Logic Matters

From time to time I do get more than a bit critical here about books of one sort or another: so it’s good to give praise for once!

Over the last couple of days I’ve been reading the first volume of Winfried Just and Martin Weese’s Discovering Modern Set Theory (AMS, 1996), with an eye to moving on to the second volume. Well, I just loved the style, and think it works very well. I don’t mean the occasional (sightly laboured?) jokes: I mean the in-the-classroom feel of the way that proofs are explored and motivated, and also the way that teach-yourself exercises are integrated into the text. For instance there are exercises that encourage you to produce proofs that are in fact non-fully justified, and then the discussion explores what goes wrong and how to plug the gaps. My grip on set theoretic niceties is patchy enough to be find this kind of reinforcement of understanding pretty helpful from time to time, even at the elementary level of the first volume. So I’ll be rather warmly recommending the book to students.

I’ve spent the last couple of days reorganizing and rewriting the reading list for the Part II Math Logic paper (that’s a third year undergraduate paper for philosophers). It was a rather minimalist affair, and I’ve taken a step or two towards its becoming an annotated study guide.

The paper is something of a Cambridge institution, pretty much unchanged in its basic syllabus since when I took it a long time ago. It rather distinctively mixes an introduction to the “greatest hits” as far as formal results are concerned, with a look at some of the philosophical issues arising.

Anyway, having had some initial comments here and from local grad students, you can now download my third shot at an updated list. All comments and suggestions for further improvement (within the current, fixed, syllabus) will still be very welcome.

Another logic-seminar regular hits the big time! It is good to see that Thomas’s “The Iterative Conception of Set”, published last year in the new Review of Symbolic Logic was judged one of the ten best papers of 2008 by the Philosopher’s Annual. Here’s a link.

A footnote to my post, Logic disappearing over the horizon. I’ve just been reading Stephen Simpson’s “Unprovable Theorems and Fast-Growing Functions” (an introductory piece in the 1987 AMS Contemporary Mathematics Logic and Combinatorics volume that contains some important papers on provably computable functions — it is a pity that Simpson’s very helpful and accessible survey isn’t more readily available, e.g. on his website). I was struck by this remark:

Like most good research in mathematical logic, the results which I am going to discuss had their origin in philosophical problems concerning the foundations of mathematics.

And that’s right: the most interesting work in mathematical logic is bound up with problems and projects of a more philosophical kind concerning the foundations of mathematics. All the more worrying, then, the seeming trend I was remarking on for logic courses to be less and less available even to graduate philosophy students. If the wonderfully fruitful long dialogue since Frege between philosophers and mathematicians (or often, between the philosophical and mathematical sides of the same individual) is to continue, then some philosophers at any rate do need to be logically well-educated!

Richard Zach, over at LogBlog, has posted this:

Exciting developments! The Association of Symbolic Logic has made the now-out of print volumes in the Lecture Notes in Logic (vols. 1-12) and Perspectives in Mathematical Logic (vols. 1-12) open-access through Project Euclid. This includes classics like

• Shoenfield’s Recursion Theory,
• Lindström’s Aspects of Incompleteness in the LNL,
• Sacks’ Higher Recursion Theory,
• Hájek and Pudlák’s Metamathematics of First-order Arithmetic,
• Shelah’s Proper and Improper Forcing,
• Barwise’s Admissible Sets and Structures, and
• Barwise and Feferman’s Model-theoretic Logics in the PiML.

I’m especially excited about the Hájek/Pudlák and Barwise/Feferman volumes, which are chock-full of useful material!

This is indeed an excellent development (I’m not sure why Project Euclid puts the books up in chapter-length chunks and then complains if you download too many chunks at once: but let’s not sound ungrateful, because I’m certainly not!).

Looking around online, you can in fact find a large number of logic books available, though most of them are there contrary to copyright. Frankly, I don’t feel guilty about having a bootleg e-book on my laptop if the hard copy acquired with hard cash is sitting on my shelves. But it would be wonderful if this is the beginning of a trend for out-of-print classics to be made freely available in high-quality PDFs.

Newly in to the CUP bookshop today, a second edition of Lawvere and Schanuel’s Conceptual Mathematics. This has a little new material over and above what was in the first edition: that looks a good move, as I found when new to category theory that the first version ended too soon, without enough pointers forward to where we were we being taken.

I’ve just had an invitation to give a talk at the University of X, a distinguished place, with a philosophy graduate community of about fifty (according to their website). So I checked out how much logic/phil maths is going on, what I could reasonably take as given. Zilch. Apart from a first year course perhaps approaching the level of my intro logic book, nothing at all, as far as I can tell. Which leaves me a bit bereft of anything to go to talk about. But more to the point, it means that for students at X a central swathe of the work of lasting value from the last hundred years has disappeared over the horizon. Which is, shall we say, a pity.

My sense is that this is happening more and more in UK universities. I’d be delighted to learn that I’m wrong.

Logicians will be delighted to see that Wilfrid Hodges has been elected a Fellow of the British Academy.

Well, the Cameleon weekend is over. Something of a success, I think (though the main CMS building at the weekend is a rather bleak empty space, not very conducive to socializing between papers, and the catering-arrangements could have been better; these things matter — don’t they? — to getting a really good feel to an occasion).

I talked about the incompleteness theorems. The exercise of trying to pack some headline news into three sessions was very useful (to me, at any rate) — though I’m afraid that I became rather conscious of some inadequacies in my book in the process. Enough of the audience seemed gratifyingly surprised by simple observations that e.g. generic Gödel sentences (fixed points for ¬Prov) can be false, and that there are provable “consistency” sentences, to make it worthwhile going over some basics again. I wrote a 43 page handout, though I think I’d like to tidy it up just a bit before publishing here. So watch this space.

Thomas Forster talked about countable ordinals, and there’s a rough-and-ready version of some notes here. I’ve heard him talk about these things before, but I like his way of thinking about ordinals, and I want to get clearer still about these things before getting back to writing about Gentzen.

John Truss presented some work on countably categorical structures, Fraissé theory, and the “classification of countable homogeneous multipartite graphs”, all with enviable lucidity. I don’t know enough — or have the right interests — to really understand why this might be interesting. I had a sense that denizens of a mathematical zoo were being pointed out and classified (giving, as it were, the natural history of part of the abstract realm). I guess my tastes run to more abstract theory.

For me, though, the high points of the weekend were Wifrid Hodges’s talks on the history of logic. His question was: why did modern logic take so long to arrive? And his tour through various episodes from the history of logic, and his diagnosis of some causes for stagnancy from Aristotle to Leibniz, was absolutely fascinating. Do visit his website and look at some of the other historical pieces there too.

Just to note that there is a new entry in the Stanford Encyclopedia of Philosophy, on Set Theory: Constructive and Intuitionistic ZF, by Laura Crosilla. I’m out of my comfort zone here, but I found this a very interesting and helpful piece. The SEP really is going from strength to strength, and the logic/phil maths entries are most certainly of a fine standard.