Thales + Friends

Alexander interviewed by Harris

Posted on 21 July, 2007 in category: Circles Disturbed Contributor Interviews, Interviews

The following interview between Amir Alexander and Michael Harris was conducted on July 21, 2007.  The interview as it appears here was transcribed and edited only slightly for clarity.

Harris: I like your way of viewing the history of mathematics in terms of guiding stories.  Whether or not they correspond to reality, they are certainly a product of their times; at the same time they provide a context for people either to fit in or not.

Alexander: It’s not so much a matter of everybody fitting those models so much as they become models. They become archetypes of what mathematics is or, in some cases, what a mathematician is. Obviously, Galois is a very extreme case of any biography which is why he’s so famous, but he becomes the model of what a mathematician really is. That’s what I’m looking at.

Harris: Had I known I was going to interview you, I would have made an effort to get a copy of your book. You are writing another one now. Is it going to be on these three periods?

Alexander: Yeah. My first book was specifically about the emergence of infinitesimal methods and what I call, exploration mathematics. It was a very specific historical study of a specific period and the emergence of a narrative of exploration, encouraging brave men to go out and possess distant lands. I then follow this narrative from its origins all the way to the actual mathematical techniques used. I find that the story emerges in the 16th Century in association with the voyages and it draws upon two different Medieval narratives. First the crusading narrative says ‘Let’s go back to the Holy Land and possess it,’ except that instead of going back to the source, this “crusade“ leads to the ends of the Earth. Second, it draws on the knight-errant romance, which dies contain this theme of going to the ends of the Earth but, unlike the crusade narrative, it has no element of possession. The knights-errant go to distant lands, but leave no mark, discovering only their own worth. It takes of both of these narratives to come up with the 16th century notion of exploration.

I start there and I see how explorers themselves used this story and how it shaped the geography of the land. I discuss how mathematicians were involved in the voyages, and focus specifically on Thomas Harriot, who was both explorer and mathematician. Overall it’s a very detailed historical study of one particular period.

Michael Harris, Thales and Friends (C)

Harris: Before you go on, I realize I should ask you about your training, how you got into history of mathematics, how you came to have the idea of approaching history in this way, and how you put together your “narrative of the narrative” from source material.

Alexander: I was an undergraduate in Jerusalem, where I studied history and mathematics. Later on I combined my two interests and started looking at mathematics historically, using some of the approaches I‘d learned in history.  I also saw this as something of an intellectual challenge because mathematics is the most abstract discipline. I wanted to see if I could make a convincing argument about how those seeming opposites relate to each other, influence each other.

Harris: It sounds to me as if you’re saying that it’s a challenge to do a materialist history of mathematics. It’s very interesting that you do that by way of narratives.

Alexander: One influence in my work that would be fairly obvious would be Foucault. Another one would be the anthropologist Clifford Geertz, who shows how people telling stories about themselves shape their own world.

Harris: And neither one of them was a historian.

Alexander: Foucault was, in a way, a historian. When you look at The Order of Things there are a lot of details one could argue with, but at the end of the day you really feel like he told you something very deep and very true about the order of knowledge in the 18th and 19th Centuries.

Historians are trained in looking at a lot of sources and reading them properly and accurately but what they are not always so good at is making sense of it. They have an ingrained suspicion of generalizations or anything that smells like it’s going to turn into philosophy.

Harris: Moving to the next project…

Alexander: The next project is more of a step back. The emphasis here is less on any particular period, but rather a gesture towards a general storied history of mathematics. I offer some broad strokes of what a history of mathematics, based on stories, would look like. As you heard in my talk, I suggest several different periods in the evolution of mathematics. I have the three periods that are most advanced right now: the 17th Century, which is the age of indivisbles; the 18th Century, in which mathematicians were viewed as “natural men” in touch with the inner harmonies of the universe; and finally those beautiful martyrdom stories of the 19th century, which appear in the 1830s practically, it seems to me, out of nowhere. They have no precedent among mathematical biographies.

