GWT2 — a second instalment

As promised, I’ll be posting revised chapters for a second edition of GWT in bite-sized instalments. So I’ve added just a dozen pages this time, covering two more chapters (just stylistic/clarificatory revisions of the old chapters 4 and 5).  I have also — for any new readers — now included the front matter with a revised Preface. So here they are, the first five chapters.

According to Littlewood in his Miscellany, the great Edmund Landau “read proof-sheets [for one of his books] seven times, once for each of a particular kind of error”. I confess I can’t claim such levels of meticulousness. So I say it, and mean it, every time: corrections and friendly suggestions are immensely welcome at this stage. Now really is the time to let me know if you had issues with the first edition!

Updated, corrected with many thanks again to David Furcy.

Popper’s logic

There is a new book just published, collecting Karl Popper’s published papers on logic (mostly from the late 1940s), together with some contemporary reviews of them by, among others, Ackermann, Curry and Kleene, and also some further unpublished pieces by Popper (including one jointly with Bernays).

How much interest is there here for us now? What Popper gives us is an early form of inferentialism, in the sense that he proposes that the logical operators (the “formative signs”) be characterised entirely by their role in deductions. And deducibility is initially characterised in terms of very general structural properties (like weakening and transitivity, as we would say) which make for what Popper calls ‘absolutely validity’. Then he can say, in his earliest paper,

An inference is valid if, and only if, it is either absolutely valid, or it can be shown, on the basis of the inferential definition of the formative signs, to have been drawn in observance of absolutely valid rules.

We have a family resemblance here to themes now more familiar to us from Gentzen and those much influenced by him like Prawitz. Though Peter Schroeder-Heister (who has now written a number of times on Popper’s logic) has in the past argued that Popper’s views are in fact closest to Arnie Koslow’s variant programme in his A Structuralist Theory of Logic (though my good friend Arnie doesn’t mention Popper at all).

The editors of this volume — David Binder, Thomas Piecha, and Peter Schroeder-Heister — give us a 72 page introductory essay ‘Popper’s Theory of Deductive Logic’. I have to report, however, that those wanting an engaging overview will probably be pretty disappointed: it just isn’t that clear and inviting. So I certainly wasn’t left with a burning desire to dig further into Popper’s logical writings beyond the couple of papers I already knew. Except that I will read the joint work with Bernays, since I very much admire the latter.

One very good thing though. Springer have published this as an open access book: you can download the whole PDF here, and then judge for yourself how much you want to explore around.

GWT2 — a first instalment

I’ve decided to get to work, putting together a revised version of Gödel Without Too Many Tears. 

I’ll obviously correct the known glitches in the first edition. But as I start to read through the opening chapters, I find myself wanting to make quite a lot of little stylistic improvements. And there will be a couple of more significant changes, I think, later in book. So this will be, while not a radically different version, still rather more than a lightly corrected reprint. So we’ll count  it as a second edition, GWT2.

I’ll post revised chapters here in bite-sized instalments, from time to time over the next few weeks. As always, corrections and friendly suggestions are immensely welcome at this stage: now is the time to let me know if you had issues with the first edition.

Here then are the first fourteen pages, based on the old Chapters 1 and 2, now three chapters. Enjoy!

Added 9 July: With many thanks to David Furcy, I’ve already uploaded a corrected version, repairing about ten(!) minor typos.

Francesco Guardi’s Venice

One of the pictures I’d most like to smuggle home from the Fitzwilliam is Francesco Guardi’s View towards Murano from the Fondamente Nuove. Small enough, quiet enough, to live with — turning its back on the declining splendour of Venice to look over a group of working boats, on a cloudy day, out over the lagoon to a distant Murano. Even in many of Guardi’s grander vedute,  the wonderful buildings are receding, some rather shabbily, into the background as a busy life continues in the foreground. Here for example is his Santa Maria della Salute and the Dogana, the setting for the bustle on the canal rendered impressionistically, the Salute almost a dream-like presence. There are other fine examples of Guardi’s work in the Wallace Collection too (you can search here): but I’m particularly enjoying this  one as my current desktop picture.

While books in English about Canaletto are many, there doesn’t seem to be even one relatively recent one about Guardi. Which is odd, as his works are a significant presence in so many galleries, and he was rightly much admired by the later French impressionists. And more to the point, I find that his paintings have a certain resonance, as we too carry on against a background of decline.

