Monday, October 29, 2007

Leopard, first impressions

Leopard arrived today, in its rather cool box. And I installed it by the book -- the book in question being the useful, confidence-inspiring, and very inexpensive e-book Take Control of Upgrading to Leopard. So I used SuperDuper! to clone my MacBook Pro's hard-disk onto a bootable external drive (well, two different ones actually), did an "erase and install" of Leopard, and then used the set-up assistant to migrate all my files back from one of the external drives. Doing things the longest way round like this, the whole business, after the initial cloning, took about two and a half hours. But much better safe than sorry. (The only hiccup was that the installer initially took a long term to recognize the presence of my laptop's hard disk, which would have been distinctly alarming, had I not seen talk of the phenomenon on MacRumors. And apart from losing the Cisco VPN Client -- the university computer services say they'll have a Leopard-compatible replacement available in a day or so -- everything seems to be working again just fine.)

What strikes you first, of course, is some of the eye-candy -- e.g. the new dock, semi-transparent menu bar and menus, the folder icons. Quite a few mac reviewers have hated all these (e.g. see the ars technica review which is very informative about the under-the-bonnet improvements). Well, I'm all for the dock and I quite like the semi-transparent menu bar; the transparent menus are I think are far too transparent; and the "recycled cardboard" folder icons seem quite out of keeping with the space-age look of the rest of the UI. Or at least, that's my two-pennyworth. And it is only worth about that much fuss (especially as my bet is that these things will be subtly adjusted in an early update for Leopard). Otherwise, the cleaned-up look of the windows across the system is all pretty neat, and the new Finder windows are that bit more useful. On the whole, things look terrific.

Still, looks aren't everything: here are four things I instantly really like about Leopard, and which even just by themselves make the upgrade worthwhile:

  1. The whole system is consistently quite a bit snappier (e.g. one bounce and iCal with seven calendars is open, similarly for Mail).
  2. Cover Flow and Quick Look are amazingly useful, as well as very pretty. For example, I have a folder into which I park PDFs and other documents as I download them. I can now just instantly browse through the folder to see what is in the various documents without opening the relevant application(s), and can eventually file them away or trash as appropriate.
  3. Spaces is a very nicely implemented way of getting much cleaner work-spaces. I'm an immediate convert. (One space for Safari, Mail, etc.; another space for TeXShop windows; etc. Very uncluttered.)
  4. Time Machine is wonderful. I was already pretty good about cloning my hard disk to a pair of external drives, one at home and one at work. But inevitably you do foul up and accidentally delete stuff. So I've set up a new big external drive to be an automatic Time Machine archive whenever I'm in my little study at home (drives have become so cheap, there's no reason at all not to err on the side of caution -- it would just be too painful now to lose everything): and I will still carry on cloning onto the other drives. Feels very virtuous!
Worth waiting for (and it can only get better).

Sunday, October 28, 2007

G. C. Lichtenberg

Fellow local blogger James Warren recently posted a seemingly depressing list of "top books" listed on Cambridge students' Facebook pages. But I'd not be too downhearted. Probably the moral is: don't believe all you read in Facebook entries! I know that when I was still a college fellow and "director of studies" and so able to get to know a few students very well over their three years here, I'd repeatedly be surprised when they eventually opened up about the books that they really loved and which meant something to them. I learnt a lot that way, about the students themselves, and about books too.

Just a couple of weeks ago -- I can't at all recall how it came up -- one of our graduate students warmly recommended to me the Hollingdale translation of excerpts from Georg Christoph Lichtenberg's The Waste Books. This version was quite new to me, and is a real delight. I had a much shorter collection of excerpts translated by Franz Mautner and Henry Hatfield almost forty years ago; and I first came across the aphorisms and their author in a favourite book that I had when a student, J. P. Stern's Lichtenberg: A Doctrine of Scattered Occasions. But the pleasure of re-discovery after a good few years is enormous.

I was moved too -- having found a number of enthusiastic reviews -- to send off for another book, Gert Hofmann's novel Lichtenberg and the Little Flower Girl (this indeed was the contents of the packet that should have contained Leopard!). But I found this really rather disappointing.

