## Gillian Bennett: on doing the brave, decent and loving thing

If you happened not to see her final testimony and plea for the legalization of assisted suicide, then do read these very moving webpages left by Gillian Bennett (the wife for 57 years of that fine philosopher Jonathan Bennett) who faced with the ravages of dementia, ended her life last month.

Posted in This and that | Leave a comment

## One little “}”

No, you didn’t need new spectacles. One little “}” missing, and the last half of the lovingly crafted TYL version 12.0 was all in the smaller font intended for postscripts and asides. Pah! Sometimes $\LaTeX$ is annoying. Just a tinsy bit.

OK, so here’s version 12.0a, which is at least easier on the eyes.

Posted in Logic, TYL | 1 Comment

## Teach Yourself Logic, version 12.0

In time for the new semester/new term/new academic year (depending on how things are chunked up in your neck of the woods), there’s a new version of the Teach Yourself Logic Study Guide and a supplementary page on Category Theory, both downloadable from the Guide’s usual page.

The Guide has been restructured into Parts in a different, more logical way, to make navigating though the 95 pages easier. There have been quite a few changes in the recommendations (e.g. on FOL and model theory). The final section on serious set theory has been restored and improved. There’s even now an index of authors’ names. What’s not to like?

The Guide seems to get used quite a lot (one previous version was downloaded almost three thousand times), which is why it seems well worth spending time on it occasionally. But I’m pretty happy with the current structure and content, so I hope that for a while the main Guide will only need minor tinkering to keep it in good shape (though there might be some more supplementary pages still to come).

As usual, please do let me know if you spot typos, or indeed if you have more substantive comments!

Posted in Logic, This and that, TYL | Leave a comment

## Model theory without tears?

Ah well, you win some and you lose some. I was writing for months about recursive ordinals and proof theory with a view to a short-ish book. And now, quite a way in, I realise that I have to go back to the drawing board, do a lot more thinking and reading if I am to have anything both interesting and true to say, and then (maybe) start over. Well, at least being retired I don’t have ‘research productivity’ (or whatever it is currently called) to worry about. Frustrating, though.

As a distraction, initially just with a view to updating one of the less convincing parts of the Teach Yourself Logic Guide in the next version, I’ve been looking again at some of the available treatments of elementary model theory. One immediate upshot is that there are new pages (briefly) on Jane Bridge’s Beginning Model Theory and (rather more substantially) on María Manzano’s Model Theory linked along with some other recent additions at the Book Notes page.

Now, the books by Bridge and Manzano have their virtues, of course, as do some other accounts at the same kind of level, But still, the more I read, the more tempted I am to put my hand to trying to write my own Beginning Model Theory (or maybe that should be a Model Theory Without Tears to put alongside Gödel Without Tears). The exegetical space between a basic treatment of first-order logic and the rather sophisticated delights of (say) Wilfrid Hodges’s Shorter Model Theory and David Marker’s Model Theory isn’t exactly crowded with good texts, and it would be fun to have a crack at. And unlike the recursive ordinals project, at least I think I understand what needs to be said! Which is a good start …

Posted in Logic | 3 Comments

## Hilbert’s Foundations/Logic Lectures

I’ve just been spending a couple of days looking at the massive volume of David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917-1933, edited by William Ewald and Wilfried Sieg (Springer 2013), which has at last arrived in the library here.

The original material is all in German (sadly but understandably untranslated), and since my grasp of the language is non-existent, what I have been reading is in fact the general editorial introduction, and the introductions to the various chapters of book covering different lecture courses and supplementary material (like Bernays’s Habilitationsschrift published here for the first time). There’s a lot of other editorial apparatus (the whole project seems to have been done to an extremely high standard), but these discursive introductions themselves amount to upwards of 130 pages. They are extraordinarily interesting and illuminating even if you can’t (or simply don’t) read the texts they are introducing. True, some of the material in these introductions overlaps heavily with Sieg’s essays already published in his Hilbert’s Programs and Beyond, but it is all still well worth reading (again). So this is a volume your university library should certainly get: and not just for you to leave it on the shelf and admire from afar!

