Thales + Friends

A discussion with Apostolos Doxiadis

Posted on 16 July, 2005 in category: Conferences

Writer Apostolos Doxiadis was one of the first people on the organizing committee of “Mathematics and Narrative”. In this informal chat, held a few days after the meeting, we ask him to give a general sense of the tendencies represented and draw some conclusions.

Thales + Friends: Let’s begin with a hard one: as one of the Mykonos meeting’s organizers, would you say that it was successful, on the whole?

Apostolos Doxiadis, Image: Thales & Friends (C)

Apostolos Doxiadis: Oh, that’s not so hard! Without meaning to sound at all immodest, I will say simply: “Yes, it was very successful – as expected!” And the reason I think my statement displays no lack of modesty is that the success of the meeting was just another instance of the Victor Hugo maxim about nothing being “as powerful as an idea whose time has come”. So, as it was very definitely not we, the organizers, who determined that the time had come for mathematics and narrative (I usually prefer the simpler word: story) to meet, I don’t think we should be getting overdue credit for facilitating the occasion. Though perhaps we can tap ourselves on the back a bit for being able to tell which way the wind blows.

T + F: What exactly do you mean by saying that the meeting of mathematics and narrative was “an idea whose time had come”?

AD: Well, I will not even begin to attempt to tell you exactly what I mean and this because I don’t think exactness should be a requirement in a new area of human activity. Tidiness is a virtue only if you’ve lived for a long time inside messiness. So, this being really an early stage in the co-existence of mathematics and narrative, it’s alright for us to be super messy. Yes, and let’s enjoy the mess! Let’s look at the matter this way and that way and then another way, and feel free to make the many mistakes you need in order to achieve the few lucky moments of true discovery, which only appear serendipitous if you disregard the huge number of failed attempts on the way to them.

So, here’s a messy answer to your question – but if you want a cleaner one and are not afraid of long words you can also look at the Statement in the meeting’s website. Here goes: The two sets ‘mathematics’ and ‘narrative’ – for both these concepts are really sets of things and not just things – have traditionally suffered from belonging to two different supersets, the first to that of the generally-called scientific culture and the second to the generally-called literary. And anyone who has read C. P. Snow’s seminal essay “The Two Cultures” – or anyone who knows anyone who has read it, for that matter – will understand what I’m talking about: from the time of Romanticism onwards there has been this most central dichotomy in our culture. Science and art are viewed as two totally distinct, even totally contradictory worlds. And mathematics, considered as it is by scientists and artists alike to be the paragon of science, has suffered from the effects of this split more than anything else. So, if you are a true child of the post-romantic era and believe, if ever so slightly, in the science/art split – which to most people really means the reason/emotion split – you probably believe that the set ‘mathematics’ and the set ‘art’ have, as a mathematician would say, a null intersection. Or, in everyday words: absolutely nothing in common.

T + F: A position with which you obviously disagree.

AD: Indeed, or else there wouldn’t be a point in working on organizing a meeting on the subject! But, you see, if anyone had gone public with a disagreement a couple of decades ago and started telling people that mathematics and storytelling are really not that far apart, he/she might have been considered a crackpot – back then! Not so today: not in the day when the biographies of great mathematicians or the narrative accounts of the solutions – or attempts at solutions, for that matter – of great mathematical problems are becoming bestsellers, when there is something called ‘mathematical fiction’, when films or plays of mathematics or mathematicians are winning Oscars and Pulitzers.

T + F: But there are some fields, such as the history of mathematics, which have traditionally attempted to move in the in-between area, where mathematics and narrative meet.

AD: Yes, though almost all history of mathematics was written, until some years ago, from a point of view that was not very interesting to non-mathematicians. With all due respect to some good work, most of the classical history of mathematics was really quite pedantic, the sort of work written by (and/or read by) retired mathematicians, those poor creatures, in G. H. Hardy’s constipated view, who are “no more able to do good mathematics” and just good enough to regurgitate its past achievements. In kinder language, they are more chronicles than history proper. And it is only in the last very few decades that the history of mathematics is becoming fascinating, and this I think to the extent that it begins to adopt the methodologies and viewpoint of the history of science: as history of mathematics becomes, in a sense, a branch of the history of science, so it also begins to profit from the full range of narrative tools and concepts available to the latter.

