Harris on mathematics

While in Paris for the 'Impact of Categories' conference in October I met up with Michael Harris, an American mathematician based at Jussieu. He's a number theorist who has worked with Richard Taylor, the mathematician who helped out his PhD supervisor Andrew Wiles fixing the holes in the proof of Fermat's Last Theorem. Harris has been asked to contribute to 'The Princeton Companion to Mathematics', which is being edited by Tim Gowers. It should be quite a book when it appears. (No website for it yet, but several contributions can be found with Google.) Harris has made available a draft of his essay “WHY MATHEMATICS?” YOU MIGHT ASK . I like it very much, not only because it discusses my work, but also because it lends support to the notion that philosophers should be thinking about mathematics as a body of ideas, rather than as a body of truths.

Gowers' Survey articles and general lectures are worth reading too.

Category theory and philosophy

The draft of a paper by Barry Mazur on what category theory tells us about identity is available: When is one thing equal to some other thing. Philosophy's encounter with category theory has not been its happiest. The trouble is that, as Mazur puts it, category theory shines its spotlight in a different direction to the traditional foundational languages. Looking through its lens it leads us to ask different questions about mathematics. The closest one will come to a direct clash is over identity or sameness. But many philosophers find it hard to understand category theory's position on this as first they want to know what 'stuff' goes to make up the things we are wondering are the same or not. Besides Mazur's piece, there's a very good account of the category theoretic position by Steve Awodey. Awodey is replying to the philosopher Geoffrey Hellman, so if you are coming from an Anglophone philosophical starting point, this should be especially helpful.

A recently establish publisher, Polimetrica, is bringing out a philosophy book at the start of next year - What is Category Theory? - including a contribution by myself. On their catalogue you will also find Generic figures and their glueings - A constructive approach to functor categories by Marie La Palme Reyes, Gonzalo E. Reyes, Houman Zolfaghari:

This book is a "missing link" between the elementary textbook of Lawvere and Schanuel "Conceptual Mathematics" and the much more advanced textbooks such as the one by MacLane and Moerdijk "Sheaves in Geometry and Logic."

Avoiding the complicated, fully fledged notion of a Grothendieck topos, whose very formulation presupposes a good deal of mathematical experience, this book introduces topos theory through presheaf toposes, i.e., readily visualizable categories whose objects result from glueing simpler ones, the "generic figures". Several phenomena which distinguish toposes from the ordinary category of sets appear already at this simpler level.

Six easy to understand examples accompany the reader through the whole book, illuminating new material, interpreting general results and suggesting new theorems.

This book is aimed (via appropiate examples) at a beginner mathematician or scientist or philosopher who would like to take advantage of the richness of presheaf toposes to prepare himself or herself either for further study or applications of the theory described.

Of course, after categories come bi-, tri-, and tetra-categories, right up to omega-categories. There are even Z-categories, where Z stands for the integers. (The spectra of algebraic topology are examples of Z-groupoids.) The point is that sameness becomes a more subtle question as we climb the ladder. The most noticeable manifestation of this is where one says two categories are the same if they are equivalent. This means that there are a pair of functors going in opposite directions between them, such that their composites are not equal to the identity functor, but that there is a natural equivalence between them. This idea shows up in many fields by the name Morita equivalence (see, e.g., A bicategorical approach to Morita equivalence for von Neumann algebras). It's a little like saying that London and Oxford are more alike than London and Paris, because I can walk to Oxford from London and then retrace my steps, and while this isn't the same as resting in London, there's a continous set of paths mediating between the null journey and the Oxford round trip.

You can read about this in chapter 10 of my book and in this paper, in Yuri Manin's Georg Cantor and his heritage, and in 101 places on John Baez' site.

The aim of mathematics

There's often much to think about in even brief pieces of mathematical writing. Take the following extract from The algebraic meaning of genus-zero, where Terry Gannon claims:

At the risk of sending shivers down Bourbaki’s collective spine, the point of mathematics is surely not acquiring proofs (just as the point of theoretical physics is not careful calculations, and that of painting is not the creation of realistic scenes on canvas). The point of mathematics, like that of any intellectual discipline, is to find qualitative truths, to abstract out patterns from the inundation of seemingly disconnected facts. An example is the algebraic notion of group. Another, dear to many of us, is the A-D-E metapattern: many different classifications (e.g. finite subgroups of SU₂, subfactors of small index, the simplest conformal field theories) fall unexpectedly into the same pattern. The conceptual explanation for the ubiquity of this metapattern—that is, the combinatorial fact underlying its various manifestations — presumably involves the graphs with largest eigenvalue |λ| ≤ 2.

