Peter Smith's Blog, page 120

Ooops …!

Having, in my last post, invited people to email if interested in seeing a late draft of the second edition of my Gödel book, I now realize that my address might have been elusive. There’s now a link added to the last post which takes you the page where the address appears at the foot, and a link to the same page restored to the sidebar.

Whoosh …!

In the immortal words of Douglas Adams, ”I love deadlines. I like the whooshing sound they make as they fly by.” The second edition of my Gödel book was supposed to be off to CUP today. But it will be two or three weeks yet. Still, the end is in sight …

I’ll be sending out the complete draft PDF to a few very kind people who have given me comments on the early chapters that I’ve posted here. But would you like to see the whole draft PDF when it is done in a couple of weeks (and maybe see your name in illuminated letters on the Acknowledgements page for the book)? Then here’s the deal …

Send me an email with the subject line “Proof reading Gödel” (my address is on the ‘About’ page) from an academic email address, with a sentence or two about you, promising

not to pass on the PDF,
to do a very careful proof-reading for typos, cross-checking of references etc., of an assigned chunk of the book [about 30 pages] as quickly as you can, and definitely within three weeks of getting the PDF,
to comment on any issues of readability etc. in that chunk.

Of course, comments on more would always be most welcome, even at this late stage!

Oh, how can you resist!?

Gödel book comments? Anyone??

The draft of about a third of the second edition of my Gödel book that I posted here a month ago has now been downloaded over 875 times.

It isn’t too late to send comments. Indeed, less than one percent of downloaders have so far given me comments (even on a single typo), so I’m not overwhelmed with stuff to work through and respond to.

Now, that could be because you all think the book is stunningly good as it is …. But somehow, philosophers and logicians being the picky bunch we are, I do rather doubt it! So, folks, if you’ve got some suggestions/corrections/remarks, now is the time to get in touch. (I’m not sure whether or not it counts as an incentive, but comments mean you’ll get a look at the complete draft too when its done at the end of the month …)

The Search for Immortality

The Fitzwilliam Museum mounts quite wonderful small exhibitions. The last major one (major in terms of its interest and the quality of what was on show) was the Vermeer exhibition last year. Now there is a beautifully mounted show of objects excavated from tombs from Han China. This continues until November, and is very well worth a visit if you are in our near Cambridge: entrance is free too. We particularly liked these terracotta figures of the musicians and dancers (click to enlarge).

The exhibition is called ‘The Search for Immortality’. And some of the commentary on the audio guide did attribute crude beliefs about the afterlife on the basis of what was found in tombs (as if the pots and musical instruments were there to be used …). But if Romeo buries his love letters to Juliet with her, is that because he must think she needs reading matter in the grave?

Cambridge Foundations, 3

I was particularly looking forward to two of the talks scheduled for the last day of the Foundations conference, those by Steven Methven and Dan Isaacson. Sadly, for different reasons, neither happened.

That left just three talks. Sam Sanders spoke about his research on the reverse mathematics for a certain system of non-standard analysis. You could get a few glimmers of what might interesting in this area from his talk, but it was rather ill-judged for the occasion: but visit Sam’s web-pages if you want to know more.

Alan Weir also gave too-rushed a tour, this time through some of the themes of his book Truth Through Proof which I’ve blogged about at length: I think it is fair to say there was nothing new here.

The remaining session, though, was a very nice review by Mary Leng of some points of difference between the sort of fictionalism that she favours and the kind of formalism that Alan defends. On the principle ‘the enemy of my Platonist enemy is my friend’, Mary found herself wanting to minimize some apparent big divergences between their positions. But the crunch disagreement, for her, comes over their respective views about logical consequence. The fictionalist talks of inferences within the fiction, the formalist of inferences within the formal game (putting it roughly). But what makes for ‘good’ patterns of inference in either case? Mary thinks she has to appeal to a primitive logical modality here. Alan eschews such modalities, but then — Mary argued — has nothing helpful to say about logical consequence (unless he smuggles a modality in the back door). Alan of course thinks otherwise.

Like a Catholic bemused by warring factions of Protestants, I’ll leave them to battle that one out …. But I think Mary probably does jab her finger on a sore spot for the formalist.

Cambridge Foundations, 2

I missed Mic Detlefsen’s talk, and arrived half-way through Steve Awodey’s “Homotopy type theory and univalent foundations”, which was a beautifully presented advertising pitch for the project you can read a lot more about here.

