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Wednesday, March 15, 2006

Mathematicians and philosophy

This survey of mathematicians' philosophical reading habits was a little small, but interesting nonetheless. It's hardly surprising when someone in their early twenties who is wondering whether to become a mathematician is drawn to read philosophies addressing lived experience. Norbert Wiener's I am a mathematician and Paul Halmos's I want to be a mathematician indicate how much this choice gets taken up into one's identity. Alexandre Borovik on his home page writes "I am a mathematician and that is what makes me interesting."

Now, as I have been insisting recently, a philosophy of mathematics ought to recognise this dimension of commitment to a tradition of rational enquiry. As Alasdair MacIntyre puts it:

…critical rational enquiry is not itself the kind of activity that anyone can undertake on her or his own. For the same reasons that I cited, when I argued that we are able to become and to continue as practical reasoners only in and through our relationships to others, we are able to engage in critical enquiry about our beliefs, conceptions, and presuppositions only in and through relationships to others. Rational enquiry is essentially social and, like other types of social activity, it is directed towards its own specific goals, it depends for its success on the virtues of those who engage in it, and it requires relationships and evaluative commitments of a particular kind. (Dependent Rational Animals, Carus Publishing 1999: 156-7) introduce the Thomistic conception of enquiry into contemporary debates about how intellectual history is to be written would, of course, be to put in question some of the underlying assumptions of those debates. For it has generally been taken for granted that those who are committed to understanding scientific and other enquiry in terms of truth-seeking, of modes of rational justification and of a realistic understanding of scientific theorizing must deny that enquiry is constituted as a moral and a social project, while those who insist upon the latter view of enquiry have tended to regard realistic and rationalist accounts of science as ideological illusions. But from an Aristotelian standpoint it is only in the context of a particular socially organized and morally informed way of conducting enquiry that the central concepts crucial to a view of enquiry as truth-seeking, engaged in rational justification and realistic in its self-understanding, can intelligibly be put to work. (p.193) ‘First Principles, Final Ends and Contemporary Philosophical Issues’ in Kevin Knight (ed.) MacIntyre Reader, pp. 171-201.
All the same, at some point philosophy must address what mathematics is about. This all too small survey reflects what I know from elsewhere that very few mathematicians read contemporary philosophical treatments of the nature of mathematical entities. The greatest appreciation seems to be directed toward their fellow mathematician Reuben Hersh's 'mathematics as the study of mental objects with reproducible properties'. If they were tempted to read about mathematical entities as structures, they'd sooner read Barry Mazur's
'When is one thing equal to some other thing?' than anything by Benacerraf, Resnik or Shapiro. It seems unlikely to me that we'll beat them at this game of clarifying their working language, although we might play the role of gadfly there.

Perhaps we might find richer pickings in answering a challenge posed by Michael Harris in "Why Mathematics?" You Might Ask that philsophers "...have a duty, it seems to me, to account for terms like “idea” and “intuition” — and “conceptual” for that matter — used by human mathematicians (at least) to express their value judgments." (p. 17). Take the term idea:

Nothing in the life of mathematics has more of the attributes of materiality than (lowercase) ideas. They have “features” (Gowers), they can be “tried out” (Singer), they can be “passed from hand to hand” (Corfield), they sometimes “originate in the real world” (Atiyah) or are promoted from the status of calculations by becoming “an integral part of the theory” (Godement). (p. 14)
Harris makes the very useful point that my own use of the term is liable to a certain slippage:

Corfield uses the same word to designate what I am calling “ideas” (“the ideas in Hopf’s 1942 paper”, p. 200) as well as “Ideas” (“the idea of groups”, p. 212) and something halfway between the two (the “idea” of decomposing representations into their irreducible components for a variety of purposes, p. 206). Elsewhere the word crops up in connection with what mathematicians often call “philosophy,” as in the “Langlands philosophy” (“Kronecker’s ideas” about divisibility, p. 202), and in many completely unrelated conections as well. Corfield proposes to resolve what he sees as an anomaly in Lakatos’ “methodology of scientific research programmes” as applied to mathematics by

a shift of perspective from seeing a mathematical theory as a collection of statements making truth claims, to seeing it as the clarification and elaboration of certain central ideas… (p. 181)

He sees “a kind of creative vagueness to the central idea” in each of the four examples he offers to represent this shift of perspective; but on my count the ideas he chooses include two “philosophies,” one “Idea,” and one which is neither of these. (p.16)
Point taken. I'll see what I can do.


dt said...

Hey. I like your "philosophy of real mathematics" blog (I'm a real mathematician) and I'd like to know something about your "philosophy of real mathematics" program; this seems like a good post to ask questions under.

What's philosophy? I can think of a couple answers:

It's a tradition. (So, like ballet? Or like judaism?)

It's a toolset. (Waaaay to many parenthetical questions raised...)

Probably neither of these answers is quite right, or even close enough to answer my followup question: Which do you mean when you talk about pursuing "philosophy of real math?"

So, maybe you can answer "what's philosophy" better. I can't really answer "what's math" in a satisfactory way, but I can define large parts of math by accident. Like "it's the stuff you care about if you care about (somewhat silly) properties of natural numbers." (And if you throw in something about PDEs and something else about compters, I think you've covered it all. But maybe add something about graphs, too.) Can you do something like this for "philosophy"?

Also: You seem to be especially interested in category theory. Is this because it's of some fundamental interest in philosophy or just because it's a good case study? Would your p. of r. m. program gotten off the ground sixty years ago when real mathematicians hadn't caught on to this stuff?

Sorry, mathematicians don't have much professional interest in being articulate, and so lots of us aren't. (But you're probably used to that by now)


March 17, 2006 1:41 AM  
david said...


Thanks for this. I'll address your questions in a post (or probably posts) over the coming days.

March 17, 2006 11:12 AM  
Anonymous said...

I think the notion of an "idea" in mathematics is very interesting, and very deep. One could say that category theory is just the working-through of the idea that one way to study a space of objects is to study the mappings in-to and out-from the space which leave the elements or structure of the space unchanged (or, variously, which change the elements or structure in interesting ways). This is a very powerful idea, and not at all obvious to someone educated (indocrinated?) in the set-theoretic ideas of collections and membership.

Mathematicians often say they wish to talk with another mathematician in order to learn how he/she thinks (about a problem or a domain). I believe this is related to the notion they have of an "idea" -- an idea can be a way of viewing a domain, a weltanschauung.

-- Peter McBurney

March 18, 2006 3:50 PM  

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