T + F: What disciplines were represented at the meeting?

AD: Apart from historians, you mean? Oh, many: writers, artists, philosophers, semioticians, cognitive psychologists – you name it.

T + F: Aren’t you forgetting anyone?

AD: Dear me, yes: mathematicians! We had some very eminent ones attending, some great pure mathematicians but also some great statisticians and theoretical computer scientists. But I guess the reason I forgot to mention them is that having mathematicians at a meeting with ‘mathematics’ in its title is not news. What’s news is having a theatre director there, a comic book artist, a psychologist-cum-literary theorist, a couple of novelists. And these people were there, of course, because they have done work with some reference to mathematics. But the mathematicians who were there were also chosen for having demonstrated an urge to transcend the big divide.

T + F: Say a bit more on this: what were the criteria by which people were invited to the Mykonos meeting?

Marcus du Sautoy, Image: Thales & Friends (C)

AD: The criteria were very clear, really: on an individual basis we wanted people who a) had a good-to-excellent reputation in their own field but also, b) had actively involved themselves with something paramathematical. For mathematicians, this meant that (apart from being good-to-excellent in their field) they had also done work either in popularization, the history or the philosophy of mathematics, or mathematics and the arts – with the element of narrative playing a significant part. Barry Mazur is a great mathematician, but he was invited because he also wrote his book Imagining numbers, where he tries to explain the notion of imaginary numbers to non-specialists, with a strong historical element –and also he has written very interesting essays on how mathematics is created. Or take Fields Medalist Tim Gowers, who has employed the time-hallowed tradition of the philosophical dialogue to explore higher mathematical concepts and, also, has written essays exploring more general aspects of mathematics. Martin Davis was invited because, apart from his great work in logic, he has done a wonderful work of exploring the history of computers in an overlap of exposition and narrative, in The Universal Computer. Robin Wilson has told a wonderfully lucid tale of the Four Color problem in his book on it. Cosmologist John Barrow wrote the play Infinities. And Marcus du Sautoy (here seen lecturing at the meeting, wearing the shirt of his football club) made page-turning drama out of something as abstruse as the Riemann Hypothesis in his book The Music of the Primes.

For the people coming from other areas, normally unrelated to mathematics, the criterion of paramathematical work meant that they had tried to approach, with some degree of success, mathematics by other means. For example, theatre director Barbara Oliver was invited for having produced and directed a play on Hardy and Ramanujan. Or (ex-mathematician and) semiotician Brian Rotman for having done distinguished work on mathematics as a sign system, work with obvious implications for narrative. In other words, we wanted people who had distinguished themselves by journeying into the dead zone, the cultural divide, the “gap” where the mysterious and yet-unfathomed intersection of the two sets, Mathematics and Narrative, inhabits. For mathematicians it meant going one way and for non-mathematicians the other, perhaps to meet somewhere in the middle.

Now, if you look at the collective criteria, i.e. those affecting the participants as a whole, these were that: a) thirty looked like a good upper bound on the number of speakers for a four-day meeting and, b) we should have as many different disciplines represented, with a good healthy stock of mathematicians as a base. Clearly, there were more people who were eminently qualified to be invited. But we couldn’t possibly fit them all in our program.

T + F: Will you name some names?

AD: I will name some fields that were not represented – and should have been. We definitely needed more artists, with the word taken in the widest sense. As also literary theorists, people working in narratology, the scientific study of narrative. As also, of course, people from AI, those trying to build story-generating programs.

T + F: What were the criteria for the historians? After all, if they’ve worked on mathematics they are by definition doing both mathematics and narrative.

AD: Ah, yes, good point. For historians we had an extra criterion: we tried to stay away from the (if I may call it so) traditional school of the history of mathematics, the people who mostly write the chronicle-type history that I spoke about already, the historians who are really only interested in recording events as precisely as possible. This is of course a worthwhile occupation – but as were trying to actively explore the nature of the mathematics/narrative connection we thought we would go for historians who adopt a more complex viewpoint. The three more or less pure historians of mathematics at the meeting, Amir Alexander (seen here speaking on “Tragic Mathematics”) Karine Chemla and Leo Corry, were people who have done work that can be seen to belong to a more general history of ideas: history of mathematics as history of ideas. And perhaps I should add Doron Zeilberger here who, although an eminent mathematician, often refers to the historical viewpoint in his web “Opinions”, and gave what was, really, a historically inspired talk.

