Logic Matters

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.

Non-philosophical events have really been rather stressful over the last six weeks or so. While I’m keeping my head above water as far as teaching and seminars are concerned, lots of other things — like sounding off here! — have been pushed right into the background. And I’m going to pressed for time this coming week too. For I’m down to respond to Luca at a Faculty Colloquium this Friday: we’re talking about Graham Priest’s animadversions against the iterative conception of sets, and I need to write a talk. How on earth this will go down with a general audience of non-logicians, heaven alone knows. But then, the whole idea of a general colloquium seems to me mildly daft. Still, the topic is a good one to think about.

Indeed, logic matters have been a lot of fun this last week, and a very welcome distraction to keep me sane. Two of our MPhil students gave very nice and very helpful presentations at the two seminars — one on Chap. 3 of Parsons’s book in the reading group on that, and one on Big Typescript §113, the section on “Ramsey’s Theory of Identity”. None of us ended up very much clearer about what Parsons’s noneliminative structuralism comes to exactly: but I’ll return to that here shortly. Lectures are going pretty enjoyably too (for me, at least!). I’m probably going too slowly talking about Gödel’s theorems to cover everything I really should, but at least people seem to still be on board. And I unthinkingly produced a frisson of mildly scandalized reaction in the first year logic course by saying en passant that, while Wittgenstein might have been a great philosopher — he’d just come up in the context of talking about the idea of a tautology — he seems to have been a bit of a shit as a human being. Still, we do need to encourage some healthy disrespect!