The novel is based on Lichtenberg's relationship with Maria Stechard, the thirteen year old girl he met selling flowers (he, a hunchback, was by then in his mid thirties); she became his housekeeper, then his lover, and died shortly after her seventeenth birthday to his intense distress (the story, though, is Beauty and the Beast, not Lolita). But the Lichtenberg in Hofmann's tale is just too far from the Lichtenberg I thought I knew from the Waste Books -- he seems a diminished and much less substantial figure than the highly successful and popular teacher, the science professor at Göttingen, who inhabits Stern's pages (he lives in too distant an alternative possible world). And die kleine Stechardin remains almost as blank in Hoffman's novel as she does in Lichtenberg's handful of references to her.

Saturday, October 27, 2007

The art of ordinal analysis

I've just come across Michael Rathjen's 2006 paper The art of ordinal analysis. A useful and pretty clear survey.

Friday, October 26, 2007

Not Leopard

Sigh. This was going to be the post where I gave you my calm, judicious, balanced, critical appraisal of the truly awesome OS X Leopard. So imagine my frustration -- or at least, fellow macheads will be able to imagine my frustration -- to find that (i) the announced package in my pigeon hole which I've at last been able to go in to pick up wasn't the expected one from Apple, and (ii) in fact the carrier tried to deliver the genuine article today, and there happened to be no-one in or around the Faculty Office to sign for it just at that time. How was that possible? Delivery rescheduled for Monday. It's not been my week.

Ah well. I'm on the mend from the earlier unpleasantness. Life goes on. And patience is a virtue. They say.

Tuesday, October 23, 2007

Mocking the pomos

I was amused to (re)discover Communications from Elsewhere's wonderful Postmodern Generator. Just refresh the link and you are served up each time with a new generous helping of a randomly generated postmodernist-style word-salad: very clever and very funny. And yes, intellectual rubbish should be mocked and parodied and satirized wherever we find it.

Much more seriously, but of course not unrelatedly, you are also served up with a constant link to Alan Sokal's page on the "Social Text Affair": if you don't know what that was, then exploring Sokal's page will be a very instructive treat. (Oh, and I was delighted to discover that Sokal has a another forthcoming book announced for next year.)

Monday, October 22, 2007

In bed with a Trollope

I've been struck down over the last few days, and - judging from previous experiences with the same unpleasantness - it will be a few more days before I'm really up and about again, and a while yet before really 100%. Still, I'm on the mend, and have gone through the stages of just about managing a Michael Dibden, then devouring a second, and now I am up to the delights of a Trollope again (so back to Barchester Towers).

And also, thanks to wireless networking, I can idly surf the web in bed. I hadn't previously noticed this excellent letter from Richard Dawkins, which ends:

Of course, university departments of theology house many excellent scholars of history, linguistics, literature, ecclesiastical art and music, archaeology, psychology, anthropology, sociology, iconology, and other worthwhile and important subjects. These academics would be welcomed into appropriate departments elsewhere in the university. But as for theology itself, defined as "the organised body of knowledge dealing with the nature, attributes, and governance of God", a positive case now needs to be made that it has any real content at all, and that it has any place in today's universities.

Spot on. It is depressing to find, though, that some comments on Dawkins would put philosophy in the same boat as theology. Now, a few ignorant remarks in this vein wouldn't matter in themselves -- but I suspect that they are symptomatic of a much more widespread deep ignorance of what analytical philosophy is about, even among those who should perhaps know. How else do we explain that the powers that be at Cambridge (of all places) think it is perfectly respectable to peg the philosophy faculty here at just twelve for the last nineteen years while our student numbers have grown by almost 80% -- and indeed, they proposed to cut the number of faculty a few years ago -- while there are twenty one(!) theologians in the divinity faculty. Sigh.

But Trollope would have been no more surprised by the oddities of ancient universities than by those of ancient churches.

Wednesday, October 17, 2007

Anjan gets real

Continuing the project of bankrupting my readers by recommending unmissable books to buy, let me add another warm recommendation, for Anjan Chakravartty's A Metaphysics for Scientific Realism which has just appeared on the new book shelves at the University Press's bookshop. On a quick browse through, it looks predictably terrific: Anjan is certainly tackling the right issues, and I've liked other things that he's written about realism. I see that I come in for stick in Chapter 7 for some overhasty stuff I wrote a decade ago, and probably quite right too.

