Kreisel’s Interests?

College Publications is indeed a decidedly peculiar outfit. On the one hand, they publish a lot of books on logic (broadly construed), and the books appear as very inexpensive print-on-demand paperbacks (excellent!). On the other hand, the quality control seems entirely unreliable, and many books must surely fall more or less stone dead from the press with tiny sales. Moreover, the press seem to make no real effort to spread the word: there is a very amateurish, ill-organized, website which often has only uselessly minimal information about the publications. As far as I can see, there is nothing like e.g. a quarterly newsletter that librarians and the rest of us can sign up for. It’s not surprising then that few of their books seem to end up even in the vast Cambridge university library system.

There are of course successes — for a couple of relatively recent ones, there is the very nice Model Theory for Beginners15 Lectures by Roman Kossak (2021), and the good-to-have-in-one-place collected Essays on Set Theory by Akihiro Kanamori (also 2021). So a few days ago in an idle hour, I trawled through the College Publications website to see if I could find anything else published in the last few years that looked sufficiently appealing …

One book that caught my eye was Kreisel’s Interests: On the Foundations of Logic and Mathematics, edited by Paul Weingartner and Hans-Peter Leeb (2020). The advertising blurb says just this:

The contributions to this volume are from participants of the international conference Kreisel’s Interests — On the Foundations of Logic and Mathematics, which took place from 13 to 14 August 2018 at the University of Salzburg in Salzburg, Austria. The contributions have been revised and partially extended. Among the contributors are Akihiro Kanamori, Göran Sundholm, Ulrich Kohlenbach, Charles Parsons, Daniel Isaacson, and Kenneth Derus. The contributions cover the discussions between Kreisel and Wittgenstein on philosophy of mathematics, Kreisel’s Dictum, proof theory, the discussions between Kreisel and Gödel on philosophy of mathematics, some biographical facts, and a collection of extracts from Kreisel’s letters.

Which certainly isn’t very informative, but was enough to pique my interest, and I sent off for a copy (at that point offered at a large discount by Amazon — I suspect they had an already-printed copy they wanted to offload!).

A grave disappointment. Working backwards, this short book ends with some fifty pages consisting on short snippets taken from letters from Kreisel to Kenneth Derus (or letters to others which Kreisel copied to Derus). The snippets are mostly from five to ten lines long: there are some fun gossipy remarks about a wide cast of characters — but not much logic. This is preceded by a twenty-five page piece by Dan Isaacson, ‘Georg Kreisel: Some Biographical Facts’. But this is all about Kreisel before 1950, i.e. before his first publications. We learn a bit about his early interests, his early contact with Wittgenstein, his war work, his return to Cambridge and his developing interests; there are hints, not really worked out, that Kreisel’s project of trying to extract finitist content from non-finitist proofs was inspired by Wittgenstein. But there is little to delay us here.

So that just leaves the preceding 86 pages of the book. There is a short piece by Göran Sundholm on Kreisel’s Dictum’. You’ll recall that Dummett attributes to Kreisel the thought that “the point is not the existence of mathematical objects but the objectivity of mathematical truth”. Sundholm notes that while Dummett attributes a variety of versions of this remark to Kreisel, none of them are to be found clearly stated in Kreisel’s writings. And Sundholm reports that when he directly asked Kreisel, he didn’t endorse any of the Dummettian versions. There follow a few pretty unclear pages on what the dictum might come to.

Quite out of place in this book, there is a very technical paper by Ulrich Kohlenbach, with the not-very-inviting title ‘Local Formalizations in Nonlinear Analysis and Related Areas and Proof-Theoretic Tameness’. I didn’t get anything at all out of this: but expert proof-theorists can read the paper here and judge for themselves.

That leaves us with just two remaining papers. The two volumes of Gödel’s Collected Correspondence don’t feature exchanges with Kreisel, as he wouldn’t give permission to use the letters. Charles Parsons was one of the editors of Gödel’s Correspondence and, although he does not say so, I imagine that his paper here on ‘Kreisel and Gödel’ was based on work done in preparing those volumes which at the time came to nothing. So now we do get brisk outlines of the preserved exchanges. But there is a tantalizing lack of detail. For example we are told that Gödel in a letter to Kreisel refers to his (Gödel’s) Russell paper but then adds remarks about intuitionism which aren’t in the paper. But we get just a one-sentence hint of their gist. A perhaps slightly frustrating read, then. (Parsons, though, does reproduce in full a rather odd final letter from Kreisel to Gödel.)

Finally, the book starts with a piece by the indefatigable Akhiro Kanamori, on ‘Kreisel and Wittgenstein’. There’s an early period, in which the impressionable undergraduate Kreisel is befriended by Wittgenstein, and becomes a companion for walks. Some seeds are sown here by Wittgenstein’s rather constructivist inclinations. Then middle Kreisel, so to speak, famously reviews Wittgenstein’s Remarks on the Foundations of Mathematics in a highly critical way (“it seems to be a surprisingly insignificant product of a sparkling mind”). Later Kreisel is somewhat more nuanced in his responses to Wittgenstein on logic and mathematics — though I’m not sure I got from Kanamori a very clear picture of Kreisel’s later view of Wittgenstein (though maybe there wasn’t a determinate view there to be clearly pictured).

Anyway, Kanamori’s was the piece in Kreisel’s Interests that interested me the most (though, frankly, that wasn’t a difficult competition to win). Again you can read the paper without buying the book, as it is available here.

A footnote. One thing I learnt from Kanamori’s references is that Piergiorgio Odifreddi has edited some of Kreisel’s non-technical writings About Logic and Logicians into two substantial volumes. These include previously unpublished lecture notes,  journal papers, biographical memoirs, etc. It says in the preface “Kreisel himself wrote all the texts, but Odifreddi has made some substantial editorial interventions, rearranging some of the material, breaking the text into sections and paragraphs, inserting titles, moving or removing some notes, and eliminating some digressions.” The result is readable and often engaging. The collection itself doesn’t seem to have been conventionally published; but drafts of the two volumes are available online here.

Too darn hot

Two ridiculously hot days. It reached 39.9° in Cambridge yesterday (that’s 103.8°F). Not fun. Fortunately, with an older house, some north-facing rooms, strict adherence to the rules about closing windows/blinds/curtains, we were able to keep much cooler indoors. But it was a great relief to get outside into a much more temperate morning today and walk by the river.

But it was too darn hot for Logic Matters. Siteground, who host us, use servers provided by Google Cloud. The cooling system for the Google data centre in London failed yesterday, and the servers went down. The outage lasted 16 hours for Logic Matters. The logical world coped, I’m sure. But there were a lot of very annoyed/worried businesses, hotels, care providers … A burning straw in an over-heated wind of change.

Yes, too darn hot, as Ella Fitzgerald sings …

GWT2 — a third instalment

I was prompted to start working on a second edition of Gödel Without (Too Many) Tears by discovering that there was a significant muddle in Chapter 5 (I had inconsistently wavered between taking Baby Arithmetic, so called, as having just a negation connective, and having other connectives too). Thanks again to Ben Selfridge for pointing out that embarrassing glitch.

This new instalment of GWT2 corrects that unfortunate mess, and makes a number of other small improvements for clarity/readability, in what is now numbered as Chapter 6. So here it is!

To save readers having to dart between different PDFs, I have included the revised versions of the Preface and earlier chapters (with only trivial changes from the previous posting).

Need I say it? No: but I still will! Corrections and friendly suggestions are immensely welcome at this stage.

Already updated with many thanks to David Furcy and Léon Probst. (You may need to force your browser to reload the file to get the version dated 18 July.)

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.

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