I at last hit the “publish” button for Gödel Without (Too Many) Tears as an Amazon print-on-demand book; and within less than a day of its going live, I received a list of corrections and suggestions based on a late draft. Of course.

This is the kind of thing which would be so very very annoying with a book published the old-school way — I’d be kicking myself repeatedly for missing the obvious typos that couldn’t be corrected until a later reprint, perhaps years down the road. But in this case, I had a corrected version done within hours, and Amazon had approved it within another few hours. No more than two dozen very early adopters will have the original version (sorry!); from now on you should receive a copy which says on the verso of the title page “This revision: 5.xi.2020” [added: now “9.xi.2020].

In fact — rather a relief! — the caught mistakes mostly turned out to be minor, a few obvious typos, a few clumsy errors like using “then” twice in a sentence. There’s just a couple of places you could be led astray, and a footnote it would be good to add. Here’s a list of corrections.

A general comment though. Corrections to any of the three Big Red Logic Books are still welcome (or at least, corrections which e.g. note obvious mistakes, or stylistic infelicities, or uses of English which are opaque to someone who isn’t a native speaker). Dealing with such corrections is very straightforward: so you can keep them coming!

In another world, we would have gone to Florence again this year before spring, before the tourists really return. Out of season, the city becomes a delight, the galleries and churches peaceful, the cafés and restaurants recovered by the locals. But it was not to be.

So, among other distractions, I’ve had to be reading art books instead. One that I have very much enjoyed is Megan Holmes’s Fra Filippo Lippi: The Carmelite Painter (Yale, 1999). (That detail is from the Annunciation that Lippi painted for Le Murate around 1443, and forms the cover.)

This is rather beautifully produced, as Yale’s large art books usually are. Indeed, I confess I initially bought it for the many illustrations. For Lippi is one of my favourite artists (if I could smuggle just one painting home from the National Gallery in London, it might well be the Annunciation there). So there was huge pleasure to be got just from a slow look at the reproduced paintings in Megan Holmes’s book. But then I found myself becoming thoroughly caught up in her project of trying to understand how Lippi’s work is bound up with his ambiguous relation to his friary. This is perhaps not the best written art monograph ever: it can be a little repetitive and sometimes overdoes the scholarly detail at the expense of the narrative flow. But still, I not only learnt a great deal about the particular artist recorded in the Carmine’s records as ‘Frate Filippo di Tommaso dipintore’, about his use of Carmelite imagery, and about the relation between his paintings and their particular varied religious settings. I also learnt — very late in the day for me! — some important lessons about how to look at early renaissance paintings more generally. So, warmly recommended.

A footnote: You can still get Megan Holmes’s book second-hand, but significantly more expensively than when I bought my copy a couple of years ago. Moral: get art books when you are first tempted by them (whether exhibition catalogues or monographs like this one)! They far too readily go out of print and then aren’t reprinted. For example, about the time that I bought this book, I also bought Jean Cadogan’s Domenico Ghirlandaio: Artist and Artisan (Yale, 2001): that now costs at least four hundred pounds second-hand. So if there’s an art book you’ve been hankering after, don’t hesitate, snap it up now for a life-affirming lockdown treat!

Having thought a bit more about Kripke’s short note on diagonalization, linked in the last post, it seems to me that the situation is this, in rough headline terms.

How do we get from a Diagonalization Lemma to the incompleteness theorem? The usual route takes two steps

(1) The Lemma tells us that for the right kind of theory T, there is a fixed point G in T for the negation of T‘s provability predicate.

(2) We then invoke Theorem X: if G is a fixed point for the negation of the provability predicate Prov for T, then (i) if T is consistent, it can’t prove G, and (ii) if T is omega-consistent, it can’t prove not-G.

