathematical composition. It’s a huge difference.

**Herman**: So, you are still comfortable using ‘quest-story’ given that disanalogy.

**Doxiadis**: Quest-story is a metaphor, as is the photos-and-movies language. When I say photos, I am not carrying a camera. When I say movies, I am not carrying a camera recorder. Quest is really a metaphor that helps with understanding. It is understandable by all and it’s applicable at many levels. What it boils down to in this analogy is a graph language. This is the profit of using models where you move from quotidian terms (such as “love,” “quest,” “human being,” “freedom,” etc.) whose meaning you could argue about indefinitely, to a more sparse symbolic language where meaning can be agreed on a la Euclid, because it’s obvious, or because, a la Hilbert and Formalists, that’s all there is. This world only exists in the rules I give for it. When we are talking about human beings, we are talking about actual quests. When we are talking about P-space, we are talking about graphs. I am happy to talk of quests at every level because most of the things we talk about are quests.

**Herman**: This is would allow you overcome a point I raised in my written objections, concerning particularity and generality—i.e., the way stories are ground in the specific.

**Doxiadis**: Mathematics is about generality; but proofs are not about generality. Let me give you an example. A general mathematical statement is the Pythagorean theorem. When you get into the nitty-gritty of a proof, every step is at the level which is much more analytical and much more detailed than such general statements. Take for example Euclid’s proof of the infinity of primes. The thing to be arrived at, that primes are infinite, is a very general statement. But the way Euclid constructs his proof has him at one point assuming that primes are n, a precise number, multiplying all these numbers and getting a very precise number, and then adding 1 to it and attempting to divide that number by each of the n primes and seeing that there is a residue of 1. Conversely, narratives can involve generalities. A general statement at the human scale is that people must obey a divine decree. In a story, particularly ancient ones, you often see statements like that. Especially in tragedy, you have this sense of the narrative moving through a world of divine decrees. You can have both levels of realities—divine and human–and the story itself tracks a course between the general and the particular, which makes it dramatic.

**Herman**: If we have time, I had a few questions about your concept of outlines.

**Doxiadis**: Let’s use outlines to address the concept of independence of levels. As a practicing writer I do outlines, like a painter who is painting big paintings; a novel, too, is big in scope in temporal or spatial scope, and like a painter when I want to understand a composition I cannot do that working on the detail but have to move away. In Arts school students are constantly told to move away, to hang the canvas upside down or turn down the lights or do other things that give them another perspective. Likewise, when I do long texts, I always do outlines and I have discovered that there is nothing like the experience of outlines to show to you certain aspects of a text that cannot be seen when you are working on the detail.

Let’s take as an example a crucial scene from a known narrative—say, the killing of Clytemnestra in *The Libation Bearers* by her son Orestes. If I say that in the central scene of that play Orestes kills Clytemnestra, I have produced a very rough outline. The way that scene is written will not change that outline. If we go into Sophocles’ *Electra*, which is a legend, that scene is not in the play. We hear about it but we don’t see it. But in the fabula, there is a scene called “Orestes kills Clytemnestra.” That does not constrain either Aeschylus or Sophocles when it comes to writing it–not even the decision of whether or not to include it visually in the play. So, if we get into the writing of that scene, there are other choices. There is a very moving moment in *The Libation Bearers* where Clytemnestra bears her breast and says “strike over this breast which fed you when you were a baby”. Orestes was not moved. Sophocles was so far removed from this level of detail as not to include the scene. The independence of levels would tell us that any such decision pertaining to the writing of this scene doesn’t affect its structural importance as a basic element of the play. So outlining, here outlining the fabula, tells you something that is important in itself, and much more important than any of the details realized there. This is a precise example of the independence of levels.

The independence of levels is perhaps best incarnated in my paper by the discussion in section 5.2 of moving from one city to the next. These two cities are only connected by a line. When you are leaving here to go there, you are only searching to get to this highway. What is important is to get to the highway. If you get to this highway, you are done because you’ll waste two hours going to another city. Once you are there, it doesn’t matter how long you searched to find it. All that matters is that you are on it. It depends on levels. This whole approach comes from my experience as a writer. Often, I have a sense that a given problem is at a detailed level, at a medium level, or at a rough outline level. Of course, this is a statistical. Because, if at the moment when Orestes is about to put his knife into Clytemnestra’s chest, Orestes changes his mind, and says ‘Ok, live Mother’ and departs, that is a very detailed action. It’s two lines. Yet because of it the whole play would come apart: the whole structure would be affected. That kind of detail is carried over in any outline. But I should note that this is an exception vis-à-vis independence of levels. Usually, levels operate independently in story. And the same in proofs.