A silly thinko in GWT

Oh drat. There’s a very silly thinko in Gödel Without Tears. On p. 28, I say that the logical vocabulary of BA, basic arithmetic, comprises just the identity predicate and negation. On p. 29 — yes, the very next page — I’m using the conditional in giving a schema for the axioms of BA; and the conditional features essentially in e.g. a BA derivation on  p. 32. But then on p. 33 — yes, the very next page — it is plainly said that the only wffs of BA are equations and their negations, and the conditional has been forgotten about again.


I think I can see that happened. I wanted to slightly simplify the presentation of BA from that in An Introduction to Gödel’s Theorems and in earlier versions of GWT; and I inattentively did so in an internally incoherent way. How annoying.

The current version of GWT with the flawed argument has been downloaded some 4500 times, while over 1400 pointed copies have been sold, and no one has noticed — or at least noticed and thought to tell me — until yesterday. So many thanks then to Ben Selfridge for pointing out the foul-up.

I’ll think about the neatest way of clearing up the mess on the corrections page for the book. But then it will be time, given that there is already a handful of other known typos, for (at least) a corrected reprint for the book. So if you know of any other glitches in the book, now is the time to tell me!

“To make no mistakes is not in the power of man.” Too true, Plutarch, too true. “But from their errors and mistakes the wise and good learn wisdom for the future.” Let’s hope.

Added: the corrections page for GWT has now been updated.

Trying to shed a bit of logical light

As I’ve said before, it seems that I just can’t resist the pedagogic imperative. So over the years I have sporadically contributed to that admirable and heavily used question-and-answer site,  math.stackexchange. I’ve learnt a lot from it myself, as there are some first-rate logicians who contribute there. And I occasionally try to do my bit, when the spirit moves, to answer some (usually pretty elementary) logic questions, trying to  Get Things Right.

I’ve now just hit my tenth anniversary there, and simultaneously just ratcheted up 50k “reputation” points (gosh, wot fun! — like getting a gold star at primary school). But is contributing to a question-and-answer site like this really worth doing?

I’d say yes. For a start, there are far worse ways of procrastinating on the internet! But anyway, I’ve just checked the estimate for the number of readers for my answers. And while we all know that idle browsing doesn’t in general mean that we are paying much attention, someone is unlikely to be visiting math.se and clicking on the link to an answer without some level of interest. I hope. Anyway, the stats are that approximately 1.8 million people have now viewed my answers there. Heavens above! Actually, I don’t really believe that figure: but even if the site’s algorithm heavily overcounts that’s still a goodly number of readers. And more than enough to encourage me to continue dropping by from time to time, trying to shed a bit of logical light.

Elisabeth Brauß at Wigmore Hall

The extraordinary Elisabeth Brauß played again last night at Wigmore Hall, to the warmest of receptions. The concert last night was live-streamed, and is available to watch for 90 days here. In an engagingly varied programme she offered us some rarely performed Hindemith, Brahms’ late four Klavierstücke, and Schumann’s Faschingsschwank aus Wien, all done with such verve and then wonderful delicacy, as variously called for — just a delight.

But the recital started Beethoven’s Op. 109 Sonata, which inspired Elisabeth to quite mesmerising playing with heart-stopping moments: transcendental music, and a performance to more than stand comparison with the very best I’ve heard. Extraordinary, as I say.

Proofs & Theories

I have been reading Proofs & Theories. This is not a weighty logical tome, but the (oddly titled?) very slim volume of short essays and occasional pieces by Louise Glück.

How little, I would have thought, there could be in common between the writing experiences of a Nobel-prize-winning poet and of someone putting together a few elementary logic books! Yet her words resonate. At the very outset, she talks of the anxieties and frustrations: “wanting to write, being unable to write; wanting to write differently, being unable to write differently.” It requires poetic licence, perhaps, to write here as she does of “various kinds of torment” (torment?)… but yes, I recognize the constant sense of striving, “not made serene by sensations of achievement”. When you make the writing public, “The work stands as a reprimand or reproach”, difficult to connect to. And then, very soon, “What strikes me is how far away all this work seems” — and I don’t necessarily mean the particular logical content (in my case) but often the tone, the style.

