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.
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.
Went to the one-day Tractatus workshop here in Cambridge (the last in a series that has mostly taken place in Stirling). I was there in my role as the village sceptic.
Julian Dodd and Michael Morris kicked off with a joint talk on Making sense of nonsense. What are we to make of the fact that Wittgenstein officially seems to think of his claims in the Tractatus as nonsense (yet in the Preface he says ‘the truth of the thoughts communicated here seems to me to be unassailable and definitive’)? One line is that claims of the Tractatus communicate truths that can be shown but not said (the ‘truth-in-nonsense’ view). Another line is that actually not all the claims are non-sensical (the ‘not-all-nonsense’ view). Julian and Michael think there is a third way. All of the claims of the Tractatus are nonsense and they don’t communicate any genuine truths indirectly either (the ‘no-truths-at-all-view’); the prefatory remark is just another bit of philosophical nonsense.
This was interestingly done, though they also seem to want to suggest that the movement of thought in Tractatus, read their way, naturally leads to its mystical conclusion. I just don’t see that. Somewhere in the middle of the 6.somethings, sensible readers of the Tractatus can perfectly well think “Oh come off it, Ludwig, lighten up!”. The mystical guff about feeling the “world as a limited whole” is no more an upshot of what’s gone before than would be, say, something like Lichtenberg’s wryly amused attitude to the scattered occasions of his life.
Next up, Fraser MacBride and Peter Sullivan talked about Ramsey, Wittgenstein, and in particular the argument about complex universals. Peter hinted at, but didn’t in this talk really explore, an interesting thought. Ask: ‘How much do the principles of logic reveal about the nature of things/the constitution of facts?’ It seems Frege answers “a great deal” (logic reveals the deep distinctions between objects, properties, properties of properties etc.) while Ramsey answers “next to nothing” (e.g there isn’t a deep object/universal distinction reflected in the logical subject/predicate form). Peter suggested that there is a lot in the Tractatus that comes from Frege and a lot that feeds into Ramsey’s position. Which suggests that Wittgenstein’s position might be an incoherent mixture.
Finally, Mike Beaney talked about the chronology of the interchanges between Frege and Wittgenstein. And Michael Potter talked more specifically about when W. might have learnt from F. to distinguish sharply complexes and facts (early, according to Michael).
Which was all mildly instructive, though the discussions sometimes became bogglingly theological, in the way that Wittgenstein-fests can do. It was occasionally like listening to rounds of ‘Mornington Crescent’ without the jokes (and no, I’m not going to explain!).
I leave it too long between visits to the Fitzwilliam. But since the really excellent new café started up, I’ve been going rather more often. Take a book, read over a coffee, then take a break to look at just a few pictures (that is surely much the best way to “do” a gallery). I was very struck again today by The Holy Family by the seemingly rather unregarded Sassoferrato. I just wish I was clearer in my mind about how an unbeliever should regard religious art, without double-think or sentimentality.
Reading the Guardian isn’t always good for my blood pressure. Today there is an article under the name of Tony Blair, no less, saying how important it is that “we maintain and improve the high reputation of higher education in Britain” (note, it is the reputation that has to be improved). But not, of course, because education might be a good in itself; no, it’s because we want to sell the product and make the most money possible out of overseas students.
But I wonder who is going to staff these high reputation universities? Some of our brightest and best might enjoy a year of graduate study; but even here in Cambridge they seem increasingly reluctant then to launch into a PhD. And who can blame them? It could be six years more before they are in the running for a permanent academic job. Getting one is a very chancy business (since employment numbers are kept down by ludicrous staff-student ratios). And the pay is then dreadful, at least compared with what they might hope to get elsewhere. Oh well …
Maybe it is the advancing years, but occasionally there are those moments of panic. I think “Surely it must be the case that P“, guess I can see how to prove it, check out an obvious source or two, google around, and then am perhaps a bit surprised not to come across a straight proof of P. And sometimes my nerve fails: just occasionally I’ve asked on FOM whether indeed P. I’ve invariably got helpful replies. A couple of days ago, I was asking — in effect — how far up the Friedman/Simpson hierarchy of second-order arithmetics we have to go before we can prove Goodstein’s Theorem (something not mentioned in Simpson’s wonderful book). Before the day was out, I got a couple of really useful private responses, and there are now two brief but equally helpful replies on the list from Dmytro Taranovsky and Ali Enayat. What a fantastic resource this is: I’m not sure how my current book project would be going if it weren’t for FOM and its archives, and I’m immensely grateful.
Oh, and the answer to my question is that ATR_0 is enough. Which I should have got from Sec. V.6 of Simpson (which tells us that ATR_0 is good at handling countable well-orderings).
I’m ploughing on as fast as I can to get my Gödel book finished. I try to keep in mind the good advice that C.D. Broad used to give. Leave your work at the end of the day in the middle of a paragraph which you know roughly how to finish. That way, you can pick up the threads the next morning and get straight down to writing again. So much better than starting the day with a dauntingly blank sheet of paper — or now, a blank screen — as you ponder how to kick off the next section or next chapter. Instead, with luck, you face that next hurdle while on a roll, with the ideas flowing.
Well, it works for me …
I’ve been using my old desktop (well, under-the-desk) G4 Mac less and less recently, so I’ve reorganized things so that I can mainly use its fairly new and very nice LCD monitor with my laptop when I’m at home. Wonderful. I can have the TeXShop window with the PDF of the book I’m working on displayed on the external monitor (a full page at 150%), and the TeXShop drawer open too, with all the section hyperlinks: and then the source file of the current chapter and other stuff like BibDesk is on the PowerBook screen. Why on earth didn’t I think of doing this before? It’s LaTeX heaven!
So message to myself: no more lusting after 17″ laptops — keep to a 15″ one for portability, and get the additional real estate when you need it by plugging an the external monitor.
It all seems a very long way from thinking that WordStar on an ACT Sirius was really, really neat …
A blog with that title just has to be worth a link! (It’s by a group of University of Connecticut grad students — interesting content, and there are some nice links out into the philosophical corner of the blogosphere.)
The world inhabited by the philosophy graduate student has been changing fast in the last few years (and in very good ways). Blogs, on-line forums, and the rest obviously can do a lot to counteract the depressing sense of isolation that used to bug people writing their PhD. If local experience is anything to go by, those of us involved in running grad programs need to be thinking more about how best to help students make use of the changing world. Though, on second thoughts, they seem — as usual — to be doing pretty well without us …
The Advanced Book Exchange is simply terrific, isn’t it? Search over thirteen thousand second-hand book sellers, and — more often than not, in my experience — you can find what you are looking for, and frequently at a decent price.
Of course, there’s a downside. Booksellers can now easily check on-line what is rare and what is not, and check what others are charging. It’s not that many years since I picked up the complete Principia Mathematica for £30: I can’t imagine a bookseller now being so ignorant of its true worth. Still, plenty of bargains are to be had: a copy of Wolfram Pohlers’ Proof Theory has just dropped through the door. I paid all of $5.95 plus postage.
It’s a bit disturbing, then, to read a paragraph in Private Eye which reports that abebooks have been hiking the commission they charge to booksellers and are about to add more charges. It would a great loss indeed if they price themselves out of having such a wide coverage of booksellers.