**Herman**: Could you just define levels in this context?

**Doxiadis**: Yes. Degree of resolution. The cinema is the best example. Close shot. Long shot. If for example, a close shot is a very analytical level showing reactions and facial expressions; however, if you moved away in a battle scene, you would just see the two battles, not the soldier’s eyes. I would define it perhaps as the ratio of the degree of detail in what is narrated to the degree of detail in the original fabula.

**Herman**: And how exactly would this notion map onto proofs?

**Doxiadis**: Let’s not talk about proofs in the abstract but in the contextualized proof, just as we are talking about specific stories. I think in your definition of story, in the talk accompanying your own paper for the volume, you suggest that narratives involve events that must be interpreted in a light of a particular context of telling. Let’s speak of a proof in the same sense because often when people talk of proofs it’s only an ideal, platonic sense of proof, which never occurs in reality. In fact, one is always dealing with a proof article, one mathematician talking to another mathematician. Whatever.

**Herman**: So, proofs are interactionally tailored.

**Doxiadis**: Yes, when they are published. At this level, mathematicians may assume a certain level of sophistication, and also some shared knowledge and basic principles. Plus, once it’s published, I do not have to repeat the proof. My interlocutor or reader can go and look it up. In short, when you are explaining a proof the degree of knowledge available to your audience can vary. In developing a proof, you can say, ‘Look. There’s a crucial step here where we jump from modular forms to elliptic curves called the Taniyama – Shimura Conjecture, which Wiles proved in an immensely complex and difficult proof. OK. Forget it; you cannot understand it. Let’s use it to go from here to there.” That tale scopes twenty years of the history of the field to one line. You do the same thing in narratives all the time. A narrator might say, “She looked at the window. And a tear flowed down her cheek. She closed shut the window. Ten years passed. One morning in October, a bell rang.” You can talk of those ten years in two seconds. And we have the ability both to compose and interpret a story at all levels at once. It’s amazing that a child, having listened to a very detailed story, can describe it to you in a few words. It’s the same with proofs. I can tell you an outline and if I know the details of Wiles’ proof, I could detail it to you in one page or 20 pages or 1,000 pages.

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]]>Margolin: …

The post LaNave interviewed by Margolin appeared first on Thales + Friends.

]]>**Margolin**: Let me start with a classical question everybody asks: why is the history of science very active in the last thirty years or so while the history of mathematics seems to be dormant or marginal?

**LaNave**: To begin with, there is a social factor: a student goes to a department of history of science and very rarely, if ever, finds a professor that teaches history of mathematics. What he will normally find are historians of medicine, biology and genetics, and sometimes, but less often, physics and chemistry. Should a student decide to focus on the history of mathematics he/she would be strongly discouraged: there are no career opportunities, and departments do not want to create future unemployed scholars. But why are there no career opportunities for historians of mathematics? One reason is that the vast majority of people, I would say, are not comfortable with mathematics. In elementary school they are taught that mathematics is all about memorizing and calculating fast, so when they are introduced to more complex concepts they find themselves without the mental attitude necessary to understand the subject. The result is that people are afraid of mathematics.

**Margolin**: May I interrupt? People are much interested in the history of physics, and you cannot do history of physics without good knowledge of mathematics.

**LaNave**: And that is the other point I wanted to make: many mathematicians focus strongly on mathematics as a discipline outside history. Sometimes during their career there will be some interest in the history of mathematics, but they will focus on a very technical approach, an internal history of mathematics, which will not be accepted in the historians’ environment. So, history of mathematics is mostly left to mathematicians that do this internal history. But if you are a historian of science with a background in mathematics what you want to do is to work on both the mathematics and the social/historical environment around it. You would like to study the historical context, and also speak about mathematics. But if you do the mathematical side you have a public of historians of science who will not listen (and professional journals will not publish your papers with too many formulas in them). Conversely, if you put in the context without the mathematics, then you have a public of mathematicians that will not listen. Physicists are more likely to be willing to listen to their science put into a different context.

**Margolin**: Well, I think there may be another explanation. (To be honest, I’m just guessing because I’m not in this field). Amongst physicists there is a broad agreement that physical theory is a human construct and its specific nature is very much context- dependent, whereas many mathematicians are ultimately Platonists who believe that what they do isn’t constructing something which is good for a certain period of time but discovering eternal truths. Am I right?

