End of APA

There were a bunch of other talks I went to, but some I didn’t understand the material enough, some I didn’t take good enough notes, and some I was just too tired for. Also, I’ll let Greg Frost-Arnold discuss his own paper (which was an interesting suggestion for semantics of non-uniquely-referring singular terms, in a sense somewhat dual to that of free logic, for non-referring singular terms). The session on Jody Azzouni‘s book (with Mike Resnik, Gideon Rosen, and Otávio Bueno criticizing, and Mark Colyvan chairing) gave me a lot of material to think about, and I’ll probably mention it several times over the course of my next several posts, rather than making one post about it.

And the “informational session” on epistemic modals by Kai von Fintel, Thony Gillies, and John MacFarlane was also excellent. I’ll let one of them with a blog discuss it first (especially since Brian Weatherson had to fill in at the last minute for Kai von Fintel – which he did excellently), unless they choose not to. At any rate, they discussed three very different aspects of the semantics of epistemic modals (“might” and “must” in claims like “oh, I see your umbrella is wet, so it must be raining” and “as far as I know, he might be in Boston”). Thony Gillies tried to claim that the models of information states he was talking about were very sexy, but I think he was beaten by John MacFarlane, who pointed out that his theory of relativism about truth might better be described as “bicontextualism”.

And it was also great to meet the many people that I did in different sessions and at the “smokers” in the evening (for non-attendees, that’s apparently what everyone calls the “reception” in the main ballroom on the middle two nights), including a bunch of readers of this blog, as well as several other bloggers. I can only hope that this conference is marginally close to as fun as this in the year that I have to go on the job market and wear a suit all day, and run back and forth between interviews instead of sessions, and impress people to hire me rather than just meeting people more casually.

APA Blogging: Dorr, Bennett

Cian Dorr‘s talk, “Of Numbers and Electrons” on Thursday morning made me realize that we’ve got a lot of the same metaphysical goals. The point of his talk was to show that a weakened (and therefore tractable) version of Hartry Field’s program will be able to support realism about theoretical entities of physics and anti-realism about mathematical entities. The scientific anti-realist might suggest a theory like the following:
BAD: As far as observable matters are concerned, it is just as if T
where T is our actual scientific theory, that talks about electrons and other unobservables. However, almost everyone agrees that such a theory is bad (hence the name Dorr has given it). The mathematical realist then claims that the mathematical anti-realist would have to give a theory like:
AS-IF: As far as the concrete world is concerned, it is just as if T
where T is our actual scientific theory, that talks about functions and numbers and other abstract entities. Dorr proposes an alternative.

APA Blogging: Rabin

Michael Rabin started his talk by mentioning that the traditional picture of proof says that in principle a proof is formalizable, can be verified in an automated way, is reproducible, publishable, and can be transferred (that is, if I have a proof, then I can give it to you and then you will have a proof). In practice, this isn’t entirely correct, because actual proofs are rarely formalized, and are normally checked by a social process rather than an automated one. Apparently there have been case studies showing that even restricting attention to the Hilbert problems, a large number of results have been “proven” once, and then years later reproven in a way showing that the initial proof was incorrect.

He argued that with the rise of new methods of proof (some of which I discussed earlier), there are even more differences – now, there can be proofs that are non-transferrable and non-publishable.

APA Blogging: Hamkins

Joel David Hamkins gave a talk last night on the modal logic of forcing, based on work he had done with Benedikt Löwe. (The talk isn’t listed here yet, but several related ones are.) He said that the aim was to do for forceability what Solovay had done for provability. The idea is that if M is a model of ZFC, and G is some generic filter over some partial order in M, then M[G] is a model of ZFC that is accessible from M, because M has names for all the elements of M[G], and can prove many of the logical relations between facts about M[G] (in fact, in many cases it knows the truth value in M[G] of every sentence with no free variables). Using this accessibility relation, with the “worlds” being models of set theory, we can then define a modal logic. I’ll now write []p to be the sentence of set theory (which is in fact expressible standardly) that says that p is true in every generic extension of the actual universe, and <>p the sentence that says that p is true in some generic extension.

APA, holidays

I’ve noticed that readership dropped this week, understandably, with people traveling for the holidays, and the semester ending in North America. (I suppose it ended a while ago in Australia – I have no idea how it works in Europe, and especially Latin America, Africa, and Asia, though I don’t get many hits from those areas.)