Harris: The stories of the period of the Age of Discovery are well known and people learn about them still. In the 18th Century, the stories are rather different; it’s the Enlightenment and the French Revolution…

Alexander: They are different stories. I do have some examples of stories of exploration in the 18th Century and it’s very interesting how different they are from the stories of the 17th Century. Maupertuis leads an expedition to Lapland in Finland and the point is to determine the shape of the Earth; to measure the degree of latitude. (He wanted to) compare the degree of latitude at a higher latitude to one closer to the equator.  Another expedition went to the Andes of Peru to measure a latitude there. What I find interesting about that is that it’s a totally different notion of what exploration is. The mathematicians in the 17th Century were interested in the exotic other world, penetrating it through forests, rivers and mountains, and seizing its gold. When you go into the 18th Century, their focus is not to discover new facts about Lapland or Peru. What they want to do is show that wherever you go, you’re still subject to one rational system. That is, they want to show things are everywhere the same, whereas in the 17th century explorers wanted to show how different they were. It’s very much about spreading this rational grid around the word and making it homogenous rather than penetrate it to detect hidden secrets.

Harris: Let’s talk about some alternative narratives. In the 17th Century, another popular narrative centers around Fermat and Pascal. France is a point of stability in the 17th Century when there were wars everywhere else. It’s the period of stabilization of the centralist monarchy. Do Fermat and Pascal fit in with the exploration theme?

Alexander: My sense is that the tradition in France is very different, starting with Descartes. When Descartes was young, he experimented with indivisibles and infinitesimals and tied to mix various calculations with those. Later on, as he developed his more systematic philosophy he ruled against it. Infinitesimals for him were not sufficiently clear and distinct. They’re paradoxical, problematic, obscure, and he ruled them out of bounds of acceptable mathematics. Instead, he developed analytic geometry, which seems to me to express an alternative tradition to the calculus. Infinitesimal methods are based on a materialist intuition — they are a mathematics abstracted from what we feel that matter is like. It’s “mathematics from below,” — you start with materialist intuition and you abstract from there. That’s something that Descartes was very unhappy with. Mathematics was supposed to be a model of rationality. It was the anchor you could rely on in the face of the uncertainty of our intuitions. If mathematics itself is an abstraction from sensible matter, what is left? So Descartes created a method in which you start with mathematics and then define the world accordingly. That is, you take a mathematical expression, an algebraical expression, and then you draw what it is like in the world. So instead of mathematics being an abstraction of the world, he much preferred the world to be an expression of rational mathematics. For Descartes, the world is rational because mathematics is rational and the world expresses mathematical principles.

I’m not saying that all of French mathematicians were Cartesians, they were not, but these kinds of concerns about mathematics being independent of the world were much stronger in France than they were in England or Italy. There was a really wonderful exchange between Fermat and John Wallis in England. Wallis published it because he published everything. They understood each other because they both wrote good Latin but apart from that, they understood nothing about what the other was doing.  On the one hand, there was Wallis doing experimental mathematics. He had this idea that to prove something (you needed to say) ‘Try one case. Try two cases. Try three cases. You see a pattern here. Proof.’ He would have no problem dividing by zeroes and multiplying by infinity. It was all very wild, his kind of mathematics, but he reached the results so, that’s all he needed. Mathematics was an experiment, an exploration. ‘Go there and see what we come up with.’ That was perfectly fine for him. Fermat of course was appalled. This was not mathematics. He showed him his number theory and all the speculations leading to Fermat’s theorem, and Wallis says ‘Well, you can do that but, what’s the point?’ I find this symptomatic of the cultural divide between the kind of mathematics that was being created in France and the contemporary mathematics in England. My sense is that the Galilean school in Italy was conceptually close to the English school.

Harris: Were there exploration poems and dedications in other kinds of literature as well? Was the imagery more concentrated in mathematics?

Alexander: Those kinds of poems and that kind of imagery was very common in other fields of science during this period. ‘Natural philosophy is exploration’ was a very dominant theme of the scientific revolution. In some ways, I find that the mathematicians felt like they were being left out.  in the 16th century the bounds of the old world were shattered and there was a sense that a whole new and unknown world was out there. The reformers of knowledge, like Bacon and Galileo, promoters of Empiricism and Experimentalism say ‘well, let’s go out and do the same for the natural world. There are great continents of knowledge to be discovered elsewhere as well. We are going to do exactly what those explorers do. We will go there and look, and try and see what works.’ Exploration was a wonderful model for experimentation and empiricism. You don’t just stay home and speculate. You actually have to go and explore and see what you can find, with no preconceptions. Mathematics was left out because, according to the Euclidean ideal, all geometrical truths are in principle implicit in the assumptions. There are no surprises. Geometry was considered a known and completed field that left no room for exploration.