(It's a handsome bit of book production too, as is usually the case with CUP. That's more than can be said for the "Schilpp" volume on Dummett which has just arrived: the typography is an insult to the eyes.)

Tuesday, October 16, 2007

Harmless geekery

I was going to sound off on the subject of tripos reform: but on second thoughts, it's probably safer to indulge in harmless geekery instead. So ... (roll of drums):

  1. There's now another much bigger, much better edition of the LaTeX Graphics Companion. It has, inter alia, some worryingly enticing/timewasting things to do with Beamer presentations ... I suspect this might be fun.
  2. And from today you can pre-order Leopard which will of course make us all* so very much more productive, clear-thinking and happier. I just know this will be fun.
*For "all" read: all right-minded Mac users.

Friday, October 12, 2007

Semantics, Toyota style

Our math logic reading group is going through Manzano's Model Theory as therapy/revision before tackling Hodges's Shorter Model Theory. I'm not sure that Manzano was, after all, a good choice; though equally it isn't clear what would have served our purposes better.

Anyway, I was struck again by the still-standard logician's habit of treating the formal semantics of first-order languages by explaining how to extend an interpretation by assigning values in the domain to each and every variable of the language -- and then later proving that e.g. different assignments to variables other than those that appear in the wff being evaluated don't make any difference. I know this is how Tarski did things, but isn't there something inelegant about stocking up on assignments of objects to variables only not to use infinitely many of them?

It is reminiscent of the bad old overstocking habits of industry! Toyota-style "just in time" production, where we only stock up with what we actually need next, is better!! Likewise, surely, giving a semantic story where we deal with e.g. "AzFz" by talking about alternative ways of extending an interpretation by assigning an object to "z" (treated as a parameter/temporary name) is more elegant and more intuitive. That way, we just talk about alternative extensions of an interpretation to cover particular variables as and when we need them.

Is there a good reason, other than historical piety for doing things the first, Tarski, over-stocking way, rather than the Toyota way? We couldn't think of one.

Gödel: chapter one online

I've put Chapter One of my Gödel book -- very short, and hopefully accessible -- online at the book's website here. Perhaps not of terrific interest to too many visitors here, as it is very introductory, but you can always tell your students!

Wednesday, October 10, 2007

Eating ice cream in Naples

There are noticeboards immediately outside my office, where seminars in other faculties are advertised. It's often rather jaw-dropping what other people think it worth getting up to. For example, the History Faculty is running a series on “Consumption”, and the final meeting of the year is on “Paradigms of enlightenment and the consumption of ice cream in late eighteenth-century Naples”. Gosh. That sort of thing must be really demanding to do research on (and the library visits to Naples must be a bit of a pain too)!

Actually, I can see the topic might be quite amusing (and to be fair, it doesn't sound a positive intellectual disgrace, like some of the post-modern bullshitting that goes on around here). But still ....

I've talked to a couple of our grads in the last few days about the significance of the proof-theoretic ordinals for various extensions of first-order Peano Arithmetic, about whether you can diagonalize out of the hypercomputable functions (on various natural understandings of hypercomputation), and about Kripke semantics and set theory (the issue at dispute between Jonathan Lear and Alex Paseau). Or, let's be honest, the students in question sent me stuff/talked to me, and I nodded along trying to look slightly intelligent. I guess that outsiders would think this is an odd way for us to be spending our time too. But there is a difference, for all that. For this stuff is really difficult, requiring serious technical knowledge of the relevant maths, plus an even more serious amount of hard disentangling of intricate conceptual puzzles, and relates immediately to genuinely deep questions -- in this case, in the foundations of mathematics and computation. It requires a heavyweight amount of intellectual firepower to get to say anything sensible about these matters. While ice-cream eating patterns ...?