**LaNave**: Yes.

**Margolin**: And if it’s a question of discovering an eternal truth, then who cares when, who and what, the main thing is that it has been discovered. So in mathematics the status of your claims is not that of a hypothesis and a construct but of a discovery and a proof. And therefore the history and the context of reaching them are less important. Would you think this is an explanation?

**LaNave**: Yes. Assuming we have a straightforward definition of what is science. Is mathematics a science? People aren’t comfortable when you refer to mathematics as a science because the word “science” carries this idea of a form of knowledge that applies, refers, is related to nature and so mathematics would not look like a science. At least following a Platonic definition of mathematics.

**Margolin**: Probably because mathematics has no experimental basis.

**LaNave**: I do not think so. I think that mathematics is experimental. People judge mathematics from the way in which results are presented in its system of knowledge: there is the theorem and the proof and all appears to work purely in virtue of deduction. However, this is only the final narration, the piece as it is presented to the world. Mathematicians do not work that way but follow a path in which deduction, induction, intuition and many other factors interplay with each other just as in any other science. Above all, once we stop thinking of the objects of mathematics as belonging to some unidentified spiritual world then mathematics is not so different from physics. Unfortunately, given the official fable told about what mathematics is, it’s probably closer in people’s minds to theology than to science.

**Margolin**: Fair enough. So let us go now to the other major question arising from your paper. There are in fact two issues here. One is beliefs, the change of beliefs and the causes for changes of beliefs. The other is the role of belief in theory formation, especially when theory formation is concerned with innovation, with production of new knowledge and the transition from problem to solution. Your particular case is that of Bombelli and thinking about imaginary numbers, but the general problem is the dynamics of belief and the role of belief in theory formation.

**LaNave**: Yes. That’s well put.

**Margolin**: In Greek Philosophy truth is defined as correct belief with reasons or grounds. This suggests a sort of belief-knowledge dichotomy and a continuum from one to the other. So we can take the totality of the propositions that are held or asserted by a certain person at a certain time and divide it into two subsets. One would be the set of knowledge claims: those propositions for which the person thinks there exists sufficient evidence or support. In the case of mathematics it would be a proof. And the other subset is the subset of belief claims, namely, those propositions for which there does not yet exist sufficient evidence or support or, in the case of mathematics, a proof. And the sum total of these two will represent the epistemic profile of this individual.

**LaNave**: Let’s go back to beliefs and their role in the general way of knowledge. The paper on Bombelli and the imaginary numbers is part of a more general study in which I consider different mathematicians in different historical periods and their process of belief formation prior to proof. In dealing with this, I was putting together a historical question and a philosophical question. The philosophical issue is how do beliefs work in the process of knowing, especially in mathematics. I wanted to analyze this philosophical question in history and focus on different cases and periods, on mathematicians from different cultural contexts and with different ideas. What I wanted to know was what constituted the knowledge they had and with what methods they justified it. The result is a process of mapping the beliefs of the mathematician at work. There are various kinds of beliefs involved, and one can really see how much this building up of beliefs is going to create a group of pieces of evidence sustaining and maintaining the person’s conviction, creating what a person perceives as a justification of his or her claims to knowledge. What I could see was that belief had a particular function in the process prior to proof. In philosophy, when we speak of knowledge or of belief we really can have many different positions and definitions, and this process of interconnected beliefs constituting pieces of evidence for knowledge claims doesn’t necessarily match the classical concept of justified true belief. However, it does constitute a justificatory structure for knowledge claims. After all, Bombelli does not find a visual representation for these numbers, yet his belief in the practical evidence for their true nature seems to be a justification for his acceptance of them. This is part of the scientific process of knowing, although we may want to be careful in calling it knowledge. Furthermore, this justificatory structure, this belief, is working as a kind of engine creating new research. Thus, there is a precise function that belief seems to have in the process of knowledge.

**Margolin**: So that belief is an engine for knowledge production.

**LaNave**: Yes exactly. This system made out of beliefs justifying knowledge claims is a set in relation with the set of accepted knowledge. But the borders of these sets are quite blurred. It is very hard to say what will be in one and what will be in the other.

**Margolin**: So, for some claims it will be easy to place them in the knowledge subset because a full proof for them exists.

**LaNave**: Yes. If there is a proof satisfying the accepted standards of proof in a particular period (these standards have not been always the same).

**Margolin**: While other ones are definitely just beliefs or hunches or intuitions

**LaNave**: Yes.

**Margolin**: And some of them just straddle a kind of a fuzzy line between the one and the other.

**LaNave**: Yes, and these are the beliefs having this particular function of knowledge production.

**Margolin**: And I guess that if in the course of an intellectual activity you find a proof for a given claim, then it is moved from the belief subset to the knowledge subset.

