π, τ, and Quine's pragmatism

Some readers may already be familiar with the π vs. τ debate. If not, I recommend checking out Michael Hartl's τ manifesto and this fantastic short video by Vi Hart. An attempt at a balanced evaluation of π vs. τ can be found here.

For those who don't know, τ is just 2π. The defenders of τ argue (as seen in the above links) that using it instead of π makes many things much clearer and simpler/ more elegant.

Let's assume for present purposes that the τ-proponents turn out, in the end, to be right. I want to ask a further question: what would this then say about Quine's denial of the analytic-synthetic distinction? Quine's denial, virtually all agree, is the claim that all rational belief change is pragmatic, i.e. there is no principled difference between questions of evidence/justification on the one hand, and questions of efficiency and expedience on the other (= between external and internal questions, i.e. between practical questions of which language-form to adopt, and questions of whether a particular empirical claim is supported by the available evidence).

So here's my question: if Quine is right, then is our old friend C=2πr simply wrong (and C=τr right)? If not, how can a Quinean wiggle out of that consequence? And if so (i.e. C=2πr really is wrong), does the Quinean have any way of softening the sting of this apparently absurd consequence?

antimeta in the house

One of my favorite bloggers, Kenny of Antimeta, was in Vegas last weekend and gave an interesting talk on philosophy of mathematics to our department. His basic aim was to find criteria that separated probabilistic proofs from other proofs (including, hopefully, proof sketches and computer-aided proofs). I'm not going to discuss that directly here.

I'm interested in a related claim Kenny made: that in mathematics, a theorem will be accepted only if the proof does not (he put it variously) appeal to authority/ depend on the reliability of other people/ rely on the testimony of others. That is, for a specialist in the field, they should be able to start out as serious skeptics of the theorem's truth, but end up at the close of the proof as believers. The contrast with experimental science is pretty clear: even specialists in a sub-field of experimental science have to trust (to some degree) the experimental reports of their fellow-workers, or the field would grind to a halt.

Question: Is there such a thing as mathematical fraud, of the sort we hear about periodically in experimental science? If not, that fact looks like evidence for Kenny's distinction being important and robust (since fraud is much harder in the absence of trust).

Comment 1: Some of the posters on FOM endorse Kenny's idea to the extreme: someone suggested that Fermat's Last Theorem will not really be proved until it is written in a way that average mathematics PhDs (whoever that is) can work through it themselves. I don't think Kenny wants to say anything nearly that strong, but the fact that such a strong position exists is a sign that the sentiment Kenny claims to discern really is there in the mathematics community.

Comment 2: At the end of the talk, Kenny suggested that philosophy may be closer to mathematics than experimental science in this regard. He may be right, but one thing that distinguishes philosophy from math in this regard is that in philosophy far more than in mathematics, one person's modus ponens is another person's modus tollens. This is just a direct result of mathematical axioms' being widely accepted throughout the mathematical community, whereas philosophers will challenge any premise, no matter how obvious or fruitful.

Is arithmetic empirical?

One of the questions I've been wanting to think about (in part because of my interest in the Quine-Carnap relationship) but haven't really got around to yet is: Is there any important sense in which arithmetic is empirical? I know there is some good literature on the subject, but I've thus far only perused it without really digging into it.

For me, one consideration that makes me think it might not be crazy to think of arithmetic as empirical is what happened with geometry and general relativity. If Einstein can show that the space in which we live is non-Euclidean, isn't it at least imaginable that some future scientist will show us that the 'true' arithmetic of our physical world is non-classical (which I suppose means: it does not obey the Peano axioms). [There could still be a mathematical structure that obeys classical arithmetic, just as Euclidean space is still a mathematical object that obeys all five of Euclid's axioms.]

However, I've always had a hard time imagining what possible observation could cast doubt on classical arithmetic. In last week's Science, there's a report that at least might merit consideration as a candidate. Researchers found that if you add one photon to a light beam and then take one away, you observe a different end-state than if you reverse the order of operations, i.e., first remove one and then add one. In other words, x + 1 - 1 does not equal x - 1 + 1. Even stranger, the authors find that "under certain conditions, the removal of a photon from a light field can lead to an increase in the mean number of photons in that light field," that is, (roughly) that x-1>x. The summary and background for non-specialists is here, and the full technical report is here (both behind subscription walls).

