I mostly wrote this post three weeks ago, when Hilary Putnam was at Berkeley to give the three Townsend lectures, but I didn’t get around to finalizing it until just now.
Putnam gave three talks, which were all quite interesting. The first discussed his own type of realism, and tried to counteract false impressions people have had of his views changing drastically with regards to this question over time. (He admitted that they had changed at a few points, but several other apparent changes were just due to some poor choice of terminology on his part. In particular, he had used the phrases “internal realism” and “scientific realism” in a few different ways by accident.)
The second discussed the Lucas/Penrose arguments trying to use Gödel’s theorems to undercut mechanism about the mind – he claimed that although these arguments couldn’t really work, there is an interesting modification that should undercut Chomsky’s thesis that there is a shared innate “scientific competence” that underlies the notion of scientific justification. If there were such an algorithm, then we could never recognize that algorithm as the true one – but since this thesis has to be an idealization (because of the infinities involved) it doesn’t make much sense to argue that our competence is represented by a Turing machine without our being able to say which one. However, I wonder if there’s still some sense to be made of the question – perhaps the idealization says that our scientific competence is given by some arithmetically definable set. But in that case, it is an interesting question whether the best such set is itself Turing computable (even though we can’t know which Turing machine gives it), or more complicated.
The third discussed his picture of “mathematics as modality”. What I found most interesting about this talk was his discussion of the so-called “Quine-Putnam indispensability argument”. He pointed out that when he gave the argument, it had a very different form and conclusion from Quine’s – while Quine used it as an argument for “realism in ontology” about mathematics (that is, that mathematical objects actually exist), Putnam claimed merely that it established “realism in truth-value”. Putnam’s argument was intended to show that there is a serious tension between scientific realism and verificationism about mathematics, on which mathematical claims don’t have truth-values unless they can be proven or disproven. The scientific realist says that there are some sets of equations (say, equations of motion, or wave-equations) that correctly describe the physical world. Now, if these equations have enough complexity (I suppose a three-body system under Newtonian gravitation suffices), then given the truth of certain initial conditions, there will be some interval (a,b) and some time t such that the question of whether one of the objects is in that interval or not at that time is undecidable. But if there is a fact of the matter for the physical claim, and the equations correctly describe the physical world, then there must be a fact of the matter about the mathematical claim, despite its being undecidable.
Putnam took care to point out that this only establishes that there is a fact of the matter about the mathematics, and not that the mathematical objects actually exist. Also, he took care to point out that he was not trying to make an epistemological point – nothing in this argument establishes that this is how we know the truth-values of mathematical claims, just that there must be such truth values. Since Quine concluded that mathematical claims must be about something (because of his picture of ontology as being given by the existential commitments of the best theory of the world), he also concluded that this is how we know which mathematical axioms are true.