Anyway, I’ll be in New York next week for the APA Eastern division conference. I told Richard that I’ll try to see what I can do about liveblogging some of the talks (well, during free periods in the afternoons or evenings, not actually during the talks), though it may end up having to wait until after the thing’s over. So I’ll either post nothing next week, or a whole bunch between the 27th and 30th – we’ll see.

Happy Holidays!

Forcing

I’ve now put up a draft of my expository talk on forcing, which I gave a month ago to the math grad students at Berkeley. I’m hoping that it should make basic independence proofs intelligible to math grad students, and to interested philosophers with a fairly technical disposition. If you read it, certainly leave comments here, or e-mail me (easwaran AT berkeley DOT edu), if there are any questions, unclarities, inaccuracies, comments, or anything else. I think the first page has a pretty pompous tone, so I should probably change that, but I’m not sure about the rest.

Anyway, the reason I decided to write it up (apart from being able to explain this material more clearly to people in future discussion and talks), is because it’s seemed to me that there’s no accessible introduction to this stuff available (EDIT: since making this post, I noticed that Ars Mathematica mentioned forcing for dummies by Tim Chow – I’m reading it now). All the set theory books seem to either just do basic stuff with ordinals and cardinals (excluding forcing and large cardinals and determinacy and the like) or put forcing at least 100 or 200 pages in and rely on a lot of the material that’s discussed in earlier chapters. However, I recently discovered (thanks to the latest issue of Phi-News, which I ran into through a post by Gillian Russell) John Bell’s book Set Theory: Boolean-Valued Models and Independence Proofs. This book presupposes some amount of familiarity with set theory, but it just jumps right into this material right away. And I’ve found it much easier to read than the relevant chapters of Jech or Kunen (but perhaps that’s at least in part because I spent so much time in August and September going through the relevant chapters of Jech). So there’s not as much of a hole to fill as I thought, but I’m trying not to presuppose any theorems of set theory beyond Russell’s paradox.

Bell’s book is also quite interesting to me because it presents the material entirely in the framework of boolean-valued models, rather than forcing. The results and the methods are almost entirely equivalent, but the formulation is different. The method of forcing requires the existence of countable transitive models of ZF (which aren’t guaranteed by Con(ZF)), but then gives a standard model-theoretic consistency proof by explicitly creating a model of the new theory. The method of boolean-valued models works on the universe as a whole, rather than on some subset of it. But as a result, it doesn’t actually construct the domain of some structure for the theory – instead it gives a proper class with some boolean-valued (rather than true/false-valued) relations on it representing identity and set membership, and shows that the set of formulas receiving value “1” must be consistent, and can be made to include theories of appropriate sorts. The boolean-valued approach has the advantage of making all the calculations of “truth-values” for sentences much easier, but the disadvantage of making the model somehow “blurry” and indistinct. Forcing, on the other hand, gives a clear model, in exchange for some extra calculational difficulties.

I’ve always felt more comfortable with an approach highlighting boolean-valued models much more than Kunen does, and probably even a bit more than Jech, but Bell’s approach has felt alien by the fact that it doesn’t mention what seems to be the more standard approach at all until page 88. At any rate, it’s been an interesting read so far.

Joyce on Evidential Weight

A new paper by Jim Joyce, “How Probabilities Reflect Evidence” discusses more clearly some of the issues I mentioned in a previous post, where I suggested that Henry Kyburg opposed subjective probabilities because he misunderstood some of the ideas of Bayesian epistemology. What I was talking about there looks just like the contrast between what Joyce calls the “balance” and the “weight” of evidence. He also mentions “specificity” of evidence, and shows that Bayesian epistemology can deal with this as well. All three of these distinctions are made in examples where the agent has hypotheses about objective chance processes, but I’m sure that some of this can eventually be generalized beyond those circumstances. Anyway, it’s quite an interesting paper – I was especially intrigued by the mention of some attempts to avoid something like Bertrand’s paradox for the Indifference Principle (which Joyce calls “The Principle of Insufficient Reason”) about randomly generated squares.

(Thanks to Brian Weatherson and Jon Kvanvig for pointing me to the relevant issue of Philosophical Perspectives. I might as well also mention now that I’ll be on a panel about philosophy blogging with both of them, as well as Gillian Russell, at the Pacific APA in Portland in March.)