10/25/2007

proxy bleg for a textbook

A post by request: one of my colleagues will be teaching a course for philosophy majors called "Contemporary Philosophy" focusing on what is current in the discipline now. Does anyone know of any good textbooks/anthologies that would work well for such a course?

10/22/2007

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.

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10/15/2007

Yet another way to think about Quine's critique of Carnap

Several of the blog entries here have been about the Quine-Carnap debate over the status of analytic truth. Generally, I don't feel the force of Quine's arguments as they are usually presented, either because his interpretation of Carnap is unfair or inaccurate, or the arguments just aren't that persuasive. Multiple commentators on the Quine-Carnap debate have suggested that the two are 'talking past each other,' at least to some degree. So, I am constantly trying to find a way to make Quine's view make sense to me, AND simultaneously really disagree with Carnap. This seems like installment 19 or so in that endeavor.

Carnap and Quine agree that language can be studied at various levels of abstraction. Using Carnap's taxonomy, we start at the level of pragmatics, where we study how individual speakers use expressions under particular circumstances. This level contains the most detail: speakers, their circumstances, plus the meanings of the words for particular speakers under particular circumstances. At the next, more abstract level, we have semantics, which abstracts away from particular speakers and particular circumstances. And at the highest level, we have syntax, which abstracts away the meanings of words, leaving just the symbols, the way they are put together, and which strings follow from others.

In each transition from pragmatics to semantics to syntax, some information about language is omitted/ discarded. (Like the move from Euclidean geometry to neutral geometry, which drops the parallel postulate.) Now, we can conceive of Quine's indeterminacy of meaning thesis (the radical translation thought experiment) as critiquing Carnap in the following way: Carnap is importing or introducing new information at the semantic level, because the semantic facts Carnap includes in a semantically-characterized language [a "semantic system"] cannot be 'read off' even the information contained at the pragmatic level. The analogy in the geometry case shows why this is clearly an unacceptable maneuver. It would be: thinking that there exists some claim that could be proved in neutral geometry (= Euclid's first four postulates only) but couldn't be proved in Euclidean geometry.

This may not be Quine's actual worry; his concern may stem from the fact that applied semantics (or whatever branch of language study) underdetermines pure semantics (or whatever). However, Carnap is perfectly happy to accept that claim: Creath says this is why Carnap's copy of Word and Object Ch.2 (Indeterminacy of Translation) has no marginalia. [But how does the geometry analogy fare here? Would Carnap admit that applied geometry underdetermines pure geometry? My guess is yes; and that that's not so bad...

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10/10/2007

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?

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