Tuesday, October 09, 2007

Blackburn, religion and respect

I have mentioned Louise M. Antony's Philosophers without Gods before. But I have only just discovered that perhaps the best of the essays in the book -- though it is a close run thing: the collection is consistently good -- is available in draft form here. It is Simon Blackburn's piece 'Religion and Respect', which is simply terrific. Here he is, for example, writing about the way that a proper human respect for emotions can be hijacked by those urging that we should respect religious beliefs:

I have said that holding a false belief does not give anyone a title to respect. Insofar as I cannot share your belief, I have no reason to respect you for holding it -- quite the reverse, in fact. But the same is not true of emotions. If I happen upon the funeral of a stranger, I cannot feel the same grief as the close relatives and mourners. But I don’t think they are making any kind of mistake, or displaying any kind of fault or flaw or vice. On the contrary, we admire them for giving public expression to their grief, and if they did not show this kind of feeling they would be alien to us, and objects of suspicion. It is fair to say that we ought to respect their grief, and in practice we do. We may withdraw from the scene. Or, we may inconvenience ourselves to let them go ahead (we turn down our radio). Or, we may waive demands that would otherwise be made (we give them time off work). Similarly a birth or wedding is a happy occasion, and it is bad form to intrude on them with trouble and grief (let alone prophesies of such, as in many fairy stories). ... Peoples’ emotions are important, and whether or not we can empathize with them, we do accord them time and space and a kind of shelter.

Unfortunately, it is a gross simplification to bring the essence of religion down to emotion. The stances involved are far more often ones of attitude. And it is a fraud to take the space and shelter we rightly offer to emotional difference, and use it to demand respect for any old divergence of attitude. The relevant attitudes are often ones where difference implies disagreement, and then, like belief, we cannot combine any kind of disagreement with substantial respect. Attitudes are public.

Suppose, for example, the journey up the mountain brought back the words that a woman is worth only a fraction of a man, as is held in Islam. This is not directly an expression of an emotion. It is the expression of a practical stance or attitude, that may come out in all sorts of ways. It is not an attitude that commends itself in the egalitarian West. So should we ‘respect’ it? Not at all. ... I think it is a dreadful attitude and it is a blot on the face of humanity that there are people who hold it and laws and customs that express it.

The whole essay is shot through with the same straight-talking humane good sense. Read it! Even better, buy the book.

Monday, October 08, 2007

ACA0, #6: ACA0 vs ACA, and more

Rather than keep posting longer and longer contributions on ACA0 here, I've put together some thoughts on this and related stuff into the beginnings of a long draft essay. This is very much work-just-starting rather than work-in-progress, and I'm not really sure where the arguments are going to end up. But anyone interested will find the current version here. It goes without saying that feedback would be very welcome!

Friday, October 05, 2007

ACA0, #5: An aside on PA + Th versus T(PA)

Suppose we bolt onto to first-order PA the axioms of Th, the arithmetization of a natural theory of a new truth-predicate T, in a way that – prima facie – shouldn’t upset even a deflationist/minimalist about truth. As we'd expect, the composite theory PA + Th proves each biconditional T("φ") ↔ φ for sentences φ. And as we’d also expect -- though it isn’t entirely trivial to demonstrate it -- PA + Th is conservative over PA: it proves no new arithmetical truths.

But now let’s reflect. Bolting the arithmetical truth-theory onto PA ‘externally’ leaves us, in particular, with just the same induction axioms as we started off with. However, you might say, why stop there? It would seem that the line of argument that I sketched a few posts ago for being generous with induction will apply again, and will motivate extending the induction axioms from instances involving just the original L-predicates (L is the language of PA) to instances involving the new predicate T too. So let T(PA) be the theory we get taking PA plus Th plus the closures of all instances of the first-order induction schema for new predicates constructed from T as well. Then, the argument seems to be, T(PA) should be as compelling a theory as PA + Th. But as is well known, T(PA) is not conservative over PA (it proves Con(PA) for a start).

However, the inflationary argument for generosity with induction can be resisted. But how exactly? A deflationist might be tempted to say: ‘As a deflationist, I don’t accept that truth is a genuine property: more precisely, ‘T’ in the theory Th doesn’t express a genuine property, so we can’t use it in inductive arguments.’ But this isn’t in fact terribly helpful, unless augmented by an independent account of the initially murky notion of ‘not expressing a genuine property’. So let’s proceed more carefully.

