2008

It’s around now that the reviews pages and the literary supplements carry lists of their Books of the Year. So what really worthwhile books on logic matters were first published in 2008? Alan Sokal’s Beyond the Hoax is amusing in parts but added less than I’d hoped to Intellectual Impostures; John P. Burgess’s collected papers are worth having together in Mathematics, Models and Modality (but content-wise, that doesn’t really count as a new book); Graham Priest’s second edition of Introduction to Non-classical Logic is a great textbook but isn’t exactly full of news for old logic hands! So ok, what can I particularly commend that pushes ideas on in a novel and thought-provoking way?

Ermmm …. This is really rather embarrasing. I can’t think of anything to suggest! The books that I have read and most enjoyed recently seem all to have been published in previous years.

Ah, Hartry Field’s Saving Truth from Paradox was published this year, but it is still sitting on my shelves waiting to be read. But what have I missed? Perhaps I’m forgetting or have just not registered the publication of some terrific books over the last year in the areas of philosophy of science/philosophy of logic/philosophy of maths. What would you recommend from the class of ’08?

I’m still working away on Parsons’s book — and I’m rather stumped by his claims about “intuitive knowledge”. One worry is this: he introduces the notion in cases where we acquire propositional knowledge by, as it were, “just seeing” e.g. that “|||” is the successor of “||”. But he fairly rapidly wants to extend the notion of intuitive knowledge so that it is preserved under some, but not all, logical inference, and some but not all applications of arithmetical induction. And I just can’t see what the constraint on the notion is that rules some cases in and others out — for that constraint certainly isn’t implicit in the cases which introduce the notion in the first place.

Well, be all that as it may. For light relief — and to see if any sideways light can be thrown on Parsons’s on intuition more generally — I’m reading Marcus Giaquinto’s Visual Thinking in Mathematics (OUP, 2007). The first few chapters already show that, unsurprisingly, the book is written with Marcus’s customary clarity and good sense. I’ll report back in due course as I read more: but already I’m having to backtrack a bit and slightly rethink things that I earlier wrote on Parsons.

There has just been a national petition started, supporting a proposal to make Charles Darwin’s birthday (12th February) a UK Bank Holiday. Well yes, let’s celebrate the great man — and just possibly send a signal marking some opposition to the noisy fringe of know-nothing, anti-science, religious loonies.

Why not (1) spend just a moment to sign the petition at http://petitions.number10.gov.uk/Darwins-day/ (if you are a British citizen or resident)? (2) Warmly encourage/nag a couple of friends to sign too. And then (3) spread the word to other groups/e-lists that you belong to — e.g. by distributing the text of this post.

(For here, perhaps, is a meme worth propagating!)

Delighted to report that Fraser MacBride has accepted the offer of a job in the Cambridge philosophy faculty. A terrific outcome for our recent appointment process. And a real strengthening of philosophy of maths as well as metaphysics here (as the post replaces an ancient philosopher).

Without giving anything away, knowing the shortlist, things are looking good for the Knightbridge chair replacement too.

The way things are developing, at this rate I won’t want to retire …

Ok, ok — I know that, by now, it is a such a banal thing to say: but I’m still frequently bowled over by the quality of information that there is out there on the net, freely available, the result of the labours of amateur enthusiasts. I wanted to check whether a recording of Richter playing D960 duplicated one I already had. Within three minutes I had the answer: one Paul Geffen maintains a Richter discography, and here’s the Schubert page. Terrific.

Gregory Landini is talking at our Logic Seminar next week about Frege and Russell on cardinal numbers. Since our students tend to know a lot more about Frege than Russell, we had a preparatory session on Russell last week, in which I got a chance to show off my stunning historical ignorance. But it was fun to re-read (after a long time) the opening chapters of the Introduction to Mathematical Philosophy. These were, as much as anything, the pages that got me interested in philosophy and the foundations of mathematics when I was a maths student.

Fun to re-read, but also oddly very disappointing. Chapter II starts with stirring words which I well remembered: ‘The question “What is a number?” is one which has been often asked, but has only been correctly answered in our own time’ (meaning, of course, in 1884 in the Grundlagen). But I’d quite forgotten this passage, later in the same chapter, where Russell writes

We naturally think that the class of all couples is different from the number 2. But there is no doubt about the class of all couples: it is indubitable and not difficult to define, whereas the number 2, in any other sense, is a metaphysical entity about which we can never feel sure that it exists or that we have tracked it down. It is therefore more prudent to content ourselves with the class of couples, which we are sure of, than to hunt for a problematic number 2 which must always remain elusive.

So Russell’s stirring words are misleading: he isn’t after all claiming to have located, thanks to Frege, the one true metaphysical story about numbers (as classes of classes). It’s rather that here we have one way of replacing a problematic entity with something clear and sharply defined that can do the job. And then, of course, Russell is cheating. There isn’t such a thing as the class of all couples. Far from there being no doubt about it, he doubts it himself: his official story has a type hierarchy, with classes of couples at each level of the hierarchy above the bottom two. The seductive clarity of the opening chapters IMP is sadly only superficial!

What are we to make of this passage?

Among the triumphs of set theory are Gödel’s Incompleteness Theorems and Paul Cohen’s proof of the independence of the Continuum Hypothesis. Gödel’s theorems in particular had a dramatic effect on philosophical perceptions of mathematics, though now that it is understood that not every mathematical statement has a proof or disproof most mathematicians carry on much as before, since most statements they encounter do tend to be decidable. However, set theorists are a different breed. Since Gödel and Cohen, many further statements have been show to be undecidable, and many new axioms have been proposed that would make them decidable.

Well, we might complain that this is at least three ways misleading:

1. Gödel’s Incompleteness Theorems are a triumph, but not a triumph of set theory.
2. Gödel’s Incompleteness Theorems do not show that “not every mathematical statement has a proof or disproof”.
3. Cohen’s and Gödel’s results are significantly different in type and shouldn’t be so swiftly bracketed together. While the Cohen proof leaves it open that we might yet find some new axiom for set theory which settles the Continuum Hypothesis (and other interesting propositions which can similarly be shown to be independent of ZFC), Gödel’s First Theorem — perhaps better called an Incompletability Theorem — tells us that adding new axioms won’t ever give as a negation-complete theory (unless we give up on recursive axiomatizability of our theory).

I’m slightly embarrassed to report, then, that the rather dodgy passage above is by a local Cambridge hero, Timothy Gowers, writing in the Princeton Companion (p. 6). Oops. I rather suspect he didn’t run this bit past his local friendly logicians …

In 1997 — before I got the job back in Cambridge — I was travelling up and down to lecture: and I happened to be here the morning after the Blair victory. And, a bit bleary-eyed from a late night in front of the election results show, I bumped into a friend in town on a lovely morning and we sat outside a café drinking coffee and thinking how much better the world seemed. We knew Blair was an unprincipled opportunist, but he was — so to speak — our unprincipled opportunist, and there were, we hoped, enough decent people around him to keep things on track. That was, it turned out, wildly over-optimistic.

It is easy to lose heart. And perhaps (I wish I could say “unbelievably”) 56 million Americans did just vote to try to put an egregious cartoon character within a heartbeart of the presidency. But, at least for today, let’s be a bit optimistic again.