The usual proof for the Diagonalization Lemma invoked in (1) is, as Kripke says, (not hard but) a little bit indirect and tricksy. So Kripke offers us a variant Lemma which has the form: for the right kind of the theory T, there is a fixed point in TK for any T-predicate where TK is T augmented with lots of constants and axioms involving them. The axioms are chosen to make the variant Lemma trivial. But now the application of Theorem X becomes more delicate. We get a fixed point for the negation of TK‘s provability predicate and apply Theorem X to get an incompleteness in TK. And we then have to bring that back to T by massaging away the constants. Not difficult, of course, but equally not very ‘direct’.

So you either go old-school, prove the original Diagonalization Lemma for T in its tricksy way, and directly apply Theorem X. Or you go for Kripke’s variant which more directly uses wffs which are ‘about’ themselves, but have to indirectly use Theorem X, going via TK, to get incompletness for the theory T we start off from. You pays your money and you takes your choice.

For a worked out version of these headline remarks see the last section of the revised draft Diagonalization Lemma chapter for Gödel Without Tears. Have I got this right?

These troubled times make music all the more important. So here are the Pavel Haas Quartet at the Janáček festival in Brno a few days ago. Immensely enjoyable. They play Martinu’s 7th Quartet (starting at 2.15); Janáček’s ‘Kreutzer Sonata’ Quartet (at 27.30); and Dvořák’s String Quintet No. 3 (at 53.30). These are characteristically fine performances, and well filmed too. The PHQ were asked by John Gilhooly of Wigmore Hall to do a Martinu cycle, and were planning to perform the first two concerts there later this month; but those concerts are now postponed because of covid travel restrictions.

Followers of the quartet’s fortunes will know that Jiří Kabát parted abruptly from the quartet at the beginning of the year. So their violist for the few concerts they have been able to play since, not straying far from Prague, has again (temporarily?) been their founder member Pavel Nikl who so sadly had to leave the quartet for family reasons a few years ago. I think the additional player for the Quintet is the violist of the Zemlinsky Quartet.

Added: Sad to relate, the video did not stay online for very long. I hope other PHQ fans also took the chance to see it/download it.

The Teach Yourself Logic study guide has, as I said a couple of posts ago, grown over the years in a really rather haphazard and disorganized way. Looking at it again, more carefully, the guide really need to be rewriten from the ground up. And, to add to the guide’s usefulness, it would be very good to begin each chapter/major section with a short essay (up to half a dozen pages, say) giving some orientation, briefly surveying the relevant area of logic.

So all that is what I plan to do. And it should be fun to put it together. However, it will be really quite time-consuming, writing the essays and revisiting the large literature to re-assess my various current recommendations. So I intend to work on TYL’s planned descendant — Logic: A Study Guide — in intermittent stages over the coming months, posting the new chapters for comments as I go along.

I’ve made a start. And now will be a very good time to make suggestions for improvement for the early chapters on the more elementary material (i.e. the core math logic topics covered in what are now Chapters 4 and 5). TYL is downloaded a great deal: so tell me what what you think! — all comments and advice will, as always, be very gratefully received.

Of the recent streamed performances from Wigmore Hall, Elisabeth Brauß’s lunchtime concert of Beethoven (Op. 10 No. 3), Mendelssohn (Variations sérieuses), and Prokofiev (Piano Sonata No. 2) really stood out for me — and not just for me. Astonishingly good playing, without bombast or exaggeration. One of the BBC New Generation Artists showcased over a weekend, she is surely on the threshold of a stellar career. To be watched and re-watched. Enjoy!

The Teach Yourself Logic Study Guide usually gets the most downloads here — recently, a fairly consistent seventeen hundred or more downloads a month (with occasional upward spikes). The Guide has grown by accretion over the years to the current 93 pages, and to be honest it is by now a bit of an inconsistent mess, in terms of levels of detail and coverage. So given how much it is used and recommended, and given the absence of any obvious alternative, I suppose I really ought to settle down to re-thinking it and re-writing it. Which could be fun in its way but is slightly daunting.

