Year: 2007

Issacson Day

My colleague Michael Potter organized a one-day workshop today focusing on issues arising out of Dan Isaacson’s paper, “Arithmetical truth and hidden higher-order concepts”, which appeared 20 years ago.

It’s quite difficult to tease out the full content of Dan’s claims about the status of first-order PA in his paper, but a core claim — I call it Isaacson’s Thesis — is surely this:

If we are to give a proof for any true sentence of the language of first-order PA which is independent of PA, then we will need to appeal to ideas that go beyond those that are required in understanding PA (and more generally, but more vaguely, beyond those that are required to understand the structure of the natural numbers and thus elementary pure arithmetic).

Dan wants to say more (though that’s where things get a bit murkier), but the core Thesis is reasonably clear and interesting in itself — and if not even that is true, then Dan’s stronger claims won’t be true. In my talk “Ancestral Arithmetic and Isaacson’s Thesis” (NB the link is to an unrevised version!), I suggested that we can construct a formalized arithmetic PA* with a primitive ancestral operator which in fact is within the conceptual grasp of someone who understands PA. If PA* enabled us to prove new theorems in the language of PA that would be a strike against the Thesis. But I outlined a proof that PA* is in fact conservative over PA, a result not just in harmony with the Thesis but giving it positive support. (The talk went pretty well, but I made a bit of a hash of the questions, but what’s new?)

The other talks were by Ben Colburn (on of our PhD students) and Hannes Leitgeb, who talked more about what we might reasonably mean by saying with Dan that PA is sound and complete for “arithmetical truth”. And Luca Incurvati (another of our students) and Dan Isaacson himself talked about analogues for set theory of Dan’s thesis for arithmetic. In a way, the day ended too soon, as the discussions after the final paper by Hannes were just seeming to get to the heart of issues about arithmetic when we broke up. But a very productive and enjoyable day all the same.

Paul Teller in Cambridge

Paul Teller is in Cambridge, and it is very good to see him. He gave a talk in HPS on “Provisional knowledge”. Here’s his abstract (there’s more on his website).

Physics, and science generally, rarely function according to the mechanist tradition of founding all scientific knowledge on ‘shaped matter in motion’ of the parts of a system. Rather we employ a vast range of explanatory strategies a great many of which work in terms of ‘stripping detail’ when detail is not relevant to the problem at hand. Most of these strategies involve some level of idealisation, inaccuracy, or distortion, which raises the worry: When accounts in science involve distortion, how can they count as knowledge? This problem motivates reconstruing knowledge, and in particular its requirement of (exact) truth in its content component, in terms of the kinds of standards that require something less than perfect precision and accuracy, much as the context and interest dependent standards that we apply for representational accuracy of things such as maps and pictures. Since, logically, no evaluation by comparison with an unrepresented reality is possible, evaluation of any representational scheme can take place only on the basis of some other (in general) relatively precise and accurate scheme; and the scheme that functions as our platform for the moment can only be evaluated pragmatically.

I think, though, that I’ve really lost my taste for philosophy done at this level of stratospheric abstraction, jetting over the terrain at great speed. All those years editing Analysis evidently have taken their toll. We had a fun evening arguing over dinner, but I’m not sure that my exhortations to take things slowly got us very far!

Serendipitous Kreisel

abebooks.com is a wonderful thing — you can so often find cheap copies of in-print books that you want, and (often not-so-cheap) copies of out-of-print books as well. The trouble is that booksellers can of course use the site too to find out what particular books in their stock are worth, and these days you rarely pick up real philosophical or logical bargains sold at silly prices because the seller has little clue of their value (I not so very long ago picked up a complete Principia in excellent condition for £30 — that sort of thing wouldn’t happen now).

But you still make serendipitous little finds. In the “all hardbacks 50p” bargain tray outside a little bookshop at a National Trust house, a copy of Bertrand Russell: Philosopher of the Century. Which is very much a period piece, reminding us how the reputations of philosophers can change so markedly. But there is an intriguing 72 page essay by Kreisel, ‘Mathematical logic: what has it done for the philosophy of mathematics’, which I confess I can’t recall reading more than a few pages of before. Like other Kreisel papers, you just wish it were written in a more conventional, less allusive (ok, more flat-footed) style. Just imagine what Kreisel’s reputation would be if he had been able to write with e.g. Putnam’s directness and lucidity (Putnam has a piece in the collection too). So often, he seems to have got there first, but we — well, most of us — can only really see it with hindsight.