**LaNave**: Yes, exactly.

**Margolin**: As for a hypothesis in the natural sciences, once you’ve had enough experimental support or evidence for it, then it’s no longer a mere hypothesis, but rather a well supported theory.

**LaNave**: Yes. And, as for mathematics, that is obviously in relation with the particular historical period in which a mathematician lives. What counts as a proof may be different in different periods. The system constituted by what is deemed enough for considering something justified is what puts claims into the knowledge set. But the situation in which Bombelli finds himself is not merely intuitional. The belief in question is different from a mere intuition because there are supporting pieces of evidence for it. Bombelli says these are numbers, and although he does not have a geometrical representation for them, he has a proof of their existence. Thus, there is something increasing the confidence in the justification of the belief in question. This kind of belief will be, as we said earlier, in an intersection set between a set of claims constituting knowledge and a set of mere beliefs. As I said, this belief is in this border situation because Bombelli has a proof of the existence of these numbers and, although he does not have a geometric construction for them, he is capable of visualizing an application for them in the trisection of the angle. The fact that Bombelli can think of an application for these objects (whose existence he has proved) makes him feel more confident in the nature of this objects, in their being real objects and not fictitious entities. He does not know where exactly these objects are and does not have a spatial way to imagine them, yet he has a theoretical way of imagining them. This theoretical glimpse Bombelli has of their visualization constitutes a hook for his belief that these numbers have a representation somewhere, a position in some kind of space, although he doesn’t know which one. So it’s something a little bit more than simply believing and less than having an exact formalized justification for a belief.

**Margolin**: So it’s a continuum, not a binary.

**LaNave**: In a way yes, although it can be going up and down…Are you referring to a continuum in the mind of a person? That is, in this case, in Bombelli’s mind?

**Margolin**: No. I mean that in general propositions can be ranged on a belief-knowledge continuum. But it seems that as you said the movement is not always one way. Sometimes you’re thinking of moving from belief to knowledge and then something doesn’t work and the claim goes back to being just a belief.

**LaNave**: Yes, that may happen.

**Margolin**: But if you think it’s knowledge and then you’re proven to be wrong, then the claim is not just downgraded to a belief ; it’s eliminated altogether.

**LaNave**: The strength of a belief changes and the more things you find that can give strength to this belief, the stronger this belief becomes. Sometimes a belief that seems to be quite well supported can indeed find its strength weakened if related claims lose their supporting strength.

**Margolin**: So do you think there are degrees of support?

**LaNave**: I think one can map the degree of this support of belief (and in a relatively precise quantitative way). Think of Bombelli: first of all he has this open status of belief because he wants all the equations of the third degree to be solvable. This puts him in a situation in which he is willing to change his belief (the belief that imaginary numbers are not real numbers) because there is some strong stimulus behind this idea that it is already opening the belief status to change. And then he finds in Barbaro’s “Commentaries” on Vitruvius the construction for the duplication of the cube attributed to Plato. This is the geometric construction that he will use in the proof. He sees some similarities between this construction and the irreducible case of the cubic equations. This is a visual way to build a bridge between the two. This strategy, this building of a network of interconnected justified claims, makes the belief stronger and stronger. The more similarities and supporting situations and propositions a mathematician finds for his/her belief, the stronger the belief gets. This is why I think we can speak about mapping, a map in which the degree of confidence in a belief increases proportionately to the increase of supporting propositions (that is to say pieces of evidence) related to the belief in question.

**Margolin**: But, once again, as you have said before, it may also go down, as when you discover counter evidence or if you hit a boundary of some kind.

**LaNave**: Yes.

**Margolin**: And then you go oops, I believed this and it was getting stronger and I thought this is almost knowledge but no, it isn’t.

**LaNave**: Yes, but the situation is a little more complex. Bombelli’s belief fluctuates in its strength. Yet, once the confidence mechanism was triggered, Bombelli looked for more and more supporting pieces of evidence (despite the doubts). Bombelli did not give up for twenty two years: he thinks he is right, despite everybody else disagreeing with him.

**Margolin**: But at what point does belief turn into dogma?

**LaNave**: I would think it would be unreasonable for a mathematician/scientist to give up his/her beliefs at the first non supporting evidence. However, I agree with you, the dividing line between a strong belief and a dogma is a dangerous boundary. Yet, if the belief is not stronger than the experimental evidence, then the danger of turning into a dogma, although still present, is somehow controlled. Particularly in Bombelli’s case the danger of sticking irrationally to a belief was very low given that he had an amazingly practical mind.