I take it that few people by now cleave to the old myth — propagated e.g. by Ramsey —  that Hilbert was a gung-ho naive (or even not-so naive) formalist. But that myth (already surely fatally damaged by Sieg and others) should be well and truly buried by the publication of these various lecture notes which witness Hilbert’s developing positions over the 1920s as he explores the foundations of arithmetic.

Let me just remark on two things that struck me — not about the development of Hilbert’s program(s) and the search for consistency proofs, however, but about his contribution to the modern logic. First, it is now clear that the wonderful book by Hilbert and Ackermann published in 1928 isn’t the fruit of a decade’s intensive work after Hilbert’s 1917 return to thinking hard about foundational matters. Rather, the early sections of that book are based very closely on notes for a 1920 lecture course ‘Logik-Kalkül’ prepared by Bernays, and then the core of the book is equally closely based on Bernays’s notes for a 1917/1918 course ‘Prinzipien der Mathematik’ (Ackermann’s contribution to the substantive content of the book indeed seems to have been markedly less than that of Bernays). So suddenly, within seven years of the first volume of Principia (which seems now to belong to a remote era), Hilbert has the makings of a logic text of a recognisable form can still be read with profit. That really is an astonishing achievement.

But second, there are more key ideas about logic in those early lectures which are left out of the book. Thus, in the 1920 lectures, the editors tell us

The logical calculus seems to have been designed to present propositional and first-order logic in a purely rule-based form which allows logical calculations to be presented as they naturally arise within a mathematical proof, and thus to furnish an analysis of logical inference and of the activity of mathematical reasoning.

This aspect of Hilbert’s logical investigations is lost from view in the later book Hilbert and Ackermann 1928 , where the rule-based version of quantificational logic is omitted altogether, and where the canonical versions of both sentential and first-order logic are presented axiomatically. But in these lectures the goal is to obtain a more direct representation of mathematical thought. In the 1922/23 lectures, Hilbert would formulate a calculus that presents, axiomatically, the elimination and introduction rules for the propositional connectives. Here, in early 1920, … Hilbert describes his rules for quantificational logic as ‘defining’ or ‘giving the meaning’ of the quantifiers. (pp. 279-80)

So here then already are intimations of ideas that Bernays’s student Gentzen would bring to maturity a dozen years later. Remarkable indeed.

There’s a lot more of equal interest in the editors’ wide-ranging introductory essays. Warmly recommended.

Posted in Logic | 1 Comment

## Cutting the TYL Guide down to size

The Teach Yourself Logic Guide was getting rather ridiculously bloated — 138 pages in the previous version. Oops. That was getting distinctly out of hand. I was losing sight of the originally intended purpose of the Guide.

Time to re-boot the project!

So there’s now a new version of the Guide available that weighs in at a much trimmer 78 pages (OK, that probably still sounds a lot, but the layout involves small pages and largish print for on-screen reading, and the first quarter is very relaxed pre-amble). Some of the now deleted material has been re-packaged as supplementary webpages. So I hope that the resulting Guide looks a lot less daunting both in size and coverage. It should certainly be easier to maintain, having divided the core Guide from the supplements which can be updated separately.

The Guide and the add-ons can be accessed here. Spread the word to your students (or if you are a student yourself, I do hope you find something useful here).

Posted in This and that | 4 Comments

## Category theory in two sentences

Tom Leinster’s book Basic Category Theory  arrived today on the new book shelves at the CUP bookshop.  I just love the opening two sentences, which seem about as good a minimal sketch of what category theory is up to as you could hope for:

Category theory takes a bird’s eye view of mathematics. From high in the sky, details become invisible, but we can spot patterns that were impossible to detect from ground level.

That’s a brilliantly promising start: and thirty pages in, the book is still proving a really good, if moderately taxing, read.

Posted in This and that | 5 Comments

## Parsons #5: Gödel on analyticity

The delay in getting round to talking about the next couple of papers in Charles Parsons’s Philosophy of Mathematics in the Twentieth Century signals, I’m afraid, a certain waning enthusiasm. I’m still hopeful that the essays in Part II of the book, on his contemporaries, will prove more exciting, but I found the two pieces on Gödel that end Part I of the collection rather frustrating. Though perhaps the problem is more with Gödel as a philosopher rather than with Parsons. Gödel’s more general remarks — e.g. about platonism, or the role of intuition in mathematical knowledge — too often seem un-worked-out (to put it charitably), and Parsons is too careful a philosopher to leap to giving readings which attribute to Gödel a more developed (and hence more interesting) position than the texts really warrant. So we are left, after reading Parsons’s discussions, not much clearer about what Gödel’s position comes to than we were before.