Amir Alexander, Image: Thales & Friends (C)

But we also invited some eminent historians of science who have dealt with mathematics – in fact, one of them, Ted Porter, was also on the organizing committee. And there were Norton Wise, Mary Terrall or Joan Richards. It is a well-known fact that the history of science has become an enormously more complex and interesting field in recent years, a field much informed by advances in neighboring fields (the sociology of ideas, anthropology, cognitive psychology, etc.) and by these influences achieving a much greater degree of narrative complexity. What I mean is: if, for example, we define a well-written, many-layered novel as giving us something quite close to the highest form of what psychologist Jerome Bruner calls “narrative knowing”, then the history of science comes much closer to that in recent years than straight, chronicle-type history of mathematics.

T + F: So, let’s get back to the meeting and the talks. It was obviously – one only has to look at the list of participants to be convinced – interdisciplinary in intention. But was the expected outcome really, to use set-theoretic terminology, to go for a union or an intersection? In other words, was the hoped-for basic message of the meeting of the form: “There is x and y and z and some more things that fall under the ‘mathematics and narrative’ banner”? Or something more like: “After considering x and y and z and some more things we conclude that there is a new entity, a set MN, that can express the essential way in which these two sets overlap”?

AD: Well, I think that as in all successful interdisciplinary venues there is a bit of both. On the one hand, going interdisciplinary for any field usually comes when there is a growing realization that “hey, this problem can also be approached/looked-at in ways other than the traditional.” And there was definitely this feeling at Mykonos. Not just the other participants, but even the organizers (who must have suspected that there was some point in bringing all these people together, else why did we do it?) were pleasantly surprised – some more surprised than others, of course – by the existence of so many different viewpoints. It’s like that old expression that’s often used referring to different religions as being “many different paths leading to the same summit”. It’s one thing to know a few other paths, though – other than your own path, that is – and/or suspect their existence and a totally different thing to actually meet people who have traveled them. Talking to many of the participants outside the meetings, I heard again and again, always in variations of the same pleasantly surprised tone, exclamations of joy that the meeting was happening – that all these different viewpoints were in existence and also… there! So, this partly answers the ‘union’ aspect of your question: yes, the element of a varied menu of approaches is a necessary element of interdisciplinarity and it was there, with very different people coming from very different directions and all saying “yes we too are dealing in some way with a combination of mathematics and narrative in our work”. Now, regarding the ‘intersection’ part, i.e. going beyond the friendly cohabitation of various approaches to look for deeper connections, going from “both x and y are useful and interesting approaches” to “you know, x and y have a lot in common!”: yes, there was an element of that too. And let’s thank our luck (or whatever) for that, for without at least some deeper connections interdisciplinary projects do not ultimately fare well. The centrifugal tendency is definitely the first one, the necessary motivating force of any interdisciplinary effort, for it is this that makes a field break out of its shell and seek to meet others. But unless there is also a strong centripetal force to counter it – this is simple physics, really! – the effort will eventually lead to chaos and/or dispersal. In fact, as the underlying similarities of mathematics and narrative was really the main subject of my own talk, I could go on and on about it. But I’d rather let you read my views in the write-up of my talk.

T + F: Were you alone in advocating this view?

Timothy Gowers, Image: Thales & Friends (C)

AD: Oh no, far from it! Denis Guedj, Christos Papadimitriou, Tim Gowers, Amir Alexander, Barry Mazur and Norton Wise, to mention some, had a lot to say along similar lines, though some from very different viewpoints. And this is really quite amazing, to have such a significant overlap of opinions, as there are really common references that generated these views. The speakers were really all speaking about how they had all independently reached, each adopting a different viewpoint, elements of the same basic truth. In fact, Tim Gowers said during his talk – even though it only came at about the midpoint of the program – that he (already) had a sense of a “general theory bubbling up” at the meeting, and he meant in this direction: a general theory of the deeper relationship of mathematics to narrative (on the right is Tim during his talk.)