Likewise, the real challenge of Monstrous Moonshine wasn’t to prove Theorem 1, but rather to understand what the Monster has to do with modularity and genus-0. The first proof was due to Atkin, Fong and Smith [25], who by studying the first 100 coefficients of the T_g verified (without constructing it) that there existed a (possibly virtual) representation V of M obeying Theorem 1. Their proof is forgotten because it didn’t explain anything.

By contrast, the proof of Theorem 1 by Borcherds et al is clearly superior: it explicitly constructs V = V natural, and emphasises the remarkable mathematical richness saturating the problem. On the other hand, it also fails to explain modularity and the Hauptmodul property. The problem is step (iii): precisely at the point where we want to identify the algebraically defined T_g’s with the topologically defined J_g’s, a conceptually empty computer check of a few hundred coefficients is done. This is called the conceptual gap of Monstrous Moonshine, and it has an analogue in Borcherds’ proof of Modular Moonshine [3] and in H¨ohn’s proof of ‘generalised Moonshine’ for the Baby Monster [15]. Clearly preferable would be to replace the numerical check of [1] with a more general theorem.

Even with the advances Gannon describes in the paper:

...the resulting argument still does a poor job explaining Monstrous Moonshine. Moonshine remains mysterious to this day. There is a lot left to do — for example establishing Norton’s generalised Moonshine [24], or finding the Moonshine manifold [14]. But the greatest task for Moonshiners is to find a second independent proof of Theorem 1. It would (hopefully) clarify some things that the original proof leaves murky. In particular, we still don’t know what really is so important about the Monster, that it has such a rich genus-0 moonshine. To what extent does Monstrous Moonshine determine the Monster?

Clearly you have to see the original paper to understand what he's talking about, but already there are some clear opinions about the aims of mathematics and the means to achieve these ends. Gannon is a very good expositor, see, e.g., Monstrous moonshine and the classification of CFT.

Bourbaki is presented as opposed to his view of the aim of mathematics, but perhaps this understanding of Bourbaki is gleaned only from their textbooks. Leo Corry, the historian of mathematics whose views I have been discussing in previous entries, has written extensively about aspects of Bourbakian philosophy in his Modern Algebra and the Rise of Mathematical Structures. Remember it was Andre Weil's views about how mathematicians were best equipped to write history of mathematics that form the departure point for Corry's paper. An example of a Bourbakian history is Jean Dieudonne's A History of Algebraic and Differential Topology, 1900-1960. Presumably, for some people a book like this answers the problem of how to write a history of a research programme which runs over decades, and involves a host of different contributors. Given its scale it could hardly deal with individual research philosophies, or the practices of institutions, even in the unlikely event that the author would have wanted to do so.

By the way, returning to reality and 'reality', the other day I was helping my 11-year-old daughter with some mathematics homework concerning adding volumes in litres and millilitres. The advice to the helper began: In 'real life' we often have to deal with measured quantities. What on earth are those scare quotes doing there? Is there a hint here of a pristine, unspoilt life in which we didn't have to deal with measured quantities? Perhaps life in Eden, before we were forced out and had to build things like Arks.

Mathematicians' histories and historians' histories

Returning to Leo Corry's thoughts on the history of mathematics, I think it is important to recognise that a corrective against simplistic, triumphalist story-telling was very necessary. Historians of science were many years ahead of their counterparts in mathematics, and have undertaken their own corrective work with zeal. A good example of this is one mentioned by Corry. How many times have you read about how the 1919 expeditions to measure the bending of light by the sun confirmed Einstein's theory of general relativity? It all sounds so simple: Einstein predicts a phenomenon which runs against Newtonian theory; scientists observe the phenomenon; and, anyone with a shred of rationality gives up on Newton's theory. However, historical research into this episode paints a very different picture, or perhaps I should say it paints very different pictures. The best known of these questions the way the data was selected (see John Waller (2003). Einstein's Luck: The Truth behind Some of the Greatest Scientific Discoveries. Oxford, England: Oxford University Press). However, in Matthew Stanley's “An Expedition to Heal the Wounds of War”: The 1919 Eclipse and Eddington as Quaker Adventurer, Isis, 94 (2003), 57-89, we read that:

The 1919 eclipse expedition's confirmation of general relativity is often celebrated as a triumph of scientific internationalism. However, British scientific opinion during World War I leaned toward the permanent severance of intellectual ties with Germany. That the expedition came to be remembered as a progressive moment of internationalism was largely the result of the efforts of A. S. Eddington. A devout Quaker, Eddington imported into the scientific community the strategies being used by his coreligionists in the national dialogue: humanize the enemy through personal contact and dramatic projects that highlight the value of peace and cooperation. The essay also addresses the common misconception that Eddington's sympathy for Einstein led him intentionally to misinterpret the expedition's results. The evidence gives no reason to think that Eddington or his coworkers were anything but rigorous. Eddington's pacifism is reflected not in manipulated data but in the meaning of the expedition and the way it entered the collective memory as a celebration of international cooperation in the wake of war.