**Margolin**: In general, I think belief turns into dogma when you have evidence that goes against it, and you don’t want to see this evidence or you’re trying to explain it away.

**LaNave**: That’s true, although, as I mentioned earlier, it seems to me quite rational for a scientist not to give up on a belief if there are only some pieces of evidence against it. In this particular case, Bombelli didn’t really have any strong counter evidence. What was against him was the accepted opinion about the nature of numbers coming from mathematicians before him and in his community. The other mathematicians would tell him: ‘Look, numbers need geometrical representations and you cannot represent roots of negative numbers. What are you going to do with this? They are not numbers.’ So, the only thing stopping him was the traditional concept of numbers. For Bombelli, it was more a question of putting this concept of number under discussion than not adding real counter-evidence. So, once he abandons this state of mind in which he ‘cannot look any farther’ (because it is not allowed) he is capable of opening new possibilities of research. Cardano too played with these numbers, but for him they were “sophistic” entities, nothing more than the result of a pleasant mathematical game. For him, the nature of these numbers (their being real or not) was solely related to the accepted concept of number as a geometrical/quantitative entity. So, the simple fact that he could not find a geometrical place for these numbers was enough to make them not real, but a bizarre mathematical joke. Cardano (and the contemporary community of mathematicians) had an assumption so strong that they were not going to see any other possibilities because this assumption was like a dogma. Bombelli was in a more relaxed mental state in relation to this dogma. He said “What if…?” Bombelli was an engineer after all and had a practical way of approaching mathematics, while Cardano had a rigid position regarding the relations between symbols and reality.

**Margolin**: I was thinking of a different typology of beliefs. Every scientist, every mathematician has a belief system. Some of these beliefs are more specific and some more universal. For example: some of them would concern the very nature of mathematical objects (Platonism, Intuitionism) while some others would be narrower, being about specific mathematical issues. So there are various kinds of beliefs depending on how fundamental and how general they are. One could also think of the degree to which beliefs are shared. Some beliefs are held by just one individual, some are held by part of the community and some others are held by most of the community. One can thus speak of degrees of sharing, asking how widespread certain beliefs are within a community.

**LaNave**: I think this is very important. Look for instance at the Riemann Hypothesis, which is something I am working on now and will be the next step in this study of the process of knowledge formation in mathematics, of which Bombelli’s case is part. Most mathematicians feel confident that the hypothesis is provable. The more they tackle it, the more they prove similar related propositions, propositions that make the Riemann Hypothesis more likely to be true, the more they believe it. There are some doubts about the Riemann Hypothesis, but the vast majority of the community believes in its provability. It is a matter of approximation: the more similar things are proved, the stronger the probability of this hypotheses, and the belief in its soundness gets stronger and stronger.

**Margolin**: Let me repeat something you have already mentioned: the strength of a belief can also be ranked on some kind of a scale, where the minimum would be an initial hunch or an intuition and then, as we have more and more claims to support it, the belief becomes stronger. Still another question would concern the originality of the belief. Is a belief held by a given individual an original one or is it taken from others? Because some of the beliefs we hold are simply opinions taken from others. Belief need not be original. It may be traditional and so on. We can also speak about the degree of novelty of an original belief held by a given individual. How radically does it deviate from the opinio communis, from the generally held system of beliefs about the subject?

**LaNave**: I think that it is truly rare that such a belief will be far away from the shared beliefs of the scientific community. A more likely situation is one in which some parts of the system of shared beliefs are abandoned by one or more mathematicians. Let us look at Bombelli’s case again. What he is saying is that these new roots are real numbers because, beside proving their existence, he can calculate with them (which, for him, gives them their status of being real). He is not making the revolutionary claim that numbers are abstract/formal entities. Bombelli does not detach himself completely from the contemporary concept of number as a geometrical/quantitative entity, and so for him these new roots are real because they are quantitatively valid entities. However, accepting these new roots as numbers despite the lack of a rigidly formal background, Bombelli starts breaking parts of the shared belief of the community of mathematicians about the nature of numbers. He is not claiming a conscious and elaborate break from the tradition (as it would be to claim that the defining nature of numbers is their being abstract symbols). He’s just saying that there is some kind of numbers with a strange nature. These numbers are, for him, not the same as regular numbers (or, at least, not yet). Nevertheless, he says, one can use them. It is possible to calculate with them. It’s a less radical change in belief than saying: “We made a mistake about the nature of numbers, their real nature is completely different.” It would be quite rare for someone to have such a complete awareness at a stage where things just start to change. Such a radical change in belief would be more likely after years have passed and many mathematicians have continued researching these new ideas. At that point one could find more of a common awareness about the change in the nature of the particular problem, the opening of possibilities and the consequent radical change in the beliefs shared by the community of mathematicians. But when a new question is first opened, it is all very primitive. Thus, I would say, the change in belief status is not so radical.