Logical options

There’s an afternoon planned soon to review the Faculty’s teaching, so it will be a good occasion to rethink some of our current arrangements for logic teaching. At the moment this looks to me to be about the right pattern, given the calibre of our students (good) and our resources (limited).

  • First year, propositional and predicate logic by trees (up to and including completeness for propositional trees: this is in fact what we do at the moment, using my Introduction to Formal Logic). We should also perhaps have a few lectures on set notation etc. (again, something we do at the moment). That’s compulsory for all students.
  • Second year, we could have five units corresponding roughly to the five main chapters of Logical Options by Bell, DeVidi and Solomon. So that’s a unit on other ways of doing logic, using propositional logic as the illustration — in particular, natural deduction. A unit treating the semantics for predicate logic more carefully, and an explanation of the completeness proof. Something on axiomatic systems built using a first-order logic. A unit introducing modal logic. And a unit on non-classical logics, in particular intuitionistic logic. (Again nearly all those are already on the syllabus, but we don’t teach them in a methodogical and integrated way. Using Logical Options as a course text could be a way of imposing order on the current slight mess. The book strikes me, having just got a copy for the first time, to be very good: it goes quite snappily and needs support from lectures, but it is the right kind of coverage and the right kind of level. This too is for a compulsory paper, though students can avoid answering too-technical questions by concentrating on some associated philosophical logic.)
  • Third year, mathematical logic. This could remain pretty much as we do it at the moment. Some basic model theory, comparisons of first and second order logic, etc. Then Michael Potter’s course using his Set Theory and its Philosophy, and my course using An Introduction to Gödel’s Theorems. We could also make the technical parts of the current paper available as a graduate course for those who haven’t done enough logic when they come to us from elsewhere.

It is my impression is that even good and large departments elsewhere in the UK are dropping serious logic teaching (so that e.g. a Single Honours student may have access only to one optional honours module using Tomassi’s very, very elementary book). But before getting too upset about this and bemoaning the Dumbing Down of Courses and the general Decline of the West, I am minded to try to organize some kind of survey to find out the state of logic teaching in UK philosophy departments. I’ve put a message out on Philos-L to check that no-one else is in the middle of doing this (and check too that a survey hasn’t been done recently and I’ve just failed to notice!). Watch this space.

At long last …

I’ve just delivered the final version of my Gödel book to CUP (sent the PDF, taken round a print out). So that’s that, delayed a couple of months by family matters, though I hope the publication date won’t be much changed. I’ll have to restrain myself now from looking at it again before it appears.

At the moment I’m tolerably happy with the result. That has a great deal to do with Richard Zach who sent me some quite extraordinarily helpful late comments, including a handful that saved me from embarrassing errors. I am enormously grateful to him.

Of course, as is always the way, once you’ve finished a book you can see an alternative way of organizing it overall, patches that should have been done differently, things you could have included and other bits you could have dropped. Somewhere in Popper’s third world there is the better book I could have written. But the sublunary version will have to do.

While writing it, I’ve acquired a great stack of things I want to, need to, read, and ideas for a couple of follow-up books. But first, I’ve immediately got to dive in and write a talk for a workshop in ten days time on themes from Dan Isaacson’s paper, “Arithmetical truth and hidden higher-order concepts”. Should be fun.

Fixed point logics

At CUSPOMMS today, the audience was depressingly tiny, which was a great shame because Anuj Dawar gave a terrific introductory talk about the problems he is working on in finite model theory. In a particular, he said a little about the expressive power of logics which have a ‘least fixed point’ operator, and the relation of this to the question whether P = NP.

The topic of fixed point logics links up very closely with something I need to think about soon, for I’ve promised to put together a talk on Dan Isaacson’s Conjecture for a workshop next month (I mean the claim that, if we are to give a rationally compelling proof of any true sentence of the language of first-order 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). What’s the connection? Well, it is plausible that our understanding of arithmetic involves grasp of the ancestral of the successor relation (“the natural numbers are 0, the next number, the next one, the one after that, and so on, and nothing else”). So it is natural to think about theories of arithmetic which are embedded in stronger-then-first-order logics with a primitive ancestral-forming operator (i.e. a transitive closure operator, which is closely related to a least fixed point operator). We know that such arithmetics semantically entail more than PA but are unaxiomatizable. But what about partial axiomatizations of them? I know that the very natural partial axiomatization that goes back to Myhill gives us a theory which is in fact a conservative extension of PA (so that result is in harmony with Isaacson’s Conjecture: trying to pin down a concept arguably essential to our understanding of arithmetic which isn’t definable in the language of PA still doesn’t take us beyond PA). But I need to discover more about fixed point logics to see what else interesting is to be said here.