T + F: Did he give any indication of what this theory is?

AD: No, but he was really extemporizing, as it were, giving his immediate, online reaction to what was going on around him, being very much in the midst of it – it was impossible to be more precise at this stage. His talk was about “describing mathematical proofs without losing the plot”. And I think that even the use of the word “plot” in the title – used in reference to proof! — by a mathematician of his caliber, is headline news. I was personally extremely interested in this talk. It had such an enigmatic structure of which the most telltale feature was a big hole in the middle, a hole that Tim was very much aware of, of course. And I was really thunderstruck when I saw this structure practically repeated – obviously, without this being intentional in any way – in Barry Mazur’s talk. A true meeting of great minds! The fact that two of the most eminent mathematicians at the meeting gave talks which were almost isomorphic I found extremely significant.

T + F: What was the nature of this isomorphism?

AD: I’ll tell you what the structure of the talks was and you figure out the isomorphism: both began by saying something like: “I believe there is an important connection between story and mathematical thinking. So, my talk has two parts. In the first part I’ll tell you a few things about proofs. And then I’ll tell you about stories.” (Actually, in Barry Mazur’s talk it was the other way round, first stories then proofs – but this was a symmetric structure, so that’s OK.) And in both talks it was in fact implied by a variation of the post hoc propter hoc, the principle of consecutiveness implying causality, that the two parts of the lectures were intimately related, the one somehow led directly to the other.

T + F: And the hole?

AD: This was exactly at the point of the link, going from math to narrative – or the other way round, in Barry Mazur’s talk. There is this very well-known, Sidney Harris cartoon, also used by the American Mathematical Society on its T-shirts, where two huge arrays of formulas on a blackboard are connected by the sentence “THEN A MIRACLE OCCURS.” And one of the two mathematicians standing before it points at this and tells the other: “I think you should be more explicit here at step two.” Both Gowers’s and Mazur’s talks were one half fascinating expositions of lay narratology – in fact, I was exhilarated to hear the two most purely narratological talks at the meeting coming from number theorists! – and one half a discussion of a purely mathematical kind, the two parts separated by a conjunction roughly synonymous to “this is very similar to this”. But the similarity was not clearly explained: the hole, you see, the “miracle”. Of course, both Gowers and Mazur are brilliant men, and honest too, and so they were very clear about the location of the hole, they did not try to fool us by saying that there was no hole were there was one. But that is OK: as Doron Zeilberger says in his 39th Opinion, it is alright to publish incomplete proofs as long as you indicate which are the complete parts of the argument and which need filling-in. And incomplete proofs are the best you can expect in a totally new field.


T + F: So, were these talks high points of the meeting?

AD: To me they were, certainly. But I can’t be objective, because they were very close to what I was trying to say, i.e. that there is a strong similarity between story and proof. And also there were so many interesting things said at the meeting that it’s unfair to single out anything, except as a very personal, very subjective reaction. Yet since you used – you did, not me! – the expression ‘high point’, let me tell you about the highest point to my mind, the one moment in which I had a sense that, if it had gone a milli-unit of insight deeper, the whole auditorium would have jumped up crying “Eureka”!

T + F: How dramatic!

AD: This was in Barry Mazur’s talk, very aptly called “Eureka and other stories”. So, Barry Mazur began the talk speaking about stories – this was the narratology part, you see. And he presented us with his own taxonomy, with items such as ‘purpose stories’ (i.e. didactic), ‘origin stories’, etc. And then he said “and now I’ll tell you a story.” And he started his story – what he called a story, you see — by saying: “I will begin with the punch line.” And then he showed the graph of a function which he said “looks linear”, though Persi Diaconis interrupted to say “it does not look linear to a statistician” (you can see it in the background, in the picture, and form your own opinion). And of course it was clear to the audience that this picture showed in fact the punch line – with no puns on the word ‘line’ intended.

