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I have characterised the Philosophy of Real Mathematics in my book as the study of "what leading mathematicians of their day have achieved, how their styles of reasoning evolve, how they justify the course along which they steer their programmes, what constitute obstacles to these programmes, how they come to view a domain as worthy of study and how their ideas shape and are shaped by the concerns of physicists and other scientists".
My latest thinking is contained in these papers: How Mathematicians May Fail to be Fully Rational, now called 'Narrative and the Rationality of Mathematical Practice'; Reflections on Michael Friedman's Dynamics of Reason; Categorification as a Heuristic Device; Mathematical Kinds, or Being Kind to Mathematics; Review of Omnes' 'Converging Realities'; Smoke Rings, a history of early knot theory (large files of figures, here and here); Blending Philosophy of Mathematics and Cognitive Science (slides); Why and How to Write a History of Higher-Dimensional Algebra (notes).
The time is right for the philosophy of mathematics to reconnect with mathematics since a revolution is in the air. The French mathematician Pierre Cartier remarks in an interview:
When I began in mathematics the main task of a mathematician was to bring order and make a synthesis of existing material, to create what Thomas Kuhn called normal science. Mathematics, in the forties and fifties, was undergoing what Kuhn calls a solidification period. In a given science there are times when you have to take all the existing material and create a unified terminology, unified standards, and train people in a unified style. The purpose of mathematics, in the fifties and sixties, was that, to create a new era of normal science. Now we are again at the beginning of a new revolution. Mathematics is undergoing major changes. We don't know exactly where it will go. It is not yet time to make a synthesis of all these things—maybe in twenty or thirty years it will be time for a new Bourbaki. I consider myself very fortunate to have had two lives, a life of normal science and a life of scientific revolution.
An important factor behind the timing of this revolution has been the flood of Russian mathematicians coming to the West after the collapse of the Soviet Union. Certainly, the revolution is young, but this fact should not be used to argue that no philosophical resources be devoted to the cause. Russell wasn't just tidying up after the dust had settled during the last revolution.
Part of the job of the philosopher of X, and a considerable part of it at that, is to understand how the values of practitioners of X operate in their discipline. In Lecture 2 of The Empirical Stance (Yale University Press) Bas van Fraassen claims this for the natural sciences. I do so for mathematics. This is partly a descriptive task and partly an evaluative one. So much effort has been devoted to a thin notion of truth, so little to the thicker notion of significance. To say that scientists and mathematicians aim merely for the truth is a gross distortion. They aim for significant truths. What we can do as philosophers is to assist in the articulation of a "strongly evaluative language", to use a term of Charles Taylor (Philosophical Papers I, CUP 1985: 25), one which will be qualitative and contrastive. In the case of mathematics, for example, we need to articulate the notions of "significance" and "importance" with a view to influencing the ways language is used in the decision-making of the mathematical community.
This kind of philosophical work is far removed from much contemporary philosophy of mathematics, and here lies a source of our present troubles. Much of the training required to become a philosopher of mathematics concerns its representation in logical languages which do not run along its grain, but which are supposed to reveal its 'ontological commitments'. The value of truth as applied to a statement is the only one possible to treat in this way, and it sits at the core of the discipline. The rub is that this tends to switch off mathematicians' interest. After all, they do not gain their doctorate, promotion or respect from their peers for producing merely impeccably logically correct mathematics. The values at stake for them at the beginning of this century are ones connected with the conceptual organisation of the discipline. Now, I don't discount the possibility that in the future some of these other values may be fruitfully treated as searches for timeless solutions to problems, but I believe I may be allowed my doubts on this score. Jamie Tappenden has shown us that Frege viewed his Begriffsschrift as a means not only to secure the truth of propositions, but also to "carve out" concepts correctly. It is very clear a century later that this hope has not been realised. The only plausible candidate to achieve a similar task at present is category theory, a language I'd be only too happy for some more philosophers to learn, but going beyond a judgement of category theory's present power to get oneself straightened out, conceptually speaking, to a timeless conclusion would seem to me foolhardy.
Of course, we could rejoice in the fact that our philosophy of mathematics does not speak to mathematicians. Have we recovered yet from our discipline's dogmatic advocacy of Euclidean geometry? Safer then to talk amongst ourselves. But we are permitted to aspire to more subtle forms of influence. About values in general, Charles Taylor has written:
Our attempts to formulate what we hold important must, like descriptions, strive to be faithful to something. But what they strive to be faithful to is not an independent object with a fixed degree and manner of evidence, but rather a largely inarticulate sense of what is of decisive importance. An articulation of this 'object' tends to make it something different from what it was before. (Philosophical Papers I: 38)
That we can find resonances in the writings of philosophers who have never been close to mathematics may appear surprising at first. But consider how a similar trend may be detected in contemporary philosophy of science when, for example, in The Empirical Stance Bas van Fraassen uses Sartre to help him think through revolutionary change in science. I'll end this introduction, then, with a quotation from the mathematician Hermann Weyl, an avid reader of Heidegger and interlocutor of Jaspers, which suggests why we might follow van Fraassen's example:
Mathematics is not the rigid and petrifying schema, as the layman so much likes to view it; with it, we rather stand precisely at the point of intersection of restraint and freedom that makes up the essence of man itself.
(p. 136, 'The Current Epistemogical Situation in Mathematics' in Paolo Mancosu (ed.) From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s, Oxford University Press, 1998, pp. 123-142).