Suppose someone with a taste for formalizing his knowledge – call him Kurt – accepts PA (this is the apparatus he uses in fixing his arithmetical beliefs). Suppose we now offer him the axioms Th as a partial characterization of the uninterpreted new predicate T. If we give Kurt any particular number which happens to be the Gödel number of a sentence φ then he will in principle be able to prove the corresponding theorem T("φ") ↔ φ. What he can’t do, since he doesn’t yet have induction axioms for T, is prove anything general about T to the effect that, for every n, if n = "φ" for some φ, then T(n) ↔ φ, or else T(n) ↔ ⊥. So, while still working from inside PA + Th, Kurt has no way of knowing whether T(n) has been defined for all numbers n. In this sense, then, Kurt doesn’t yet know whether T expresses a determinate property of numbers. So, for a start, he isn’t entitled to employ universal quantifier introduction applied to complex expressions involving T. Hence, he won’t be in a position to establish the quantified antedecent needed to make use of an instance of the induction scheme involving a predicate embedding T. In other words, Kurt is not entitled to make any use of the extended instances of induction allowed in T(PA). In sum, a suitably cautious Kurt – so far – is in no position to inductively inflate PA + Th.

And now there’s an added wrinkle. For we can see in retrospect that talk of the theory PA + Th in fact glosses over an issue that matters. In bolting the axioms Th onto the theory PA, were we intending these new axioms to interact merely with sufficient logical rules governing the elimination of quantifiers and the use of conditionals to enable the extraction of the information packaged in those axioms? Or were we intending that the whole weight of first-order logic can be brought to bear on axioms from either pool – so that we can trivially prove, for example, ∀x(T(x) ∨ ¬T(x))? The issue doesn’t arise for practical purposes. But now the question has been raised, we see that the second alternative overgenerates in enabling us to deduce more than we are entitled to in just being given the axioms Th as (partially) characterizing the new predicate T. A cautious Kurt should only use quantifier elimination and the conditional rules on Th.

Note, it is not being suggested that Kurt reject the instances of the induction schema that embed the predicate T as false. How can he? As the argument for inductive generosity reminds us, if the antecedents of such an instance are true, the consequent has to be true too. Rather, as we said, the point is that Kurt so far doesn’t find himself entitled to get to the starting line for using such an instance.

Of course, Kurt can now start to ‘think outside the box’. He can stand back from PA, think about his practice, commit himself explicitly to the thought that every sentence of L is either true or false, reflect this this thought using an arithmetized truth-predicate which he takes to be fully defined, so induction must apply to it -- and so he comes to endorse T(PA). We certainly don’t want to ban Kurt from such reflections or suppose that he must make some mistake if he takes on these further thoughts, and so comes to be able to demonstrate Con(PA), at least to his satisfaction. The point to emphasize is only that they are further thoughts, not commitments already implicity accepted in rationally endorsing PA in the first place. (Cf. Isaacson's Thesis.)

We’re off!

Loads of apples
We're off! The first logic lecture of the year done. No disasters. The data projector behaved. The last slide popped onto the screen with one minute to go. I talked mostly good sense. Bits of explanation had a beginning, middle and end -- even sometimes in that order. Phew. After all these years, the first lecture is still nerve-racking. Piece of cake from now on.

But the lecture room doesn't look much like that picture (of a scene that should really gladden the heart of Steve Jobs). I noticed just one laptop -- odd, as I think a lot of our students have them, but they just haven't yet got into the habit of bringing them en masse to classes here. But give it another year or two ...

Thursday, October 04, 2007

Giving it to ’em with both barrels

There's a kind of stupidity that is, to put not too fine a point on it, morally offensive. I don't mean common-or-garden dimness and/or a propensity for making daft mistakes (we've all been there!). I mean the "only I'm right and the rest of the world is wrong", "there's a conspiracy of mathematicians to cover over Cantor's mistakes", "Gödel proves that mathematics is self-contradictory" kind of stupidity that plagues the net and that is wilfully almost impervious to all reasoned response. So what do we do about the landslides of garbage that you find on newsgroups like sci.logic?