Barry Mazur and the "linear" graph, Image: Thales & Friends (C)

And then Barry started to describe a mathematical problem relating to the counting of rational points on elliptic curves and he went on talking of the research in this field on which he has been working for the past years. And the mathematicians in the audience got increasingly excited – Mazur is one of the great experts in the field, you see – and the non-mathematicians got increasingly frustrated. And then, after he had spoken about elliptic curves for twenty minutes or so, Barry said “thank you” and stopped, leaving half the audience exhilarated and the other half perplexed. Now, because Barry is such a great mathematician and because there were many mathematicians in the room, the question and answer period following the talk developed into a discussion about the finer points of the counting of elliptic curves. And then, suddenly, I had an epiphany: I thought “Oh my God, this is it! People are talking about elliptic curves and of course they think they are talking mathematics. But are they really? Or are they talking about stories?” For, you will remember, Barry had called the second part of his talk a story. So I raised my hand and I started saying to Barry: “I don’t know if you’ve realized it, but for the past twenty minutes now you are talking mathematics…” But then Barry, being an extremely nice person, thought I was calling him to order for talking mathematics in a “mathematics and narrative” meeting and he was very apologetic, saying “yes, yes, I know, I’m sorry.” But “no, no,” I said. “This is absolutely fine! I would just like you to tell us why you called the second half of your talk, the mathematical part, a ‘story’.” And then Barry said something like: “Well, this is a story of this particular area of research”. And again I commented that this story was all about curves and numbers, not people. And again I asked him why he called this a “story” and not just “an outline of the progress of a proof”. And then Barry said, looking rather crestfallen, that “well, it’s not too clear which is which,” and I, listening to this, achieved enlightenment! I was ecstatic! Because, you see, to me this was so much the essential point of the meeting: that the line separating proof from ‘story of proof’ is very fine – in fact it’s not a line but a very fuzzy sort of grey area, which may not even exist, the two may be exactly identical at certain points. But for me the absolute peak moment came a little later – yet others did not share in it because it was during the break. It was when I went up to Barry and explained that I had in no way been meaning to criticize him with my question that he was talking mathematics, and apologized in case he had thought I had been. And he said no, it was fine. And then I pressed him a bit more, almost begging him to tell me why he called his mini talk on progress in counting elliptic curves a story. And then he said something amazing. He said: “I have been working in this field for many years. But I did not really understand what is going on, until I summed it up to present here, in the form of a ‘story’”. Reflect on that!

T + F: OK, we will. But let us proceed now with other aspects of the meeting. So, you had the artists talk about works of art, literary or otherwise, with mathematical subjects, the cognitive people speak about mathematics as a human activity – and all narrative is about human activity – and the mathematicians talk about either popularizing mathematics through stories or how narrative and proof are really connected in some way.

AD: I think that if we try and be in some way systematic about what was said during, we can classify the Mykonos talks by whether they touched upon certain topics. In fact, in my summary talk, at the end of the meeting, I identified twelve such basic themes that occurred again and again. Here they are:

  • Want the truth about mathematics? Look at its (hi)story!
  • Mathematics as biography
  • The cognitive aspect: if mathematicians are people then they must be subject to the laws of psychology.
  • Stories as a way to exposition, communication and teaching.
  • Stories of math as stories of science.
  • We need new forms to communicate/speak mathematics.
  • Mathematics (not mathematicians) as story material.
  • Mathematics and story: are there more profound connections?
  • Proof and story (or Proof as story).
  • Mathematically speaking of story.
  • Story and (as?) truth.

The future of mathematics: narratives vs. formalism. I presented twelve slides in my summary, each one with one of these twelve topics as heading and listed under it the names of the people who in some way or other referred to it in their talk. But this was a half-hour talk, it had to be done on the last day of the meeting and was composed under heavy time constraints: it’s not exactly easy to digest, compare, classify and summarize twenty-nine very interesting long talks online! And so, thinking back on the meeting now, a couple of weeks after it’s over, I can say – although I still think that these twelve themes are a pretty useful tool for a taxonomy of the talks – that there probably is a coarser classification that really outlines the main issues that were raised at the meeting. Unlike my twelve-point classification though, which resulted in a rather complex graph (a talk could be described by more than one attribute, you see) this new one is a tree, it separates all the talks in five non-overlapping classes – of course you do realize any such scheme results in distortions. But it has value, as it gives the meeting, and thus the subject, a sense of form.

Pierre Cartier opened the Mykonos meeting with a talk on mathematicians’ autobiographies, Image: Thales & Friends (C)

T + F: Let’s hear it then.