**Margolin**: One can also speak about degrees of “entrenchment” of beliefs: How reluctant you are to change or bend your belief when the new evidence goes against you.

**LaNave**: Inside communities of mathematicians there are strong forces that influence this capacity for abandoning beliefs. For instance, looking again at the Riemann Hypothesis, one can see that the community believes that the possibility of its provability is so strong that abandoning this belief will require a lot of work. The more you are part of a community that shares a belief, the harder it is to abandon that belief because the belief gets strength by being justified somehow by the standards of acceptance.

**Margolin**: You pointed out that belief has a positive role. It may act as a catalyst to the formulation of new theories, new knowledge and so on. But, even in the case of Einstein, a belief in excess is a sort of obstruction. Einstein said: “God does not play dice.” This prevented him from accepting quantum theory, even though there is a lot of evidence to support it. The belief in the determinism of the universe was so strongly entrenched in him that, in spite of the greatness and the radically innovative nature of his mind, at this point he simply refused to listen. For him, quantum mechanics may look right on the surface and function as a temporary solution to some problems, but it cannot be the final answer.

**LaNave**: That’s the situation of Cardano. He thinks there is no correspondence between these new roots and reality, while every single symbol must, for him, have correspondence to reality. This strong belief about the nature of numbers prevents him from accepting the validity of these new roots.

**Margolin**: And if there is no such correspondence, then it cannot be accepted. So, even though it works in practice and even though it is useful, I refuse to accept it as more than just a temporary trick.

**LaNave**: Despite the problematic nature of the excessive strength of some beliefs in the scientific community, I think this is unavoidable. It is a necessary part of the way in which scientists, in general, think. It would be indeed hard to work without strongly fixed assumptions. This is the “dark side” of the nature of belief. It has a positive role but it also has a very negative one.

**Margolin**: It may also influence what route you will take and what routes you refuse to take. ‘This route I shall not take because it goes against my belief that the universe is deterministic’. There is a further issue here: Scientists and mathematicians have certain strategies, certain methods or procedures they employ in order to tackle a problem. And these would be kind of methodological “do” and “don’t do” which have to do with how to proceed. How do these methodological norms relate to the knowledge-belief continuum?

**LaNave**: This is a very complex situation- at least as I see it. We have a community of mathematicians. We have standards of knowledge that determine what kinds of questions are asked and what kinds of results can be accepted. These results are going to shape what doing science or mathematics is all about. When some parts of accepted beliefs start to be abandoned (like Bombelli accepting these new numbers) some of this structure of questions that can be asked and of answers breaks down, and then new research starts which will bring to the formulation of new kinds of questions. This is the situation opened by Bombelli: if mathematicians after Bombelli want to go on accepting these new numbers, they have to come up with more formalized results. In so doing they will have to change the rules of the game and to come up with things that will not look quite correct according to the previously accepted standard of knowledge. Obviously this can be very productive in the long term. It is important that the scientists’ shared system of knowledge gets broken by beliefs that initially look not justified (or, at most, only partially justified), like Bombelli’s.

**Margolin**: You have this picture in your paper of cuts in a piece of wood. This was one kind of procedure or method or strategy for proving claims that had been employed for a long time. At this point, Bombelli’s beliefs apparently make this particular method not very practical.

**LaNave**: The first instrument he used (the one with cubes) did not work for his proof. Thus, he goes to the one using two moving L-squares and that one works.

**Margolin**: I guess there is also some kind of interplay between one’s beliefs and the methods one employs.

**LaNave**: You either have to come up with new methods or use old methods in a different way, as he does. This will be the way out in a situation in which the methods are not helping you. He was there trying again and again with the cubes and he could not prove the existence of these numbers with this instrument and that is when he changes to the instrument using L-squares.

**Margolin**: So in this particular case the belief was stronger than the methodological norm. If the old method doesn’t work, I will stick to my belief and I’ll try to find a new way of doing things.

