Posted on 23 July, 2007 in category: Interviews
The following interview between Michael Harris and Arkady Plotnitsky was conducted on July 23, 2007. The interview as it appears here was transcribed and edited only slightly for clarity.
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 Galatea 2.2?
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?