The following interview between Colin McLarty and David Corfield was conducted on July 22, 2007. The interview as it appears here was transcribed and edited only slightly for clarity.
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.
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.
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.