**LaNave**: Well, stronger, but not so strong that he was ready to leave behind all the accepted standards of which the instrument using the two moving L-squares is part. After all, in the first version of his work, the one in the manuscript of “L’Algebra,” Bombelli agrees with Cardano: these numbers appear “sophistic” to him as well. However, he also did open a possibility for them by saying that should they be actual numbers, then it will be possible to apply them to the trisection of the angle. Bombelli’s imagination was going around quite wildly. He had already the capacity of thinking: if they are numbers, then such and such. This is what Cardano did not have. This really creates an open status of belief. And then, in the course of more than two decades between the manuscript and its publication, he reads Barbaro’s work, and the more he thinks about the method/instrument using L-squares, the more he sees a connection between the duplication of the cube and the solution of the irreducible case. When he borrows the method attributed to Plato for his proof and builds up an instrument, it turns out to work. It is really nice to look at this instrument: you can see the proof much more clearly when you can move the actual L-squares in relation to each other.

**Margolin**: I believe I have by now exhausted the questions and observations I had in mind. Thank you very much.

**LaNave**: Thank you.

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]]>Image: Nature

“Mathematicians are quite shy, with very…

The post What’s the Plot? appeared first on Thales + Friends.

]]>Can mathematicians learn from the narrative approaches of the writers who popularize and dramatize their work? Sarah Tomlin is on the story.

“Mathematicians are quite shy, with very few exceptions,” announced Pierre Cartier of the Institute des Hautes Études Scientifiques, south of Paris, opening his talk on mathematicians who have written autobiographies. “And most of them are in this room!” someone shouted from the floor. It was a rousing start to a highly unusual meeting. Last month, a select group of about 30 mathematicians, playwrights, historians, philosophers, novelists and artists descended on the Greek island of Mykonos. The gathering aimed to find common ground between storytelling and mathematics, and was inspired by a profusion of books, movies and plays that, over the past decade, have dragged the subject out of a cultural wilderness. Clearly, stories about mathematics have strong popular appeal. But what can professional mathematicians learn from the writers who are now taking an interest in their work? Can narrative approaches help the increasingly esoteric sub-fields of mathematics communicate with one another? And can such approaches help mathematicians frame the abstract problems that fill their working lives? It was the hope of answering such questions that led the Greek novelist Apostolos Doxiadis to cook up the idea of bringing writers and mathematicians together. After graduating with a mathematics degree from New York’s Columbia University at the age of 18, Doxiadis turned to his first loves of poetry and theatre. But in his late thirties, Doxiadis revisited mathematics with his 2001 novel Uncle Petros and the Goldbach Conjecture, in which the protagonist’s attempt to prove this conjecture — that every even number is the sum of two primes — becomes a universal story about courage and suffering. Excited by the narrative possibilities offered by mathematics, Doxiadis formed a foundation to further explore the theme — called Thales & Friends, after the first mathematician and philosopher in ancient Greece. And with financial backing from the Mathematical Sciences Research Institute in Berkeley, California, he set about organizing the Mykonos meeting. At first, the plan was for a smaller informal gathering, but it soon snowballed. “I’m surprised, frankly, at the response,” Doxiadis told Nature. The venue was apt, given that ancient Greece was where the gulf between mathematics and story-telling first opened up. “Plato approved of mathematics, but despised poetry,” says Rebecca Goldstein, a philosopher and novelist based in Hartford, Connecticut, who has used mathematicians as characters in several novels. Other participants blamed Euclid for introducing the impersonal, logical

style that has characterized much mathematical writing ever since.

Whatever the historical reasons for the divide, it has been bridged with a vengeance in recent years. The spark came in 1995, with the solution of Fermat’s last theorem by the British mathematician Andrew Wiles. His 100-page proof made prime-time news, and was followed by a best-selling book, Fermat’s Enigma by Simon Singh. Suddenly mathematics was fashionable. Next came a host of popular books on subjects such as zero, pi and irrational numbers, the Hollywood movie A Beautiful Mind, and plays such as David Auburn’s Proof and John Barrow’s Infinities. Today, there

is even a US television series called Numb3rs, in which a detective relies on the skills of his mathematical-genius brother to solve crimes.

But if it is possible to tell stories about mathematics to a general audience, why do specialists in different branches of the discipline have so much difficulty communicating with one another? “Most mathematics papers are incomprehensible to most mathematicians,” complains Tim Gowers of the University of Cambridge, UK, winner of a 1998 Fields Medal — the nearest thing the subject has to a Nobel prize. “Publication has become just a

formal stamp of approval — it is not a means of communication anymore,” he adds. Gowers is currently editing the Princeton Companion to Mathematics, which is scheduled to appear in 2006 and is intended to provide budding mathematicians with an accessible overview of the field. “Say you decided you wanted to do research in mathematics but you didn’t know what area would appeal to you,” says Gowers. “There is nothing available right now.”