Anuj recommended Leonid Libkin’s Elements of Finite Model Theory as a good place to start finding out more. Great. I’ve ordered a copy.

Back to category theory

Back in August, I did a bit of reading on category theory. I’m now taking that up again with a group of graduate students. We’ve started working through Robert Goldblatt’s Topoi: The Categorical Analysis of Logic (an amazing bargain reprint from Dover). And this time I’m getting more of a feel for what is going on. Or at least, I have the comforting illusion of understanding — but it could all be shattered soon, when it is my turn to talk at the reading group next week, supposedly helping us through chapters 5 and 6.

Up to about half-way through chapter 3, Goldblatt proceeds at a pretty gentle pace; he then accelerates a bit alarmingly. Reading his first three chapters in parallel with the first six chapters of Steve Awodey’s Category Theory works well, however, if you have the time.

And is the effort worth it? Well, we’ll see (though my sense so far is that the answer should be very positive). But in the meantime, we’re certainly having fun …

Three cheers for Piers

It’s a familiar fact of academic life that you can go to hear a talk with great expectations of something arresting from someone who you have read with admiration, and then be bored or even irritated by a banal recycling of half-baked ideas. Equally, you can dutifully turn out to a unpromising-seeming seminar, apparently remote from your interests, feeling for some reason that you really ought to be there — and you find yourself riveted and enthused.

So three cheers for Piers Bursill-Hall, who was talking at the clumsily named CUSPOMMS last Friday. “Descartes’s Earlier Epistemology in Natural Philosophy” wasn’t exactly a title to set me alight. In the event, his talk was wonderfully entertaining but also a highly illuminating lightening tour through ancient and Renaissance takes on the problem of the unreasonable effectiveness of mathematics in its application to the material world, and Descartes’s attempt to resolve the problem without committing himself to the theologically dangerous idea of mathematics as a direct route to reading the mind of God. Enthusiasm is always engaging, and Piers has it in spades. (You can get a glimpse of his style here.)

Vice-chancellors, please note

Moscow university in the 1820s, a place of dissent and a reputed ‘hot-bed of depravity’. The Tsar appoints Prince Golitsyn Director to impose order. Herzen writes: ‘Golitsyn was an astonishing person. It was a long time before he could accustom himself to the irregularity of there being no lecture when a professor was ill; he thought that the next on the list ought to take his place, so that Father Ternovsky sometimes had to lecture in the clinic on women’s diseases and Richter, the gynaecologist, to discourse on the Immaculate Conception.’

Which no doubt improved classes no end. The university also had its own prison for recalitrant students. Models for us to emulate, surely.

So what’s it all about then?

The photographer Steven Pyke has produced another set of photos of great and not-so-great living philosophers. The style is, as with his earlier set published as a book, dramatically black-and-white, distinctly pretentious — and so, of course, in some cases the portraits are hardly recognizable.

Pyke asks his sitters to describe ‘in fifty words of so their own idea of what philosophy means’. So what’s it all about then? Ruth Millikan gives a favourite quotation of mine from Sellars: ”The aim of philosophy, abstractly formulated, is to understand how things in the broadest possible sense of the term hang together in the broadest possible sense of the term.” (I wonder how many students, at least this side of the Atlantic, read Sellars any more?)

Here are two other answers that chimed with me. David Papineau: “Some hold that the aim of philosophy is to construct theories that confirm everyday intuitions. What a dispiriting ambition. In my book, the best philosophy overturns common sense. Often, the impetus for change comes from outside philosophy, in the form of scientific or cultural innovation. The task of the philosopher is then to show how the new ideas reshape everyday thinking. Does this reduce philosophy to the status of a hand-maiden? Well, far better a hand-maiden of change than a lackey of the intellectual status quo.”

And Steve Stich: “The idea that philosophy could be kept apart from the sciences would have been dismissed out of hand by most of the great philosophers of the 17th and 18th centuries. But many contemporary philosophers believe they can practice their craft without knowing what is going on in the natural and social sciences. If facts are needed, they rely on their “intuition”, or they simply invent them. The results of philosophy done in this way are typically sterile and often silly. There are no proprietary philosophical questions that are worth answering, nor is there any productive philosophical method that does not engage the sciences. But there are lots of deeply important (and fascinating and frustrating) questions about minds, morals, language, culture and more. To make progress on them we need to use anything that science can tell us, and any method that works.”

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