AD: First, there were the people who spoke about the human drama of mathematics, either in the generic or theoretical case, as did Pierre Cartier with his talk on mathematicians’ autobiographies, Martin Davis with his account of the lives behind the growth of logic, Amir Alexander with his tale of “Tragic Mathematics” or Rebecca Goldstein talking of mathematicians as characters – and I should add here Leo Corry who discussed “limits to poetic license”. In all these talks I had the sense that the speakers were saying that mathematicians’ lives are somehow related to their work, in ways which are non-trivial – though of course those ways were not always too clear. And then we had the actual stories: Barbara Oliver, with her talk on Partition, Alecos Papadatos who made a presentation of Logicomix, a graphic novel of the human drama behind the birth of computers and Ted Porter who spoke in the cooler tones of the historian about the life of the great statistician Karl Pearson and its relation to its work. Here, theory came together with practice: we had storytelling of various sorts, theatrical, sequential and historical applied to lives of mathematical discovery.

Barbara Oliver, Image: Thales & Friends (C)

Then, the second class of talks were about the drama of mathematical ideas themselves. And Dennis Guedj gave this class a wonderful start by stressing that mathematics itself is also dramatic, without taking into account the lives of the mathematicians who create it. So, he spoke in an exploratory way on the theory of that, and John Allen Paulos also had things to say from this point of view, mostly referring to statistics, some ideas also presented in his fascinating little book Once upon a story. John Barrow, presented his play Infinities, which is very definitely a play of ideas not characters. And Marcus du Sautoy performed his wonderful one-man show, The Music of the Primes, which also referred to stories of mathematics, not people. In the same class of mathematics as drama was also Robert Osserman’s film The Right Spin: here we had a real-life, gripping astronautic drama, which though humanly dramatic – human lives were at stake – all the most crucial elements of the plot revolved around equations. And Robin Wilson did an ungrateful but necessary job, presenting us with a meticulous taxonomy of the various works that have been produced in recent years on mathematics as drama: all the narrative accounts of the various problems, a classification of the paramathematical literature.

Class number three was historical. Henri Poincaré famously said that “if you want to know the future of mathematics, look at its past”. And though I’m not so sure that his statement was right – for it is in the essence of the future of anything to be quite unknowable – many of the speakers at Mykonos seemed to be working on a variation of this: “If you want to know the essence of mathematics, look at its past”. Here, I would put Karine Chemla, with her account of the earliest forms of narrative-based mathematical exploration, in ancient China; Mary Terrall, with her strong blending of mathematics and narrative in the accounts of 18th century expeditions and, on a similar theme, Joan Richards, who spoke of mathematics as narrative (historical) and its decline by the ascendancy of rigor in the early nineteenth century; Norton Wise with his “Dirichlet narrative”, as he called it, showing how Fourier analysis became rigorous in a talk which, again, strongly tied mathematical development with historical circumstance.

M. Norton Wise, Image: Thales & Friends (C)

And, strangely, I would also put Greg Chaitin’s and Doron Zeilberger’s talks in this class of historically-minded (not historical pure and simple) accounts. I only say “strangely”, of course, because they are both mathematicians and computer scientists, not historians. Chaitin’s talk really smacked of Poincaré’s injunction, in the sense that he supported his arguments for a less formal, less clean-cut mathematics proposing a future mathematics based on a narrative paradigm. And whether he was talking of Leibniz or defending the French “Collection Borel” against Laurent Schwartz’s attack on “novelistic” mathematics books, both in his talk and his statements during the discussion periods, Chaitin again and again defended the ways of past mathematicians who were not so intent on axiomatic, formal structure and rigor but more adventurous in their explorations and expositions. This was also behind the presentation by Doron Zeilberger, whose subject was Rabbi Abraham Ibn Ezra, a 12th century mathematician/storyteller with a style of mathematics which diverged very much from Euclidean orthodoxy. Both Chaitin and Zeilberger declared themselves fans of Archimedes (and Al Khwarizmi, I think) rather than Euclid, closer to a more computational model stronger on combinatorics than formalism. But although this was of course to be expected of computer-minded mathematicians, it was rather surprising to find them finding its justification in the past, an Ibn Ezra or a Leibniz.