Either way the old-style history was thoroughly misleading. For Corry there are extremely important lessons to learn from the new historians' history:

A complex mixture of social, institutional, political and cultural circumstances (all of them fully legitimate and human) stand at the background of this interesting chapter in the history of twentieth-century science. They must all be taken in consideration, together with the purely scientific issues involved here, if we want to make full sense of the impressively quick and sweeping process of acceptance of Einstein's new theory on the basis of the astronomical observations of 1919.

Being this the case, one may for a moment conjecture about possible scenarios that might have ensued, had the results of the expedition not been as readily accepted as they were (under the active influence and authority of Eddington and Dyson, and for the many reasons that guided their efforts in this direction) or if the results had showed a preference for Newton's theory over Einstein's. These are by no means imaginary scenarios and they could have easily materialized had the circumstances been different. It is important to remember, in this context, that once the measurements reported by the expedition (and by implication, the confirmation of Einstein's theory with the concomitant refutation of Newton's one) were published in British, (and later in German) newspapers, Einstein was immediately catapulted into world fame. He thus turned into a cultural icon that embodied for decades to come the ideal of the scientist as a secular saint working in isolation from the rest of the world. The events of 1919 played an important role in shaping much of the course of physical science in the twentieth century as well as of its public perception.

Corry describes the difference between the old and new forms of history of science in literary terms:

Science as Drama/Greek Tragedy
a) We know what will happen: drama arises because we know that it will happen
b) Human emotions, ideas, and behavior as products of, or responses to the unfolding of the human essence
c) Universal elements of the human situation and fate

Science as Epic Theater (Brecht)
a) "Things can happen this way, but they can also happen in a quite different way" (Walter Benjamin)
b) Human emotions, ideas, and behavior as products of, or responses to, specific social situations
c) Behavior people adopted in specific historical situations

Plurality, difference, contingency are what mark modern history of science. Things could by now be so very different. But have historians gone too far? Is it not to accord too small a role to the world to claim, for instance, that "The events of 1919 played an important role in shaping much of the course of physical science in the twentieth century as well as of its public perception."?

Now when we turn to mathematics, you might think the contingentist historian would have an easier job. Even if the world as mediated through physics is taken to play an important role in determining the mathematics we study, and even if we take there to be a single correct logic for mathematical reasoning, there's still plenty of scope to believe that mathematics could have gone in very different ways. But this brings us back to the issue of bumping into mathematical reality. Sometimes it feels as though whichever way a mathematician tries to move they can't help but knock into something. The issue for me is how to write history of mathematics which is historically sophisticated, and yet alive to this experience. A good test case would be to write the history of n-categories. Baez and Lauda have begun one here which treats the physicists' input into higher-dimensional algebra. I should imagine that historians would find this a little too much of a 'Royal Road to Me' (see Corry again). But let's turn the onus around. Are they able to treat the decades long development of a body of ideas, which involves scores of mathematicians from many institutions and countries? Or would this have to represent too great a concession to old-style history in what to them would be an arbitrary delineation of a portion of mathematical activity?

Assorted

I wonder how far this dichotomy runs:

THE DICHOTOMY BETWEEN STRUCTURE AND RANDOMNESS, ARITHMETIC PROGRESSIONS, AND THE PRIMES, TERENCE TAO, math/0512114

Abstract. A famous theorem of Szemer´edi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (low-complexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemer´edi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power of this dichotomy is evidenced by the fact that the Green-Tao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemer´edi’s theorem. In this paper we survey various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different.

I began to treat the oppositional pair lawlike/happenstantial here, but there's clearly much more to be said.