Harris: There were the classical problems that Federica is talking about. The trisection of the angle…

Alexander: And there was a renewed interest in trying to resolve them in this century and this sense of mathematics making new discoveries and not just repeating its old results, as it largely did for a long time previously. These new mathematicians looked at what was going on in other fields of knowledge and said ‘We can do that too, we can explore mathematics. We don’t have to stay trapped in those old abstractions. We can go and explore and find new things. We can go and map out what a geometrical surface is actually like and what a line is like and what a cylinder is like. We can explore and discover new surprising things about them that no one ever knew before. And, we’re willing to take risks that were considered illegitimate, because exploration is always risky.’ To some extent, you can look at it as mathematicians joining the band wagon of the new empiricism.

Harris: The method of indivisibles creates the internal structure of geometric figures. But it is seen as a process of discovering hidden treasures.  Triangles had to be imagined pre-existing in order to bear the weight of exploration metaphors.  You can’t explore a triangle-

Alexander: -unless you constructed it first. That is a problem. That is why it seems so strange that you can think about Harriot drawing a material picture of what the continuum is like after imagining the continuum. It’s very strange. I agree with you. But they saw mathematics as practically material. They thought that when they were doing these thought experiments they were actually discovering what the structure of the continuum actually is; what the structure of a triangle is. Not that they put it there. Not that they imagined it a certain way, but that they are looking into it and discovering it. They, including Harriot, used paradox as a tool of discovery because it helped them explore what the actual structure was. For example, Galileo in the Two New Sciences, used a device called the “wheel of Aristotle”. There are two wheels around a common center, and when the outside wheel makes a complete turn the inside wheel also makes one complete turn. But somehow, it makes a line that is longer than its circumference! How can this happen? Galileo concluded that there are gaps between the points that make up a line, and that the gaps are bigger on the inner line than they are on the outer one. So, through the paradox, you learn something about what a line actually is.

Harris: Newton was a narrative all by himself, I guess. People tell stories about the apple. I suppose they started telling that story during his lifetime. It doesn’t fit in with either the 17th or 18th Century. He’s such a singular figure that maybe he doesn’t have to fit in a pattern.

Alexander: I’d like to fit him in. But when I talk about the 18th Century, I talk very little about England and Italy which were central in the 17th century. The dominant tradition is certainly French.

Harris: That was true of mathematics in general. It was not a particularly creative period in England.

Alexander:  The standard story, that may or may not apply, is that the English mathematicians were so devoted to the glory of Newton that they were slow in accepting continental methods. I’m certainly aware that when I talk about dominant stories, although I emphasize time period, they clearly have a geographical boundary to them.  What I hope to do is to have some sort of scheme of what this kind of history will look like. I don’t mind qualifying it geographically or tying-in other narratives.

Harris: The first obvious point about the 18th Century is that your chapters are almost exclusively about France.  Even Euler is seen from the standpoint of Condorcet.  How important was the image of the natural man outside France, for example in Catherine the Great’s St. Petersburg?  Was Euler appreciated as a natural man?

Alexander: Good question. The Russians were certainly Francophiles, and the Russian aristocracy all spoke French, But I can’t actually say that he was viewed in that way. By all accounts he was a very down-to-earth, practical man; not playful. Just a solid Swiss-German bourgeois citizen. I would imagine he would not think of himself in any sense as a romantic, natural man. But for those who, like Condorcet, wished to view him on this manner, he would in some ways fit the mould — especially in his suspicion of the sophisticates and all of their refined manners. The main guy for the story of the “natural” mathematician is clearly d’Alembert, who left a very long record of non-mathematical writings. In his “Preliminary Discourse” to the Encyclopedie he also wrote a concise description of what he thought mathematics was about, and 18th Century mathematicians really followed it (his view) very closely. I would say he gives a very good description of a contemporary understanding of what mathematics is: an ultimate abstraction of nature that never leaves nature. Algebra is the final frontier of mathematical abstractions, but still remains anchored in the actual world. Leaving the world would be a disaster that would deprive mathematics of content.