Which is a feeling that struck me again just now, when I had occasion to re-read a chapter of IFL. I didn’t like the authorial voice very much. I suppose I originally wrote most of that chapter some twenty years ago, but I have revised it in the last five years; but I’m sure I could now do better. Or at least write with a lighter touch. I’m increasingly tempted to have one more bash at that book! (Try again. Fail again. Fail better.)

The occasion for dipping into IFL was very belatedly putting online one more set of answers to exercises, and also adding to the  online page of corrections for typos. I’m probably getting to the point where I should update the file for the printed and downloadable book. (That’s in one way not that much of a palaver; the more time-consuming bit will be checking that I don’t inadvertently make unintended changes in the process!)

I have also updated the corrections page for GWT. 

Have I mentioned here Akihiro Kanamori’s relatively recent book Essays on Set Theory which brings together nineteen previously published essays?

Apart from a handful of more technical pieces, these are mostly historical and/or philosophical essays, or papers focussing on the development and work of particular set theorists of note. It is very good to have these widely scattered essays brought together like this. They are typically readable, interesting and enlightening. Lots about proofs & theories …

And three cheers to College Publications for re-publishing them (as a 600 page book!) so very inexpensively.

Beginning Category Theory: Chs 1 to 16 (etc.)

A short post, just to announce another update of Beginning Category Theory. I have significantly improved the chapter on equalizers and co-equalizers, and I have also split into two, and expanded, the old chapter 15 which covered both limits in general and pullbacks/pushouts in particular. So here is a version of BCT including these newly revised chapters.

Just as before, to keep things simple, there is one long PDF here, with both the reworked chapters up to Chapter 16 and also the remaining unrevised chapters from the 2015/2018 Gentle Intro, with the division between old and new clearly flagged. The same remark applies as before: revised chapters get posted when I think they are an improvement on what went before, not when they are polished perfection! All comments and corrections as always most welcome.

My plan over the next few weeks is to rework/expand the last four chapters in Part I of BCT , and then to go back to the beginning in order to try to smooth out the (sometimes considerable) unevenness in the level of exposition/style of presentation which inevitably creeps in as you are concentrating on local revisions. What fun.

Logic Works?

I can hardly complain about people adding unnecessarily to the over-supply of introductory logic books, having done it myself. But here’s yet another one, Logic Works: A Rigorous Introduction to Formal Logic by Lorne Falkenstein, Scott Stapleford and Molly Kao (published just six months ago by Routledge). I’ve been asked what I think of it. Having now taken a look at the book, I’ll save you the trouble of doing the same. It’s pretty bad. Not that I’ve struggled through all 645 pages. But you’ll forgive me that: life is short and patience limited.

That’s a strange subtitle, no? As if introductions to formal logic aren’t usually rigorous. Or at least, as rigorous as they need to be — and as they say, “sufficient unto the day is the rigour thereof”. You might be tempted to worry, then, that a book that especially advertises itself as “rigorous” is likely to be unnecessarily laboured. You’d be right. And actually it is worse than that. It’s not just heavy-handed in explaining the technicalities, but quite generally the long-winded prose is depressingly clotted and terminally uninviting. I pity the poor students who have this inflicted on them!

Two sample episodes. Chapter 6 presents a Fitch-style deduction system for propositional logic. Good choice (though the system isn’t as streamlined as it could be). But the authors plod through a turgid presentation, without zip and zest, making very heavy going of things. It is really pretty difficult to imagine a reader coming to appreciate that by doing things Fitch-style we can arrive at a really rather elegant, natural, and highly user-friendly system. Things aren’t helped by the printed pages being a typographical mess. 

The same applies in spades to the grimly laborious chapters introducing the language of predicate logic. Who would ever guess from these longueurs that the beautiful and compelling basic idea of a quantifier/variable notation for expressions of generality is so very neat and attractive once explained that it can be introduced well enough to convey a reading knowledge to any beginning mathematics student in half a lecture? (I was surprised to see that one of the authors does have some mathematical background — yet the writing throughout gives no sense of the aesthetic attractions of rigorous mathematical ideas.)

I could go into more detail, but I won’t. A rather depressing read, then, which I can’t recommend at all. If you want a good introduction to formal logic which also ranges quite widely, I’d stick to Nick Smith’s!

[Added And see Phil’s comment!]

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