Now, this effect depends on the failure of commutation relations ubiquitous in quantum mechanics, so it is quite possible that this in no sense makes arithmetic look empirical. But I'm not 100% sure about that. Any thoughts?

Mancosu and mathematical explanations

Last Friday, Paolo Mancosu was in Pittsburgh to give a talk on explanation in mathematics. His visit gives me the opportunity to correct an oversight in my last post -- Paolo helped me improve my dissertation substantially: he read an early partial draft very carefully, and brought his learned insight to bear on it. He is the only person in the universe who has written on the specific topic of my dissertation, and his comments were extremely helpful.

The basic claim of Paolo's talk was that Philip Kitcher's account of mathematical explanation falls afoul of certain apparently widely-shared intuitions about which proofs are explanatory and which are not. But I was intrigued by something else mentioned in the talk, which came to the fore more in the Q&A and dinner afterwards. When working on explanation in the natural sciences, there seems to be much more widespread agreement about what counts as a good explanation than in the mathematical/ formal sciences. That is, whereas most practitioners of a natural science can mostly agree on which purported explanations are good and which not, two mathematicians are much less likely to agree on whether a given proof counts as explanatory.

So I am wondering what accounts for this difference between the natural and formal sciences. Might this be due (in part) to mathematics lacking the 'onion structure' of the empirical sciences? For example, the claims of fundamental physics are not explained via results in chemistry, and observation reports (or whatever you want to call claims at the phenomenological level [in the physicist's sense]) are not used to explain any theoretical claim, and so on. My intuitions about mathematics are not as well-tutored, but I have the sense that the different branches of mathematics do not have such a clear direction of explanation. (Of course, there is no globally-defined univocal direction of explanation in the natural sciences [the cognoscenti can think of van Fraassen's flagpole-and-the-shadow story here], but there is nonetheless an appreciable difference between math and empirical sciences on this score.) At least in some cases, this clearer direction of explanation probably results from empirical sciences' explaining wholes in terms of their parts -- whereas mathematics lacks that clear part-whole structure. Often two bits of mathematics can each be embedded in one another, but we tend not to find this in the empirical sciences. (The concepts of thermodynamics [temperature, entropy] can be defined using the concepts of statistical mechanics [kinetic energy], while the converse is clearly out of the question.)

Pointing to the onion structure/ clearer direction of explanation in science might be just a re-statement of the original question; I'm not sure. Or maybe it's not relevant. In any case, I have to bury myself beneath a mountain's worth of student essays on the scientific revolution...

The reasonable effectiveness of mathematics... for Ptolemy

The class I am teaching this term covers the emergence of Early Modern philosophy and science. The first five weeks are devoted to a whirlwind tour of Ancient Greek natural philosophy (plus a bit of Renaissance thought), and the last 10 weeks cover 17th century philosophy and the scientific revolution.

We spent half of the past week discussing Ptolemy, and I was struck by something that I had noticed before, but never really appreciated. It is very natural for Ptolemy to use fully 'mathematized' explanations for astronomical phenomena, but not for (most) other physical processes. Why? On Ptolemy's view, astronomical objects share more properties with mathematical objects than they do with terrestrial objects. He thought that astronomical objects are eternal and their properties are unchanging -- like the number 5, but unlike terrestial ones. We give a mathematical treatment of astronomical phenomena because they exhibit properties of mathematical objects.

The application of mathematical methods in Ptolemaic astronomy helps bring into focus the so-called problem of the unreasonable effectiveness of mathematics, which some days appears to me to be an unequivocal pseudo-problem. Ptolemy's application of mathematics to physical phenomena, I think, appears extremely well-justified compared to our own: astronomical phenomena can be mathematized because they share peculiar features with mathematical objects, features that the mundane, material objects in our immediate surroundings lack. During and after the scientific revolution, we preserved and expanded Ptolemy's mathematizing proclivities, but we apparently relinquished his justification for treating the natural world mathematically.

Update (10/02/05): Kenny over at Antimeta just put up an interesting post on the (un)reasonable effectiveness of mathematics too, and it is in (small) part a comment on my post.