Harris: My impression is that the calculus of variations is the typical field of mathematics in the 18th Century: Euler, d’Alembert, Lagrange.… Is the Principle of Least Action part of the narrative of the natural man or of the narrative of the principle of nature?

Alexander: I would say the latter. The assumption is that of the mathematical universe. The reigning assumption in the Enlightenment, among those who supported mathematics, is that mathematics is about the world because the world is essentially mathematical. For Mauperuis, the Principle of Least Action is the fundamental underlying principle of what the world is like.

There is a lot of opposition to mathematics as well.  Buffon and Diderot were both very interested in mathematics early on, but concluded that it was too abstract. They thought mathematics leads one so far from the real world and that one can never get back. Diderot has this beautiful image of mathematicians standing on their high mountains; when they try to look down, all they see is mist below, which prevents them from finding out anything about the real world.

In his ‘Letter on the blind’, Diderot talks about the blind mathematician Nicholas Saunderson, a Cambridge professor who was blind from birth and who was a wonderful mathematician nevertheless. Diderot uses him to makes the argument that the greatest mathematician is essentially blind because he doesn’t need the world, he builds his own structures. Saunderson is a brilliant mathematician, he even teaches optics in Cambridge, but he knows nothing of what the world is like because he’s never seen it. He doesn’t even know what seeing is because he’s been blind from birth. For Diderot mathematics is about building castles in the air, in isolation from the world, and is therefore useless for our understanding of the world. In the “Preliminary Discourse” d’Alembert agrees that mathematics should be about the world, and responds to Diderot by saying ‘no no, the world is mathematical, and mathematics is always about the world.’ He also adopts Diderot’s image on the mathematician standing on ‘the high mountain’ and looking down. Below is the world, and if look down and gradually you can see the harmonies that govern it. Those harmonies are visible to a mathematician precisely because he is so high up on the mountain.

Harris: I have one more question about this period. Were other scientists praised for being “natural men”?  I‘m thinking of Benjamin Franklin, for example, who was celebrated in France.

Alexander: Perfect! I hadn’t thought about that but it fits very well with the story I’m telling. That science was about the world and scientists were the ones who had a natural understanding of the world. The mathematical twist to it was the claim that mathematics also belonged in that company because mathematics is also about the world.

Harris: When I was first interested in mathematics, I was exposed to Bell’s Men of Mathematics. Of course, I didn’t know that this had been going on for so many years. Nobody else I knew was interested in mathematics or these stories but romantic stories of Abel and Galois seemed appropriate.  Did they present these characters just because we were teenagers? Because they appealed to us?  Why did they not talk about the other mathematicians?

Alexander: I think those were the stories that appealed to everyone. These stories have an interesting genealogy, beginning with the deaths of Abel and Galois to their canonization by Bell. Oddly enough, Galois was celebrated as a hero when he died not for his mathematics but because he was a hero of the Revolution. He had all those radical friends who railed against how he was an oppressed genius, how his mathematics was ignored by establishment mathematicians. His close friend wrote an obituary for him shortly after he died and he called him ‘a martyr to his genius.’

Harris: There was an Abel centenary in Paris a few years ago.  I believe they told the story of how he came to Paris and wanted to meet with mathematicians.  They didn’t have time for him but four members of the Academy nevertheless wrote for him.

Alexander: He was a rising star. It’s true. He did not have a lot of success in Paris but he met interesting people. He was friends with the scientist-politician Francois-Vincent Raspail, who years later told in a speech in the Chamber of Deputies how this poor mathematician was ignored in Paris and had to literally walk back home to Norway.

Harris: It’s significant that they didn’t mention that part. They did mention that he published his articles.

Alexander: Legendre wrote admiring letters to him in his final years. He could have had a position in Berlin years before except that he wanted to stay in Norway. Crelle never understood why on Earth he would want to go back to that snow-bound no-man’s land to Christiania (modern Oslo). Christiania was very small and very poor, but people there tried to help Abel as much as they could. Other professors added to his salary from their own pockets because they recognized he was a very special talent. At that point, they could not get him a permanent chair, but the man was just 26. And then Crelle was working very hard the whole time to get him a position in Berlin and he succeeded – but too late for Abel. So Abel was not a tragic figure in his own eyes at all. He dies just when so many opportunities are opening up for him and that is perhaps a personal tragedy. But he was not alienated or marginalized. Nevertheless, the minute he died, the story of the persecuted genius immediately emerged.  People were pointing fingers both in Paris and in Norway about “who killed Abel”. And ‘Why didn’t he get a permanent position here?’ It was kind of sad because the people who were being accused were his friends who helped him. The people making the accusations were the people who’d never heard about him and never cared, but in his death had found a hero.