And then, to my considerable surprise and embarrassment, Category Theory: A Gentle Introduction gets an equally consistent six hundred or so downloads a month. Surprise, because there are so many available good sets of lecture notes and freely available books out there (as listed here). Embarrassment, because it is a very rough-and-ready unfinished draft — though it already weighs in at 291 pages; it needs a lot of corrections and a lot of development and expansion to get it into a decent state. And that’s an even more daunting prospect given my pretty amateur and tenuous grasp of category theory! But people have said nice things about the Gentle Introduction even as it stands: and I think it is just different enough from the alternatives in level (more accessible!) and organization (more logical!) to be worth having a good bash at improving it.

I’m really not sure, though, how to juggle thinking about a possible IFL3, re-writing TYL, and diving into a lot of category theory homework. But I suppose it is good not to run out of projects that may be a bit daunting but still seem realistically manageable (at least these are finitely limited projects in a way that more purely philosophical projects tend not to be). I’ll just have to see how the spirit moves me once I’ve really got GWT done and dusted …

So I now at last have a full draft of the new book-style version of Gödel Without (Too Many) Tears. But I’m going to take my own advice and put this aside for a week or more, before returning with fresh eyes for a last end-to-end read through (and I will need to add some references, some brief suggestions for further reading, and a rough-and-ready index). I may run a final version past a few people for quick comments on some of the newer bits. But hopefully there will soon be another Big Red Logic Book. Big at least in the sense of being large format: but I’ve managed to keep it, as planned, to about a third the length of IGT2. I’m both embarrassed and pleased to say I learnt two or three things in the re-writing too!

How are the other BRLBs faring? I made IGT2 freely available as a PDF nine weeks ago. As I explained before, there was a quite crazy flood of downloads in the first two or three days — about sixty-four thousand — due to a direct link being posted for a couple of days on the front page of Hacker News (which shows, at any rate, the continuing real interest out there in an introduction to Gödel’s theorems even if not in An Introduction to Gödel’s Theorems). After that initial flurry, the rate of downloads has settled down to something much more sensible – amounting to another three thousand or so. Still a surprisingly high number, as there have been readily-accessible unofficial free PDFs floating around the internet for years. There has also been a small uptake for the (more-or-less at cost) Amazon print-on-demand version, more than enough to make it worth having arranged that possibility. I have of course been dipping into a few bits of IGT2 as I revised GWT and inevitably found some passages that I thought could certainly be improved. Still, I don’t think I’ll be rushing to write an IGT3 just yet.

Then I posted the free version of IFL2 a couple of weeks after IGT2, and this has been downloaded over two and a half thousand times. The sales of the Amazon printed copies are level-pegging with the sales of IGT2 — which is a bit disappointing, as I was hoping that people (i.e. you!) would be asking their libraries to order copies. Since this edition of IFL is now not coming from a commercial publisher with a marketing department, librarians will need to be told about it (order details at the end of this earlier post). Now, as we all know, there is a lot of intro logic books out there. This one at least has the considerable merit of being officially zero-cost, as opposed to the crazy pricing of many texts! And having done a bit more homework, I’m a bit surprised to find that — leaving aside forallx and Paul Teller’s old text — there doesn’t seem to much good free competition either. Which makes me think more seriously about putting together an IFL3.

As I have explained before, the first edition of IFL concentrated on logic by trees. The second edition, as well as significantly revising all the other chapters, replaces the chapters on trees with chapters on a natural deduction proof system, done Fitch-style. Which again won’t please everyone! Chapters on trees are still available; but because of considerations of space in the printed version, those chapters are relegated to the status of online supplements. This was always a second-best solution. Ideally, I would have liked to have covered both trees and natural deduction (while carefully arranging things so that the lecturer/student who only wanted to explore one of these still has a coherent path through the book). With e-publication, the question of length isn’t so vital. And the absence of much by way of freely-available alternatives suggests it wouldn’t be at all a waste of time and effort to put together a third edition. I think it would complement the excellent forallx for those who want a more expansive/discursive (or some would say, more long-winded) introduction but with quite a similar approach. But IFL3 is for the future, and first I need to finish putting answers to the exercises in IFL2 online. And there are two other projects which I must juggle with while working on intro logic. More about them in the next post!