James Studd (a graduate student in Oxford) then gave a paper on a bi-modal axiomatization of the iterative conception of set. Why bimodal? Start with the familiar informal tensed chat where we talk of forming levels of the hierarchy successively. We want to say things like: the new sets formed at a level must have members already formed at past levels, and once formed must persist at future levels. So it is pretty natural to try regimenting such past-looking and future-looking talk with a pair of tense-like modalities. Except, as Studd himself emphasised, they aren’t really temporal modalities. But then, of course, the problem is to make non-metaphorical sense of them. And (as Robert Black pointed out in discussion) the trouble is that the natural ways of doing this presuppose an understanding of talk of levels in the iterative hierarchy, so the modalities thus explained can’t really serve as giving an independent handle on how to understand the iterative conception. If instead, as I think Studd wants, we ultimately take the modalities as new primitives, then it is pretty unclear whether anything has been gained at all over e.g. the Boolos axiomatization of stage theory.

Next up, Michael Potter talked about whether Replacement can be justified, developing worries already expressed in his Set Theory and its Philosophy. Replacement amounts to two claims: roughly, (a) every set is a subset of a level, and (b) there is a level for every set size. In a Scott-style axiomatization, (a) is built in. But what would justify adding (b), which is equivalent to a reflection principle?

‘External’, regressive, arguments — from the supposed need to assume (b) if we are to prove some desired mathematical results outside set theory — don’t work: full Replacement isn’t needed for (enough) “ordinary” mathematics. So we need to appeal to ‘internal’ considerations, flowing from our conception of the set universe. A limitation of size principle might support Replacement, but is itself difficult to motivate. Meanwhile — and this is the key point Michael wants to press — the iterative conception doesn’t deliver the goods.

This particularly nice talk was followed by Hannes Leitgeb on ‘A theory of propositions and truth’. The idea is to model a theory of propositional functions and ‘aboutness’ on a theory of sets and membership: so the aim is an untyped but paradox-free theory, on which is grafted an untyped theory of truth-for-propositions, and then a theory of syntax and a derived theory of truth-for-sentences. Hannes ends up in the Tractarian position of having to say that the axioms of his theory don’t express propositions, a conclusion he cheerfully embraced. But I, for one, am quite unclear what are the rules of the game for this kind of constructional exercise in defining formal gadgets, and hence quite unclear whether such insouciance is justifiable.

In the final talk of the day, Alex Paseau talked interestingly about the possibility and scope of non-deductive knowledge of mathematical propositions. He had two different sorts of cases. First, there’s what we might call ‘experimental’ evidence — as when we draw enough diagrams to convince ourselves that the perpendicular bisectors of a triangle intersect at a point. And then there is mathematical evidence such as probabilistic considerations, neighbouring theorems, etc. — as in our evidence for Goldbach’s Conjecture or the Riemann Hypothesis. These cases do seem very different to me, however, and I am rather inclined (without much by way of argument) to return a split verdict. That is to say, I find it happier to say I can get to know some propositions of Euclidean geometry by the experimental method than I am to say I could get to know Goldbach’s conjecture by totting up more evidence short of a proof. Perhaps that’s because of the nagging doubts engendered by the dim recollection that there are other number theoretic conjectures which have only immense, and immensely sparse, counter-examples: so, the thought remains that for all we currently know — even augmented with more of the same — we could still be in for a nasty surprise.

Cambridge Foundations, 1

I am taking some time off from revising the Gödel book, going to (most of) a local conference, Foundations of Mathematics: What are they and what are they for?. Yes, it really is all fun, fun, fun at Logic Matters.

The conference kicked off with Tim Gowers (that’s Sir Tim to you) talking about discovering proofs by breaking down proof-tasks into simpler and simpler ones. He talked through a couple of cases: for example, how would you tackle the question “Find the 2012th digit after the decimal point in the expansion of ”? As an exercise by a wonderful teacher leading an audience to see that there is a “natural” way of getting to an answer, just brilliant. As a hopeful illustration of the kind of heuristics we could give a computer to discover the proof, unconvincing. And as far as the official theme of the conference goes, surely off-topic.

Brendan Larvor’s talk was about the so-called “Philosophy of Mathematical Practice” and the supposed rise and fall of a certain conception of the architectural organisation of mathematics as having/needing “foundations”. But this was at a level of arm-waving generality that I find entirely uncongenial.

