Saturday, January 03, 2009

Who's the Philosopher?

If good mathematics tells us that there are sets of some size, or functions of some type, or if good science tells us that there are waves or particles or forces or fields, then who is the philosopher to pipe up otherwise? As David Lewis forcefully argued, the philosopher taking any such line is apt only to make himself look foolish. Considering the case of sets in mathematics, Lewis wrote:
Mathematics is an established, going concern. Philosophy is as shaky as can be. To reject mathematics for philosophical reasons would be absurd [...] I’m moved to laughter at the thought of how presumptuous it would be to reject mathematics for philosophical reasons. How would you like the job of telling the mathematicians that they must change their ways, and abjure countless errors now that philosophy has discovered that there are no classes?
Philosophy simply has not got the track record of certainty, or utility, or progress, or unanimity, to mount any such high horse. If it is a question of philosophy versus physics, or philosophy versus maths, everyone knows which side to back.
That was Blackburn on Lewis on Philosophy, in Moore and Scott (eds.) Realism and Religion: Philosophical and Theological Perspectives (2007: 49), in the process of trying to draw a line between the expertise of scientists and that of theologians. This is a fairly common Naturalistic line, and there is, I think, a lot wrong with it.
......For a start, the Naturalists are usually pretty choosy about what they count as good science. And in justifying such choices they then try to draw a line between what scientists say about philosophical matters and what they say on scientific matters. As you might imagine, a lot of questions get begged in the process; a lot of the vanquished are straw men. But perhaps all that just goes to show how shaky philosophy is. So what I want to glance at here is what Lewis says about maths (to see how badly it fits with what Blackburn says about science), skipping over most of the fascinating nuances. After all, the argument is generally fallacious. E.g. a couple of years ago many would’ve said the same, as Lewis/Blackburn says about mathematics/physics and philosophy, about commerce and politics, and few think nowadays that markets did’t need some democratic regulation then.
......Mathematical classes necessarily go beyond standard set theory, as a result of the famous set-theoretic paradoxes of a hundred years ago. For the most part, their proper study is regarded by mathematicians and philosophers as the province of logicians and metaphysicians. And even the sets that mathematicians study are formal, axiomatic entities entirely suited to the algebraic ways of the mathematicians. Some mathematicians even work on constructive maths or category theory, for example, and insofar as they do so formally they are regarded as doing maths. In short, the pure mathematicians have passed the metaphysical buck to those applying the formal structures studied by them, while few applied mathematicians regard themselves as doing set theory.
......Lewis doesn’t accept that there are waves or particles or forces or fields, but only spacio-temporal points with arbitrary properties that either happen to fit a pattern or don’t. He’d let us keep whatever physicists say about such things, but would change their meanings (such being his idea of what philosophers can do). But those spacio-temporal points have, he presumes, a structure isomorphic to some set-theoretical structure. He might even allow that they could have any structure, but he needs some such points, and what if such structures happen to be metaphysically impossible? Standard mathematicians actually pass the buck on such applications.
......Lewis was talking about standard set theory, not mathematics (the two are often confused by American academics and those—most of the world’s leading academics—who’d like to become one), and sets have never got over the set-theoretical paradoxes. Or rather, the mathematicians—for the most part (Lewis conveniently ignores those mathematicians who’re constructivists or category theorists)—got round the problem by doing axiomatic set theory in an algebraic way, and thereby left the problem of interpretation up to those applying their maths. If there turn out to be no infinite sets really—no such metaphysically possible spatio-temporal points as Lewis presumes—then there would need to be some applicable mathematics done by someone, and whoever did it would be a mathematician. Lewis ought to be as sceptical about set theory as he is about particles and natural laws.
......Lewis may be moved to laughter at the very idea—a common response to his own hopelessly unrealistic work on metaphysics—but what was he thinking of? Brouwer and Heyting were professional mathematicians, and those philosophers (e.g. Wittgenstein and Dummett) who agreed with them were agreeing with mathematicians. So what was he thinking of, the maths that he was taught at school in America? If philosophers have something to contribute then they should contribute it, and if not then they’re philosophers in name only. Was Descartes a philosopher or a mathematician? He was clearly both, and we surely need more people like Descartes, not fewer. Scientists tend to say that to say nothing of God isn’t to say that God is nothing, and those who think otherwise do so for philosophical reasons.
......Mathematicians have as much right as anyone to think philosophically, and perhaps more right when it’s the philosophy of maths (e.g. Hamming and Fletcher). And what’s especially interesting nowadays is the increasing popularity of category theory, within maths. For the most part, that increasing popularity is not due to metaphysical concerns, but to concerns more internal to maths. But the underlying logic of category theory is intuitionistic. One can even envision a day when the professional mathematicians choose category theory as their standard foundation (one need only think of how popular Lewis is amongst professional philosophers) just because it provides the most interesting line of immediate research (cf. the dominance of string theory in physics)... cetera ad nauseum (note that there’s some difference between philosophy the professional job (cf. Lewis’s concern with having the job of saying what some of his colleagues have said), which does seem shaky, and philosophy the pursuit of truth:)