Another quotation from MacIntyre which is very relevant to the themes of How Mathematicians May Fail to be Fully Rational. This one is from The Essential Contestability of Some Social Concepts, Ethics 84(1) 1-9, 1973 (available on JSTOR)

Consider...the continuing argument between Kuhn, Lakatos, Polanyi, and Feyerbend, an argument in which what is at stake includes both our ability to draw a line between authentic sciences and degenerative or imitative sciences, such as astrology or phrenology, and our ability to explain why "German physics" and Lysenko biology are not to be included in science. A crucial feature of these arguments is the way in which dispute over the norms which govern scientific practice interlocks with debate over how the history of science is to be written. What identity and continuity are recognized will of course depend on what side is taken in these latter debates but since these debates are so intimately related to the arguments about the norms governing practice, it turns out that the dispute over norms and the dispute over continuity and identity cannot be separated. (p. 7)

Now this was written at the end of that fascinating period in the philosophy of science when the protagonists fought tooth and nail to establish their representation of science through historical case studies. Since that time there has been a steady trend of separation between philosophy of science and history of science. I don't think this is a happy state of affairs.

History and philosophy of mathematics have never come that close together, despite Lakatos's efforts. At the Mykonos conference at which I first read my paper, subtle tensions emerged which need to be treated. Leo Corry (historian) made a distinction between historians' history, on the one hand, and fictional mathematical writings (e.g., Uncle Petros and the Goldbach Conjecture) and mathematicians' histories (by, e.g., Bourbaki), on the other. Poetic licence is dangerous when a history is presented as factual. But there's a dissatisfaction that runs the other way. Barry Mazur (mathematician) suggested that we don't yet have a good history of Euclidean mathematics, despite the reams of pages written by historians on this period. So here's the question: Is there a way of writing a truthful history of mathematics which would fully satisfy the mathematician? (Both Corry and Mazur's papers are available here.)

From a very good blog in the field in which I an currently working, John Langford gives his vision of what the Web could do to facilitate research in machine learning. His remarks seem to me just as applicable to mathematics.

Mathematical reality

Contemporary differential geometry is dramatically broadening its horizons. For a taste see 'Non Abelian Differential Gerbes' DG/0511696, "We develop a differential geometry theory of non-abelian differential gerbes over stacks using Lie groupoids", and 'Higher Gauge Theory' http://math.ucr.edu/home/baez/higher.pdf (also ArXiv: DG/0511710) "We describe a theory of 2-connections on principal 2-bundles and explain how this is related to Breen and Messing’s theory of connections on nonabelian gerbes".

What is very noticeable is the number of routes which seem to be leading in the same direction. One shouldn't underestimate, however, the work of reconciling different viewpoints. Great rewards are due for work which, although it proves nothing new, performs this reconciliation well. In his 'Racah - Wigner quantum 6j Symbols, Ocneanu Cells for AN diagrams and quantum groupoids' (hep-th/0511293), R. Coquereaux claims:

Our purpose in this paper is very modest. Indeed, all the objects that we shall manipulate have been already introduced and studied in the past, sometimes long ago: 6J symbols, quantum or classical, are considered to be standard material, cells and “double triangle algebras” have been invented in [28], [31] and analyzed for instance in [5], [37], [14] or [39], finally, quantum groupoids are studied in several other places like [7], [25] or [26]. However, it is so that many ideas and results presented in these quoted references are not easy to compare, not only at the level of conventions, but more importantly, at the level of concepts, despite of the existence of the same underlying mathematical “reality”.

Now, why the scare quotes? There are two types of philosophical position that require them. One is a form of idealism which would want scare quotes to be used at the mention of any form of reality. Even the reality of chairs and tables needs putting into question. This is presumably not what Coquereaux believes. What I take it that he is implying is that just as there is a physical world which places severe constraints on what we can and can't do - we can swim in a river, we can't walk through trees, we can't jump up 10 metres, etc. - there is something not so very different which forces mathematicians to work along similar lines, even if this is not always obvious, and this something is not merely logic. In this quotation of Connes, again we see 'mathematical reality' in scare quotes. Again, mathematicians often meet each other in the same places:

whatever the origin of one's itinerary, one day or another if one walks long enough, one is bound to reach a well known town i.e. for instance to meet elliptic functions, modular forms, zeta functions

There is a danger in confusing this mathematicians' realism (remember not all mathematicians are convinced that this convergence is so important - Zeilberger's Opinion 49, Ruelle's 'Is Our Mathematics Natural?', Bull. AMS 19, 259-268, 1988), with what is at stake when analytic philosophers of mathematics take up realism. Here there is no interest in specific concepts like 6j symbols or elliptic functions. Where the mathematicians will be able to point to concepts that although consistent are not a part of their reality, philosophers generally argue for or against realism across the set theoretic board.