Philip Welch talked about a General Reflection Principle in set theory — fine, as you would expect, on the technicalities, but this got bound up with some entirely unclear remarks about why it might help to move from talking about sets qua objects to talking about the concept of sets (which turned out to be talking about a structure).

The best talk of the day was by Patricia Blanchette, essentially on the Frege-Hilbert dispute about theories, independence proofs, etc. This was extremely clear and convincing, but she has written before about this (see her terrific entry in the Stanford Encyclopedia), and I’m not really sure what was new.

Pavel Haas Quartet, once more

I have mentioned this young Czech quartet a couple of times before in this blog, to rave about their great recording of Dvořák’s American Quartet, and about some truly wonderful radio broadcasts of them playing Beethoven and Schubert. Today we went up to the Wigmore Hall to hear their lunchtime concert playing Janáček’s Initimate Letters and Smetena’s From My Life. To say it was worth the journey from Cambridge was an understatement. This is powerful music, and it was played with intense feeling and utter conviction and control. Very difficult to stay dry-eyed in the Smetena in particular. Thanks to the BBC you can listen online for the next week.

As the Radio 3 announcer says (but forgot to tell the audience in the hall), the Quartet have a new second violin Marek Zwiebel, and this was one of their first performances with the new line up. You wouldn’t have guessed: the ensemble was still as stunning, and their sound still as remarkable. If the Pavel Haas Quartet come your way on their travels, grab tickets, whatever they are playing. String quartet playing doesn’t get better than this.

Gödel, 2nd edition again

OK, here for download are the opening 22 chapters (164 pages) of a draft of the 2nd edition of my Gödel book (and there’s also a partial Table of Contents for the book here). This covers up to and including the first, Gödelian, proof of the First Incompleteness Theorem, and therefore corresponds to the opening  17 chapters (152 pages) of the original edition.

The rewriting isn’t working out as I had originally imagined. I’d thought it would be a matter of minor local tinkering with the original text, but then adding some chapters late in the book. But in fact, as I came to going carefully through the early chapters again, I felt they could do with quite a lot of work to make them more reader-friendly (and make the book more appropriate for the introductory series it is published in). So in fact there will now probably be very little space for new material in the second edition, despite an increased page budget as I expand the earlier material: but I hope the early chapters in particular will be notably more accessible.

This partial draft will be the last one I make publicly available here (though I am very open to requests from those who are keen to have the chance to comment on later chapters when they become available). That’s partly for the obvious copyright reasons: the press won’t be too happy if I give away yet more. But also I don’t envisage so many changes in the rest of the book (except locally in particular sections), whereas here there has been a lot of rewriting and rearrangement, and so there are no doubt a lot of new typos and thinkos that I’d really welcome being alerted to at this stage. So please please send me any comments, even if it is just a note of a trivial typo (you might well be the only person to spot it!).

If you want to focus on new material, then check out in particular the new arrangement in Chs 2, 3, 6; the treatment of induction in Ch. 9; revisions of material in Chs 11, 12; the significantly different Chs 15–17; the division of the treatment of the arithmetization of syntax into Chs 18, 19.

Given the publication schedule, I’d like any comments sooner rather than later (though given I’m producing camera ready copy, I can keep on editing until quite late in the process).

Inexhaustible Haydn

I’m beavering away at the second edition of the Gödel book, and hope to post a big chunk of it here over the weekend. So watch this space. Work divides between the (relatively) fun stuff of rewriting bits and pieces of the book, and then trying to proof-read the results (I’ll be providing CUP with a PDF again for printing). Judging from the typos in the initial printings of both my logic books, proof-reading isn’t my strongest suit. Certainly I find it an irksome chore. Haydn helps a lot, though, in the background. And for the last few days, it has been back to those terrific CDs of the Piano Sonatas from Naxos, as I try to spot stray commas, or irritating repetitions of ‘of course’.

I wonder: is Jenö Jandó the most listened-to pianist in the world, given his quite extraordinary quantity of often really very good budget CDs? Anyway, for the Haydn he really seems to be at the top of his game, sensitive, subtle but full of life: he sounds at home here. And the music is always wonderful to return to, an inexhaustible pleasure. I have just noticed that Jandó’s complete Haydn cycle is available at a now quite ludicrously cheap price. If you don’t know it, treat yourself.