We’ll look at Parsons’s familiar piece on Gödel’s Platonism in the next post. But first comes ‘Quine and Gödel on Analyticity’, written for a 1995 volume of essays on Quine, but in fact concentrating on Gödel.

The initial theme is that there are some similarities between what Quine and Gödel say about analyticity and mathematics that arise from their shared opposition to the views they attribute to Carnap. Both argue against the view that arithmetic, say, can be held to be empty of content (in Tractarian spirit) by noting that in developing that view, the contentual truth of arithmetic must be already be presupposed in arguing that arithmetic, as a syntactic system, has the properties required of it. (In his Postscript to the reprinted paper, Parsons revisits the question of the force of Gödel’s arguments against the real Carnap.)

But Gödel of course goes on to make a claim that Quine would strongly resist: namely that, while e.g.  arithmetic is not analytic in the sense of vacuously-true-by-definition, it is analytic in the sense that (as Parsons puts it) “mathematical truths are true by virtue of the relations of the concepts denoted or expressed by their predicates”.

The trouble is that on the face of it Gödel doesn’t have anything much by way of a theory of concepts, or any very helpful way of explicating his talk of “perceiving” a relationship between concepts except from some opaque remarks about intuition. Or so it will seem to the inexpert reader of the relevant passages in Gödel. And Parsons, as our expert reader, doesn’t seem to find much to help us out here, but rather confirms first impressions that Gödel lacks a serious theory. Which isn’t to say that Gödel must be barking up the wrong tree. Indeed, in the Postscript, Parsons goes as far as to say that “a view something like Gödel’s on this point [that mathematics is in a sense analytic] has always seemed to be the default position”, being something that developed out of his experience as a mathematician, and — which Parsons implies — should chime with the experience of other mathematicians. And I’m tempted to agree with Parsons here. But he too doesn’t do much to hint at how we might develop that view. Which leaves the reader — or at least, leaves this reader — not much further forward.

Posted in Logic | 1 Comment

## Pavel Haas Qt play the opening movement of Smetena’s 1st Qt.

Sinfini Music have just posted a rather good quality video of the Pavel Haas Quartet playing the opening movement of the first Smetena String Quartet, “From My Life”, with their usual intensity.

Apparently, the Pavel Haas have now recorded the two Smetena quartets, and the CD will be released by Supraphon early next year. It should be utterly outstanding — certainly, their live performances of both that I have heard have been so.

Posted in Music | Leave a comment

## Three requests/suggestions

1. Has your uni library a copy of the 2013 second edition of An Introduction to Gödel’s Theorem? Do please check! It’s a lot better in lots of ways than the first edition (indeed, ideally, the first edition could be quietly put into the library store), it’s still relatively cheap, and it’s very definitely what your students should be reading instead of the first version. Then I won’t find myself cringing thinking about people reading the dumber bits of the first edition! Anyway, if it isn’t already on the shelves, now is the time to order the lovely new second edition  to be available in time for next academic year. (This recommendation is of course not motivated by any concern to spread the truth but by the hope of massive financial gain …)
2. The Teach Yourself Logic reading guide to logic textbooks, aimed at beginning grad students or thereabouts, is now at version 10.1, 136 exciting pages, and a real snip at zero pounds, zero dollars, and zero anything else. The current version (and yes, I know it’s time for another update) can be downloaded via the stable URL http://www.logicmatters.net/tyl/ — students do keep saying that they find it pretty helpful, so why not check that it is mentioned in the relevant logic course handouts?
3. The LaTeX (not just) for Logicians site has been going for about 10 years — gulp! — and covers everything you’d expect plus some. It seems more relevant to more grad students than ever, given the popularity of using LaTeX (or close variants). I still tinker with it when I stumble across worthwhile additions (and please do let me know about anything I should add). This too has a stable URL, http://www.latexforlogicians.net which really ought to feature in the relevant info pack or on-line resource webpages for graduate students. Again, worth checking whether it is appropriately linked?
Posted in This and that | 3 Comments