Harris: Why was the romantic stereotype so effective all of a sudden in the 1930s when Bell wrote his book and at least through the 1960s when I was exposed to it?

Alexander: In France, the Galois story was already canonized by that point. It was a founding myth of official French science. That happened in the late 1800s when Galois became this misunderstood hero. The inspector general of the Ecole Normale himself wrote the most authoritative biography of Galois to this day. This was a great irony because Galois didn’t want to go to the Ecole Normale. He was kicked out of the Ecole Normale. He wrote horrible things about the Ecole Normale. Two generations later, he becomes the emblem of the greatness of the Ecole Normale and an icon of French science. During World War I the Belgian George Sarton, founder of academic history of science came to the United States. He knew the story from the Continent,  and he started publishing in the US and he spread the story of Galois. Bell adopted it from him, and then it really became widely disseminated. So the popularity of the story actually dates from long before the 1930s and, from my experience, it continues to this day.

Harris: It still works. It’s kind of a paradox because the romantic stereotype is still effective and yet very few mathematicians see themselves as misunderstood revolutionaries. Some do but they are usually crackpots, outside the official structure. Mathematics is a highly structured activity.  A mathematician may be misunderstood by some people in the establishment, but is rarely misunderstood by everybody. Then there is also the myth of the suffering romantic artist.  The dynamics in mathematics are not the same, but it’s pretty clear from your description that the romantic artist and the romantic mathematician arise at the same time.

Alexander: I think they do. Chopin. Byron. The tragic story is not original at all. It’s fairly standard in a mathematical version. The interesting thing is that it aligns mathematics with activities like music, art, poetry and so on and not with physical science which is what the previous generation thought about.  The Ecole Polytechnique tradition saw mathematics as an extension of physics and the study of the natural world.

You really don’t find that kind of tragic imagery among scientists. Einstein is viewed as an eccentric, but not a tragic one. The imagery of the scientist, from d’Alembert to Einstein, is that of the eternal child who is still curious about simple things and looks at the world with big eyes. Einstein is a natural man, not corrupted by social refinements. That’s why you have those pictures of him making funny faces. He’s just a kid playing around, expressing his natural wonder at the world. It’s a very different imagery than the ones you find associated with the tragic mathematicians.

Harris: Having established that the romantic myth retains its attraction and provides motivation for people to enter the field in the form of role models, I would consider the theme of self-consciousness.  It certainly fits with the broader cultural themes attached to names like Marx, Darwin, Nietzche, Freud, and with the mathematical theme of the foundations crisis. Cultural history is a little bit of a risk. You see what you want to see. In psychoanalysis, if the heroes are tragic figures, it may be because they are mentally ill. You have Cantor and Nash, Gödel is borderline, Turing and Wiles are presented as obsessive. These are the images that sell popular books.

Alexander: Herbert Mehrtens wrote a book about it which, unfortunately, is in German. It’s specifically about modernist mathematics and talking about the early 20th Century. He had an argument along the lines you were saying, seeing the foundational crisis in mathematics in the context of all the cultural fault-lines of the early 20th Century. When you say that those 20th Century mathematicians were not exactly tragic but were rather insane, you see a possible connection there between self-consciousness and…

Harris: and Freudianism.

Alexander: (That’s) something to look at. I looked at them as repetitions, as versions of the Galois story but I am willing to consider other options. I’m convinced that there is a strong element of continuity there. And that article I mentioned earlier today in the New Yorker about Perelman and Yau. I was very much impressed with that story which could be very unjust to Yau. It basically seemed to replay the old drama of Galois and Cauchy or Abel and Cauchy, the young genius who gets swept by the wayside, etc. Except that Perelman did get the Fields Medal!

Harris: The story has very much to do with traditional Russian notions of personal purity. He turned his back on the world. He was not trying to improve the world. He was not trying to make a stand.