Well, let's ignore a lot of it. But on the other hand, such newsgroups do get thousands of visitors a day and someone ought to stand up for the good name of logic! So I do think it is worth occasionally dropping by during an idle couple of minutes over a cup of coffee and giving the idiot du jour a blast or two with both barrels. A combination of argued refutation and more or less brutal mockery does (eventually) make many of the pretentious buffoons shut up. And if this means that, from time to time, even just a couple of students casually browsing past don't get taken in by superficially plausible nonsense, then why not?

Some do crosswords, some do sudoku: as an alternative way of procrastinating, I can recommend a few bouts of idiot-bashing as mildly amusing fun. Though you can in fact learn a bit from doing it, and learn more from reading the contributions of others who are batting for the sanity-and-reason team. Give it a try!

Wednesday, October 03, 2007

Of making many books (again)

Michael Potter has just told me that the Schilpp volume on Michael Dummett is out. (Of course, the Library of Living Philosophers hasn't been edited by Schilpp for nearly twenty years, but that's how we still refer to the books, isn't it?) So that's another Amazon order winging its way to me. Nearly a thousand pages too. Gulp.

Monday, October 01, 2007

Suddenly there is bustle ....

After weeks when the faculty has been almost deserted apart from the admin staff, suddenly there is bustle in the philosophy grad. centre outside my room, and the place is filling up again. It is one of the main delights of being in Cambridge that we have such bright graduate students, and I really rather miss the logicians when they are not around. Sadly, it's not obvious where the successors of the current crop are going to come from, as none of this year's M.Phil. intake seems inclined to go in the direction of the serious stuff. But Michael (Potter) and I will try to get a convert or two ...

Anyway, this term's teaching for me ought to be a lot of fun. First year logic lectures (trying to enthuse even the symbol-phobic to work through my intro book); third year Gödel's Theorems lectures (chatting about themes in my Gödel book); the Math Logic Reading Group (this year is model theory year: we are kicking off with Manzano's book as revision, then aim to do Hodges's Shorter Model Theory); Michael and my Logic Seminar (phil. logic this term -- Davidson and Dummett revisited); and a handful of modal logic lectures. I can have no complaints about that!

ACA0, #4: And what about ACA?

I've been trying to get my head around the significance of (i) the relation between ACA0 and ACA (the theory you get by keeping arithmetic comprehension, but allowing induction for any second-order wff) and (ii) the relation between ACA and T(PA) (the theory you get by adding to first-order PA a Tarski-like truth-theory AND allowing the new truth-predicate to appear in induction axioms). The technicalities hereabouts are reasonably clear: but -- as I say -- their philosophical significance is not. So watch this space.

We might be tempted by the following argument for being generous with induction in moving from (i) ACA0 to (ii) ACA, or indeed in moving from (i') PA + T, the result of taking PA and bolting on a truth-theory "from outside" while leaving the induction axioms untouched, to (ii) T(PA):

Instances of induction are conditionals, telling us that from F(0) and (Ax)(Fx -> Fx') we can infer (Ax)Fx. So we can derive(Ax)Fx. when we have already established the corresponding premiss (i) F(0) and can also establish (ii) (Ax)(Fx -> Fx'). But if we can already establish (i) and (ii) then trivially we can (iii) case by case derive each and every one of $F(0), F(1), F(2), ... However, there are no 'stray' numbers which aren't denoted by some numeral; so that means (iv) that we can show of each and every number that F holds of it. What more can it possibly take for F to express a genuine property that indeed holds for every number, so that (v) (Ax)Fx is true? In sum, it seems that we can't possibly overshoot by allowing instances of the Induction Schema for any open wff of the language we are working with. The only usable instances from our generous range of axioms will be those where we can in fact establish the antecedents (i) and (ii) of the relevant conditionals -- and in those cases, we can't go wrong in accepting the consequents (v) too.

This argument for inflating ACA0 to ACA, or PA + T to T(PA) -- and thereby getting to be able to prove new arithmetical sentences like Con(PA) -- is surely too easy! But saying exactly why isn't so easy ...