Anonymous said...

This is a concern of mine as well. I think philosophers certainly should play a role in commenting on mathematics. I had a professor once who said, "It is the role of scientists to do science, but it is the role of philosophers to question their assumptions to make sure it is good science." I think this applies equally well to math (and also I don't think their role should be only in questioning assumptions, but certainly this is a major one).

I think this began to concern me most when I would go to "philosophy of math" talks at conferences and I saw a trend. Philosophers would get up and talk about things in a way so as not to get into an argument with the mathematicians in the room since they felt they wouldn't be able to hold their own when mathematical examples were brought up (and rightly so, mathematicians tend to be incredibly defensive about the infallibility of the subject). Then mathematicians would get up and talk about philosophy in a very classical way. They really had no training or understanding of post-Descartes or really any 20th century philosophy.

So basically the division hurts philosophy of math incredibly. Philosophers are afraid to say anything and mathematicians are rehashing old philosophy (at least at the conferences I've gone to). I wish we could get past this.

jeff said...

Bertrand Russel had some interesting things to say about the appearance that philosophy doesn't actually make much progress.
Essentially, he said that many disciples begin as branches of philosophy. When they reach a certain critical mass of experimental verification, they become branches of science.
Linguistics is a pretty good contemporary example of this phenemona. Fifty years ago, linguistics would have been a philosophical topic. Certainly, we might still discuss the philosophy of linguistics. But nonetheless, within the last fifty years, departments of linguistics have sprung up. Linguists view their discipline as quite seperate from philosophy. And this view is legitimate. But with out a sort-of incubation time within the domain of philosophy, linguistics could never have gotten there.

To criticise philosophy for never showing much progress is a bit like criticising a nursery for only serving babies: by the point a somebody is no longer a baby, they don't belong in a nursery any more. Similarly, by the point a discipline is read to make progress, it begins to take on a life seperate from the body of philosophy proper.
(Clearly philosophy ought to be up to other things, too. But developing babysciences is one important task.)

Anonymous said...

I agree with the attitude. I have experienced the problem in probability and statistics. The inventers and developers in probability were for the most part philosophers, but since this philosophers may not do maths attitude, the subject has become completely confused by this infallibility faith in what mathematicians have been taught. The arbitrariness of the probability calculus has been forgotton.
As a philosopher I am well aware of the problems of considering a probabilistic hypothesis as a proposition, i.e. truth apt. The philosophers who have thought this through, Dorothy Edgington, Ramsey, Wittgenstein, Peirce, have all been perfectly competant at mathematics, Ramsey being one of the geniuses of all time. They all recognised that inferences of the form If P then Q aren't propositions. The probability attached to such an inference is therefore the probability that Q on the supposition that P. Bayes understood this, and Ramsey, and therefore defined conditional probability as basic, rather than in terms of prior probabilities.
But post Kolmogorov "Mathematicians" define conditional probability as the probability of P and Q divided by the probability that Q. This is perfectly sound, just so long as P and Q are propositions. But "Bayesians" try and apply it to hypotheses, and it simply doesn't work for hypotheses. The philosophers who have pointed this out are many, but the point is impossible to get across to someone who has been taught the probability calculus as axiomatic, which is anyone who has been trained as a mathematician.
Whenever I try and advance the idea, people accuse me of not understanding probability, and try to explain probability calculus and basic decision theory to me. But Tversky and Kahneman, Allais, Markovitz have all shown that basic decision theory is just wrong. The obvious solution is that probability statements aren't propositional. But if there is some crazy rule that only mathematicians are allowed to talk about maths, then this obvious solution is unlikely to come to light.