Alexander: He’s not peculiar in any other way. Perelman in the article seems perfectly happy. He chose this kind of life. He lives with his mother at that institute in Moscow and he seems perfectly satisfied in that life. There’s certainly nothing tragic about that. That’s certainly a difference. He cares about the truth. He doesn’t care about worldly rewards. And then, on the other hand, you have his nemesis who cares very much about worldly rewards, credit, power and so on. In basic ways, it does follow the old paradigmatic story.

Harris: This is a story written for an American audience but it concerns two figures who are not really present in the 19th Century mathematics- a Soviet-trained mathematician and a Chinese mathematician. There’s a limit to which these stories can really be exported. I have no idea if there are any figures comparable to any of the ones you mentioned in the Chinese literary tradition.  I don’t know what stories are motivating contemporary Chinese mathematicians.

Alexander: I don’t know what they are either but I suspect they are very different. This story in the New Yorker is told from a Western perspective and uses Western tropes.

Harris:  You write: “The correct results, which for Lagrange were both the purpose of Analysis and the ultimate guarantee of its viability, were for Abel merely a puzzling aberration”. Where did Abel get this idea and how do you compare it to the shift from natural history to analysis? How did this actually get realized in the case of Abel?

Alexander: Where did he get this notion? I’m sure he was influenced by Cauchy. Even while he was in Paris, in his letters, he complains about his treatment but said that he is the only one who truly understands what mathematics is. The only one who was a true mathematician there.

Harris: So how did he know that?

Alexander: He read everything. Cauchy was publishing like mad.

Harris: How did he know when he read Cauchy that he was the one who really understood everything?

Alexander: I think part of the mystery here is that you have a break with tradition, and this means that you cannot see a direct continuity with what went before. Where did Abel get the idea that 18th Century mathematics was dull mathematics? I don’t know. It’s an interesting question but it’s not the question I am ultimately interested in. Overall, you can see a general break, a clear change of directions in the understanding of what mathematics is about in a large group of mathematicians in France and then in Germany; along with this change comes a clear break in the stories that are being told about mathematics and mathematicians. Ultimately, more important for me than causality is the isomorphism between these stories: it shows that this kind of mathematics is intimately related to these kinds of stories. That is really what I focused at; telling a story that is not necessarily continuous but rather has breaks in it, that the breaks occur both in the stories and in mathematics.

Harris: Something big happened around 1800 that shook up Europe quite a lot. We can certainly see how the romantic hero arises in that situation but what’s surprising is how this romantic hero gets associated with the notion of mathematical rigor.

Alexander: In a book called In Bluebeard’s Castle George Steiner talks about this period as the generation that is after Napoleon. They were very bored after their fathers’ great generation. This produced a whole generation of alienation. He talks about the poets and how they try to look beyond this mundane, boring, depressing world. They tried to look for purity and purpose away from this world, in an alternative universe of beauty, poetry, and perfection. And I think there is something of that movement in mathematics as well. True mathematics resides in a beautiful alternate reality that is no longer part of this world. It is over there, ruled only by its pure self-referential standards, not by the limitations of our corrupt reality. So there is certainly a parallel there between poetry and literature and mathematics. The emphasis on mathematical rigor is a result of the conception of mathematics as self-referential, that mathematics is not part of the world. In the 18th century mathematics was legitimized by the fact that it correctly described the world, but in the 19th century it has to be true only to itself, not to the world around us. How do we know if a mathematical statement is true to itself? Only by following its own inner standards rigorously, because there is nothing else to fall back on. In other words rigor becomes really important because that is the only standard that you have – perfect, internal coherence. Whereas, as long as you think mathematics is abstracted from the world, rigor might be nice, but it’s not essential. The big movement is taking mathematics from our world to its own world, and the emphasis on rigor follows from that.

Harris: This morning I made the hypothesis that the alienating nature of mathematics education explains the persistence of the image of insane mathematicians, marginal figures in popular culture; the idea that people who are not mathematicians have an image of mathematicians as unstable figures.

Alexander: But you could turn the argument on its head and say that the reason why mathematics is taught in this way is because a certain image of mathematics prevails. The way mathematics is taught in college is very, very difficult and unnecessarily so. Students are basically asked to forget everything they know or think they know about math, and build an alternative new world. Students come in with a certain idea of mathematics. They know that a derivative is related to the slope of a graph. An integral is the area under a curve, and so on. When they come into college, they have to start everything from scratch. Forget about slopes and areas, instead, let e be greater than 0 and go from there. Bright students can follow a professor’s proof step by step, but have no idea why he’s doing it because they are given no context – they are operating as if in a vacuum.