**Tales of tables**

Narrative approaches can help make arcane branches of mathematics more accessible to specialists, as well as to the lay public, Gowers argues. Persi Diaconis, a statistician at Stanford University in California, agrees. “I can only work on problems if there is a story that is real for me,” he says. As one of the few applied mathematicians at the meeting, Diaconis perhaps has an easier time than most in applying narratives to his work. In Mykonos, he picked three of his papers, and told a story about each. The first was a 1987 paper entitled ‘Projection pursuit for discrete data’, which deals with a mathematical technique for finding patterns in data in a systematic way. To make this technique come alive, Diaconis chose a real-life

problem — the dating of Plato’s most important texts. Scholars had previously classified the last five syllables of Plato’s sentences as either ‘short’

or ‘long’, in the expectation that his writing style had changed over time, and that this would show up in these data. But they struggled to find

anything meaningful. By looking at pairs of syllables using his technique, Diaconis was able to find patterns in the texts that had previously been hidden, and could subsequently work out an order for Plato’s works. Diaconis learned to love the abstraction of pure mathematics from his tutor Barry Mazur of Harvard University. But Mazur admits that he used to be mystified when Diaconis would ask: “I’m lost. What’s the story?” Today, Mazur says he has woken up to the power of narrative, and in Mykonos gave an example of a 20-year unsolved puzzle in number theory which

he described as a cliff-hanger. “I don’t think I personally understood the problem until I expressed it in narrative terms,” Mazur told the meeting. He argues that similar narrative devices may be especially helpful to young mathematicians, who seem particularly poor at explaining their work to others. Mazur explained that he used narrative to help him develop a general organizing structure around the problem, which involves a conflict between theory and the large amount of data gathered on elliptic curves. Mazur did not find a solution by using the narrative device of a cliff-hanger, but it helped him to frame the question — and that, he argues, may be as important. Mazur’s explorations of narrative may sound trivial. And some of the writers who attended the Mykonos meeting admitted that they were left cold by this particular story. But Doxiadis explains that mathematicians, unlike experimental scientists, aren’t used to dealing with conflicts between data and theory in this way. They like to understand a problem at its most basic level, from the inside. So for a mathematician, Mazur’s depiction of his number-theory problem in dramatic terms might be a genuine eye-opener. “The thing that inspired me was the idea that this could affect mathematics itself,” says philosopher David Corfield of the Max Planck Institute for Biological Cybernetics in Tϋbingen, Germany.

**Maths fever **

In general, the non-mathematicians in Mykonos were passionate about the mathematical ideas that they have encountered. Alecos Papadatos is a cartoonist who is working on a graphic novel with Doxiadis and

others about the history of logicians, most of whom died tragically. He claims he never liked mathematics, but now sees that the discipline is full of exciting stories. Barbara Oliver, who is artistic director of a theatre in Berkeley, California, echoes his view. She enjoyed directing the play Partition by Ira Hauptman — which is loosely based on the lives of two very different mathematicians, G. H. Hardy and Srinivasa Ramanujan — despite her lack of specialist knowledge. “I’m untutored, unskilled in mathematics,” she says. Whether mathematicians are similarly eager to embrace the joys of narrative remains unclear — particularly if telling a good story involves compromising standards of mathematical accuracy. Some of the most heated discussions in Mykonos were about the mistakes in a recent popular book on infinity, and whether they really matter. Diaconis, despite being an enthusiastic story-teller, still sees a sharp tension between narrative and mathematics. “To communicate we have to lie. If we don’t we’re deadly boring,” he says. Diaconis waited until his 60th birthday to start writing a book on mathematics and magic, and he understands why others are cautious. “There’s no reward for expository stuff, and a bias against it,” he says. Still, the Mykonos meeting at least marked the beginning of a rapprochement between the estranged arts of mathematics and story-telling. Another meeting is planned for next year — and there is much work to be done, if the tales of the thorniest mathematical problems are ever to be told. Take the Hodge conjecture: according to the website of the Clay Mathematics Institute in Cambridge, Massachusetts, which is offering a US$1 million prize to anyone who can prove it, this asserts that “for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations called algebraic cycles”. If that makes no sense to you, don’t worry — most mathematicians regard the Hodge conjecture as ‘unexplainable’. Let’s hope, at least for the sake of the journalists who must try to communicate the essence of the conjecture to the general public if it is ever proved, that mathematicians will by then have honed their narrative skills. ■

Sarah Tomlin is a senior news feature editor for Nature.

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