Greg Chaitin posing his question on the future of mathematical theories: narratives or formal axiomatics? Image: Thales & Friends (C)

The fourth class I shall call cognitive. Brian Rotman, who is a mathematician by training and did research in the first years of his career (mathematical research I mean) spoke about the significance of gesture in the origin of mathematical understanding. Mark Turner, the co-founder (with Gilles Fauconnier) of the theory of conceptual blending, outlined the application of blending to mathematical concept formation. Martin Krieger, who unfortunately could not be at the meeting but nevertheless contributed his paper, examined mathematical discovery as everyday work, again making a case for mathematics as a human activity, both cognitively and socially determined, often culturally specific. And philosopher David Corfield – both his and Krieger’s published work adopt a similar stance of exploring “real” mathematics – applied Alasdair MacIntyre’s ideas to show how mathematical progress is a tradition-based inquiry. Interestingly, the speakers in the cognitive class were the ones that used the word ‘narrative’ or its variants least often. It seemed as if the four speakers who adopted the most strongly cognitive position, i.e. who considered mathematics mostly as a manifestation of human mental functioning, were taking narrative for granted, as a basic underlying mechanism of the human mind – as indeed it is.

Doron Zeilberger during his talk, with a text of mathematician/storyteller Rabbi Abraham Ibn Ezra’s Sefer HaMispar (Book of Number) in the background. Image: Thales & Friends (C)

T + F: Still, it’s interesting that they were the ones who used the term least often. But how accurate is your comment?

AD: Well, I wasn’t using a word counter, it’s just a very intuitive, very informal observation – for what it’s worth. But it does make sense: what you are most at home with, you feel less complied to explain – or even refer to. Interesting! As it is also interesting – and perhaps this is also explained by the same principle, being as it were the opposite sign of the coin – that the class in which the words story or ‘narrative’ and/or ‘story’ were most used was significantly, the class of talks by some of the strongest mathematicians in the meeting – in fact, four of these were by the strong mathematicians and one by a sort of mathematician manqué (me). This is the fifth, and last, of the classes: it consists of the talks which tried to explore, rather openly and more or less directly, the similarities of stories to mathematics, where mathematics in their case usually meant proof. All five began by more or less accepting the premise that there is such a similarity and that it is significant. All of them paid lip service to both directions of application, both the mathematics in stories and the stories in mathematics, though Christos Papadimitriou’s talk was the only one where more emphasis was put in the first case. He spoke enchantingly, and playfully, on the similarities of stories to computer programs, emphasizing the mathematical (=combinatorial) power in stories and also making the only real reference in the meeting to AI work on artificial story-generation – though, the truth be said, he was not exactly enthusiastic about the results achieved so far. I already spoke at length about Tim Gowers’s and Barry Mazur talks. Statistician extraordinaire Persi Diaconis’s talk was the one closest to these two. He began his talk by making the strong statement that “I can only work on a mathematical problem when I know it’s story”. And then went on, very interestingly, to equate this (i.e. a problem’s story) with “who cares about this problem”. And I think that this is most interesting because this comment forms a link between the talks where narrative was considered mainly as a formally describable structure and the more humanistically-minded ones, where it was seen mostly in context. With his comment Diaconis grounded narrative in human experience and the emotions, exactly as does Jerome Bruner in his seminal essay “Two modes of knowing”, in which he declares the narrative mode to be the only other alternative to the classificatory/deductive way of knowing, on which most science is based. Diaconis spoke of his need to discover the real story that gives rise to any mathematical problem on which he works, demonstrating his case with three papers from his opus: he picked three at random and then recounted, for each one, the real problem that gave rise to it, and how knowing the problem was instrumental for him in his search for a solution, both as motivation and inspiration.

And, finally, my own talk also belongs to this class, although it also had a toe or two in the previous one. I called it “the story of the proof is the proof”. In it I tried to show how proofs and stories are, in a certain sense, very similar beings.

Persi Diaconis, Image: Thales & Friends (C)

T + F: One final question: what’s next, what will follow the Mykonos meeting?

AD: Well, it’s very early to give details. But there will definitely be another one. And another, we hope. And another, and another… And more!

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