I think that comes from that particular understanding of mathematics which has its origins in that period that we’re talking about. Mathematics exists I an alternative world operating according to its own rules. A student’s materialist intuitions are therefore useless and he/she should start from scratch. Going back to your question, perhaps it works both ways. This flawed education causes people to view mathematicians and strange and otherworldly, but also the view itself produces the flawed education.

Harris: So, you’re thinking that the people who teach mathematics have internalized a view of themselves as separate from the world.

Alexander: Of their field, maybe not personally of themselves.

Harris: Well, yes, but their intellectual life is not connected to the world, it is not subject to the vicissitudes of material life.

Alexander: That’s mathematics.

Harris: That certainly is an attraction.

Another thing I mentioned as a possible periodization is the rise of conjecture and internationalization as a source of self-consciousness in mathematics.  When Hilbert stated his 23 problems in 1900, it was consciously an international research program in the setting of the International Congress of Mathematicians.  The act connects in several ways with self-consciousness more generally: you analyze the field and try to determine where it’s going to go on the basis of the incorporated self-consciousness with respect to foundational questions.

Alexander: So you are saying the very fact of setting up research, the self-reflection of the field is indicative of a larger move towards self-reflection in perhaps other cultural areas?

Harris:  Orientation to the future operated in mathematics in the other periods of exploration.  In that sense, it is consistent with the tradition as well.

Alexander: Those great problems remind one a little bit of the famous ancient problems, like the trisection of the angle or the squaring of the circle. It’s an interesting question seeing how those people actually talked about those questions in antiquity and how they saw them. Was there a research program to resolve the great remaining riddles? I don’t think there was the same sense of progress in mathematics in antiquity, of this open-ended research program. They basically felt that they knew what they were doing but there were few of those problematic islands which were tough nuts to crack.

Harris: There was a very self-conscious program for modern music.

Alexander: I agree, that’s the place to look – music, architecture, art. I’m really interested in Peter Galison’s analogies between modernist architecture and the Bourbaki project. It makes a lot of intuitive sense.

Harris: I had some other images regarding 20th Century mathematics. It’s harder to compare them to what you call “tropes” in popular culture.”  I already asked what “popular culture” meant in the 16th Century, but you more or less explained that by saying that in travel narratives, for instance, people had a shared culture that they could read.

Alexander: Yes, they could. In the 16th century those travel stories were enormously popular. It seems like every explorer, from admiral to ship’s boy published a book of their adventures, their discoveries, and all the hazards and the hardship they faced. This was very widespread in that period.

Harris: Bourbaki provides a new image. I don’t know whether Hilbert and the Bourbaki really should be separated as systematizers. They fit with the Puritanical impulse that you have already identified in the 19th Century, but are by no means tragic figures. At the opposite end, people like von Neumann and Ulam are in the middle of a completely different sort of adventure. Ulam has written an, I guess, not–so-popular book called The Adventures of a Mathematician.

Alexander: And then there is Turing, who is a tragic figure though. I recently saw another description of Turing’s story very much in that vein.

Harris: We’ve still got a few more topics but I haven’t got to the 21st Century yet. There is one survival of Puritanism I identified. I don’t know to what extent it goes back to the early 19th century but there is strong and irrational disapproval of giving up any research time to pursue other activities. This is something of which I was already aware as a graduate student. Students were gossiping about famous mathematicians who had accepted administrative positions — obviously because they were no longer able to do research. The only reason you would do anything other than mathematics is that you’ve lost it. There’s the myth that you can’t do mathematics after you’re 30. I don’t know when that got started.

Alexander: It started right about the time of Galois and Abel. Euler published new work well into his 70s, as did Lagrange and Legendre. The myth of youth, I think, is related precisely to those young martyrs, whose youth was a sign of their purity. They’re young. They’re pure. They’re not contaminated by the world, and as a result they are the ones who can see into the true world of mathematics. When you grow older, by implication, you become too corrupted by the world, by power, by interests, by children, by family, by all those things that pull you down. You’re no longer pure enough to access the beauty of mathematics. My sense is that this emphasis on youth is related to purity; the purity of mathematics and the purity of mathematicians.

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