Sunday, May 28, 2006

It's that time of year again ...

... when I'm buried in tripos marking. Distractions between marking sessions are necessary. So I've just finished reading Margaret Atwood's The Penelopiad which made a wonderful diversion from the usual mixed bag of metaphysics scripts.

Incidentally, the white smoke has at long last gone up from the consistory chapel window, and the Knightbridge Professorship has been offered to X. But unlike popes, who don't get to negotiate their terms, potential professors do. So we'll have to wait and see if X indeed arrives. [Later breaking news, 11 June: X = Quassim Cassam, who is indeed coming to Cambridge from Oxford via UCL for January 2007.]

Tuesday, May 16, 2006

Libraries should be circular?

When I was in Aberystwyth, I had a decent sized room in the Hugh Owen Building which is halfway up Penglais, with panoramic views over Cardigan Bay. In Sheffield, I had a huge room on the 12th floor of the Arts Tower -- and while the daytime urban view wasn't exactly a delight, on winter evenings the transformation into a glittering landscape of lights was magical. These days I have a very small room in the Faculty, with a window into the grad. centre and otherwise tiny windows too high to look out of, which isn't as bad as it sounds, but equally isn't very enticing.

So I work a lot in the Moore Library. It took me a while to really "get it", but now it strikes me as in many ways a quite splendid building, and I love being there. The reading tables run around the perimeter, so you are looking out to trees and to the modern buildings of the rest of CMS; even when the library is busy, you can only really see a few people either side of you because of the curve of the building and the book shelves which are arranged as along the spokes of a wheel. And while the bookstacks in the UL seemingly run off to infinity (so you can feel lost in a Borgesian nightmare), there is a sense that here the readers are surrounding the mathematical knowledge shelved behind them. There is a rather calming feel to the place, which draws me back especially when things aren't going well with my book. So I should get down there now ...

Saturday, May 13, 2006

Tired of ontology?

It requires a certain kind of philosophical temperament -- which I seem to lack -- to get worked up by the question "But do numbers really exist?" and excitedly debate whether to be a fictionalist or a modal structuralist or some other -ist. As younger colleagues gambol around cheerfully chattering about these things, wondering whether to be hermeneutic or revolutionary, I find myself sitting on the side-lines, slightly grumpily muttering under my breath 'And who cares?'.

To exaggerate a bit, I guess there's a basic divide here between two camps. One camp is primarily interested in analytical metaphysics, or epistemology, or the philosophy of language, and sees mathematics as a test case for their preferred Quinean naturalist line (or whatever). The other camp is puzzled by some internal features of the practice of real mathematics and would like to have a story to tell about them.

Well, if you're tired of playing the ontology game with the first camp, then there's actually quite a bit of fun to be had in the second camp, and maybe more prospect of making some real progress. In the broadest brush terms, here are just a few of the questions that bug me (leaving aside Gödelian matters):

  1. How should we develop/improve/augment/replace Lakatos's model of how mathematics develops in his Proofs and Refutations?
  2. What makes a mathematical proof illuminating/explanatory? (And what are we to make of unsurveyable computer proofs?)
  3. Is there a single conceptual grounding for the standard axioms of set theory? (And what are we to make of the standing of various large cardinal axioms?)
  4. What is the significance of the reverse mathematics project? (Is it just a technical "accident" that RCA_0 is used a base theory in that project? Can some kind of conceptual grounding be given for that theory? Would it be more principled to pursue Feferman's predicative project?)
  5. Is there any sense in which category theory provides new foundations/suggests a new philosophical understanding for mathematics?
There's even a possibility that your local friendly mathematicians might be interested in talking about such things!

Sunday, May 07, 2006

Laws of nature

I'm giving just four second-year lectures on the philosophy of science this term, revisiting Lakatos (I'm a long-time fan). Last year I talked instead about laws of nature; rather to my surprise I found myself taking exactly the opposite line from that I used to take in supervisions, and warmed to a wild Humean subjectivism. Re-reading the notes from the lectures on laws they seemed at least provoking enough to be worth handing out again to this year's class. I don't promise that I believe any of this stuff: I was just interested to see if you can play the game the Humean way.

Ancestral logic

One of the many things I want to do once I've got my Gödel book finished is to slowly trawl through the first twenty years (say) of JSL to see what what our ancestors knew and we've forgotten.

I was put in mind of this project again by finding that John Myhill in JSL 1952 ('A derivation of number theory from ancestral theory') already had answers to some questions that came up in re-writing a section of the book last week.

As is entirely familiar, we can define the ancestral of a relation R using second-order ideas: but it doesn't follow from that that the idea of the ancestral is essentially second-order (as if the child who cottons on to the idea of someone's being one of her ancestors has to understand the idea of arbitrary sets of people etc.) Which in fact is another old point made by e.g. R. M. Martin in JSL 1949 in his 'Note on nominalism and recursive functions'. So there is some interest in considering what we get if we extend first-order logic with a primitive logical operator that forms the ancestral of a relation.

It's pretty obvious that the semantic consequence relation for such an 'ancestral logic' won't be compact, so the logic isn't axiomatizable. But we can still ask whether there is a natural partial axiomatization (compare the way we consider natural partial axiomatizations of second-order logic). And Myhill gives us one. Suppose R* is the ancestral of R, and H(F, R) is the first-order sentence which says that F is hereditary down an R-chain, i.e. AxAy((Fx & Rxy) --> Fy). Then, putting it in terms of rules, Myhill's formal system comes to this:

  • From Rab infer R*ab
  • From R*ab, Rbc infer R*ac
  • From H(F, R) infer H(F, R*)
where the last rule is equivalent to the elimination rule
  • From R*ab infer (Fa & H(F, R)) --> Fb
which is an generalized induction schema. Myhill shows that these rules added to some simple axioms for ordered pairs give us first-order Peano Arithmetic. But do they give us more?

Suppose PA* is first-order PA plus the ancestral operator plus the axiom
  • Ax(x = 0 v S*0x)
i.e. every number is zero or a successor of zero. Then -- if we treat the ancestral operator as a logical constant with a fixed interpretation -- this is a categorical theory whose only model is the intended one (up to isomorphism). But while semantically strong it is deductively weak. It is conservative over PA. To see this note that we can define in PA a proxy for R*ab by using a beta-function to handle the idea of a finite sequence of values that form an R-chain, and then Myhill's rules and the new axiom apply to this proxy too. And hence any proof in PA* can be mirrored by a proof in plain PA using this proxy. (Thanks to Andreas Blass and Aatu Koskensilta for that proof idea.)

So the situation is interesting. Arguably, PA doesn't reflect everything we understand in understanding school-room arithmetic: we pick up the idea that the numbers are the successors of zero and nothing else. In other words, we pick up the idea that the numbers all stand to zero in the ancestral of the successor relation. So arguably something like PA* does better at reflecting our elementary understanding of arithmetic. Yet this theory's extra content does nothing for us by way of giving us extra proofs of pure arithmetic sentences. Which is in harmony with Dan Isaacson's conjecture that if we are to give a rationally compelling proof of any true sentence of basic arithmetic which is independent of PA, then we will need to appeal to ideas that go beyond those which are constitutive of our understanding of basic arithmetic.