Ian Hacking in his Historical Ontology (Harvard University Press 2002: 63) discusses and recommends a philosophical approach to the study of science, including mathematics, which involves "taking a look" at the practice of science. Why bother? Philosophers seem to divide fairly sharply on this issue. Those with a strong historical sense tend to see the point straight away, and would agree with R.G.Collingwood when he claims:
"There are two questions to be asked whenever anyone inquires into the nature of any science: what is it like? and what is it about? of these two questions the one I have put first must necessarily be asked before the one I have put second, but when in due course we come to answer the second we can only answer it by a fresh and closer consideration of the first." (The Principles of History: 39-40)
Collingwood intends 'science' to include any viable knowledge-acquiring practice. Clearly, then, to gain a sense of what mathematics is like, one needs to make a study of what mathematicians do. But we may perhaps arrive at the same point starting out from a squarely analytic orientation. A theme emerging in recent philosophy of mathematics is the idea that two dimensions may have been conflated in debates concerning realism. One dimension relates to the mode of existence of mathematical entities (concrete like a chair, abstract like democracy, fictional like Oliver Twist, etc.). The other dimension concerns what it is that constrains the mathematician in her choice of research. Is there something more restricting than logic, other than mere fashionability, which dictates that some concepts just fly after they've been introduced (e.g., quantum groups) while others never really make it, and which gives many mathematicians the sense that they are sculpting from hard stone rather than from butter?
We can detect something of the splitting of these dimensions in an article by Mark Balaguer ('A Theory of Mathematical Truth', Pacific Philosophical Quarterly 82 (2001), 87-114), where he discusses a notion of "objective correctness" to which one may ascribe whether realist or anti-realist:
"A mathematical sentence is objectively correct just in case it is "built into", or follows from, the notions, conceptions, intuitions, and so on that we have in connection with the given branch of mathematics."
The task of the philosopher of real mathematics is to determine how these "notions, conceptions, intuitions, and so on" are developed, and what constitute good reasons for this process.
Doesn't arithmetic suffice? Already there we have a rich mix of symbolism, concepts, intuition, applications, judgements, necessity, the infinite, pertinent formal results, and plenty of first rate philosophical spadework. In that mathematical activity presents to some degree the "link between experience, language, thought, and the world, which is at the very centre of what it is to be human" (Michael Potter Reason's Nearest Kin: 18), surely all the clues are there to be found in arithmetic. Well, I have wagered against this claim. I hold that not all of the philosophical juice to be squeezed out of mathematics will be found by considering this theory and its applications. Some aspects do not feature in arithmetic, some are only partially realised there, and some are fully realised but very unlikely to be noticed without a detour. Set theory takes us only a little further.
In my struggle as a philosopher to make my work responsive to mathematics as actually and historically practised, I have generally found it illuminating to search for similarities between mathematics and the natural sciences. Now, this strikes some people as wrong-headed. Even if there are similarities between these two knowledge-acquiring practices, why focus on these and not what makes mathematics unique, say, its distinctive use of proof. But what if this choice between searching for similarities and searching for differences in the case of mathematics and science resembled that facing the zoologist wondering about that large sea creature - the whale? It seems clear to us now that we can be led to make important discoveries about the whale from previously acquiring knowledge about the elephant (anatomy, physiology, genetics, behaviour). After locating commonalities we can start to think about which features are unique to whales, and who knows an apparently unique feature there might find an analogue back amongst the elephants. You are probably thinking that I am likening the less accessible species - the whale - to mathematics, and the more accessible one - the elephant - to science. But consider the possibility that certain features of knowledge-acquisition are easier to detect in the case of mathematics. This crazy idea was one that the Hungarian mathematician George Polya adhered to. And in fact as early as 1941 he had worked out most of the principles of the probabilistic approach to epistemology known as Bayesianism. Much effort could have been saved if philosophers of science had listened to him. Well I believe there are many other reasons we might to look to mathematics: as an example of an extraordinarily intricate web of coherence meeting hierarchical foundations; clearer examples of the heuristic effects of thinking in different ways about the same object; the relationship between rationality and aesthetics; and, the tireless working over of ideas after publication.
This having been said, enormously more work has been carried out on the natural sciences, so it is natural to look to studies of science for inspiration. In the introduction to my book I raise five debates, contributions to which should illuminate the nature of mathematics. The first three are brought over from Ian Hacking's discussion of the natural sciences in 'The Social Construction of What?' (Harvard University Press, 1999).
1. Inherent structurism/nominalism: Can we make any sense of the idea of carving the definition of a concept correctly? Can we agree with Lakatos here?
"As far as na´ve classification is concerned, nominalists are close to the truth when claiming that the only thing that polyhedra have in common is their name. But after a few centuries of proofs and refutations, as the theory of polyhedra develops, and theoretical classification replaces na´ve classification, the balance changes in favour of the realist." (Lakatos Proofs and Refutations: 92n)
Why does Frege describe the qualities of good mathematical concepts in the same terms as modern exponents of the theory of natural kinds?
[Kant] seems to think of concepts as defined by giving a simple list of characteristics in no special order; but of all ways of forming concepts, that is one of the least fruitful. If we look through the definitions given in the course of this book, we shall scarcely find one that is of this description. The same is true of the really fruitful definitions in mathematics, such as that of the continuity of a function. What we find in these is not a simple list of characteristics; every element is intimately, I might almost say organically, connected with others. (Frege, Foundations of Arithmetic: 100)
2. Inevitability: given a mathematics as sophisticated as our own, how probable was it that concepts such as natural numbers, groups, groupoids would be devised? Do any concepts 'force' themselves upon our attention, and do they dictate to us the way they are used?
3. Reasons for stability: Why do we persist in teaching certain ways of thinking about particular concepts - social inertia or because that's what they are like?
4. Connectivity of mathematics: How should we represent the connectivity of mathematics on a scale running from thoroughly fragmented to very unified? Why are there so many apparently surprising connections?
5. Miraculousness of applicability: Choose a position between the amazement of Eugene Wigner and its deflation by those who take empirical research as the source of much mathematics.