Such a wonderful writer. The Wolf Hall trilogy is the extraordinary work of our times, that only strikes you as all the greater on rereading. And the many touching tributes to Hilary Mantel’s human qualities make her untimely death seem all the sadder.

I picked up a copy of the very recently published A New History of Greek Mathematics by Reviel Netz in the CUP Bookshop a couple of weeks ago — an impulse buy, encouraged by the rave endorsements on the back cover.

This is the most irritating book I’ve read (well, not read to the bitter end) for a long time. On the positive side, it is extraordinarily interesting and illuminating about the intellectual and cultural milieus at various stages in the development of mathematics in the ancient Greek world. It told me a great deal about our fragmentary knowledge about the earlier figures, about the kinds of mathematics being pursued, when and why. That background story is told very readably, with zest and engaging enthusiasm. So Geoffrey Lloyd could be spot on when he writes that the book “brings to bear an extraordinary range of material from non-Greek as well as Greek sources, and develops original arguments concerning the fundamental question of why and how Western science developed in the way it did”.

So why the irritation, the great disappointment? Because the author, sad to say, gives no sense at all of having any real feel for mathematics. His accounts of ancient proofs (and actually there are surprisingly few detailed ones) are to my mind uniformly very poorly and unclearly done; they just don’t pass muster by normal expositional standards. I suspect that the author has zero significant mathematical background: and it shows badly. Having — only metaphorically, as I hate maltreating books! — thrown the New History across the room for the fourth or fifth time in frustration, I gave up after Archimedes. Though I took away this much: one day, I’d like to find out more about just what Archimedes knew about conics and the proofs he had available to him, as reconstructed by a competent mathematician.

Much will be written, no doubt, about the man (whom I never met), and here I remember only the impact that Kripke had on logic-minded philosophers of my generation and the next. That was immense, from the time of his absurdly precocious first papers on modal logic (the first JSL paper published when he was nineteen), through the 1970 Princeton lectures on Naming and Necessity, and the later 1970s papers — such as the “Outline of a Theory of Truth”. And there was so much more too. Those 1970s papers struck me, still strike me, as a paradigm of philosophy — imaginative but full of good sense and straight talk, with forceful arguments appealingly written with great clarity, and in the background a real depth of technical logical knowledge lightly worn.

Unlike some, I wasn’t such a fan of Kripke’s 1981 long paper on Wittgenstein and rule-following, which indeed perhaps marked the end of his extraordinarily fertile great publishing period. But there is a very large amount of still unpublished work from then and later, with significant pieces to appear in further volumes of his Collected Papers if the first volume, Philosophical Troubles, is anything to go by. I look forward to that. And look back now to so many rich hours spent in Kripke’s intellectual company.

As I have said before, it is difficult to know quite what to make of the absolute numbers given in the stats for this website (supposedly there are about 35K ‘unique visitors’ a month). But the relative numbers can surely be trusted. Every month, the largest number of visitors come from the US, followed by Germany and then GB. And there are no surprises in the next few countries down the list.

But regularly about the tenth on the list, ordered by numbers of page views, is Bulgaria. And this really is a surprise, at least to me. The population of that country is less than a fifth of that of Poland, for example, yet supplies a dozen times as many visitors. Indeed, there are over a quarter as many visitors from Bulgaria as from here in GB. In fact, relative to size, it seems that’s where Logic Matters is most read!

A little googling suggests that logic has a very substantial presence in the University of Sofia, with a large department. So maybe some students from there find their way here. And I guess free resources are always going to be particularly appreciated by those in relatively poorer countries. Anyway, warm greetings from Cambridge, if you are exploring Logic Matters from Bulgaria!

Roughly speaking, the proof has three components. (1) Randall proved over twenty-five years ago that NF is consistent if what he called Tangled Type Theory, TTT, is. Arm waving, if a sentence is a correctly typed sentence of the simple theory of types, and is what you get by replacing the type levels in with new type levels with the same order relation, then is a sentence of TTT. So where in the simple theory, for to be well formed, has to be one type higher than , in TTT is well-formed even when is more than one type higher than , so there is an relation between any lower level and any higher level. This relative consistency claim of NF and TTT is unproblematic.

(2) TTT is a seemingly rather wild theory (Holmes calls it “weird”). But Holmes now aims to present a Frankel-Mostowski-style construction that purports to be a model of TTT. The devil is in the contorted(?) detail: do we get a coherent description of a determinate structure?

(3) Assuming that stage (2) is successful in at least describing a kosher structure that a ZF-iste can happily accept as such, there then is the task of verifying that it really is a model of TTT.

Now, over the last few weeks, a bunch of maths students here have been working on a summer project arranged by Thomas (with a lot of Zoomed input from Randall) to formally verify the consistency proof in Lean, by first checking (2) that the model is coherently constructed, and then going on to check (3) it really is a model of TTT. And I understand the state of play to be this: that first of these stages is successfully more or less completed. And it has in the process become intuitively clear — said Thomas — that the defined structure is indeed a model of TTT. Dotting the i’s and crossing the t’s and implementing a Lean check that the model satisfies a certain finite axiomatization of TTT will take more time than there is left in this summer’s project (the students have lives to lead!). But with (2) secure, it looks as if Holmes indeed has his claimed proof, though its final best-form shape remains to be settled.

If that’s right, Holmes has settled one of the oldest open problems in set theory. Though quite what the wider significance of this, I’m frankly not so sure. Will a consistency proof (of a decidedly tricksy-seeming kind) really make us look much more kindly on NF? Should it?

To the Fitz, to see their exhibition (on for another twelve days) “True to Nature”. Which we very much enjoyed (rather to our surprise, much more so than the trumpeted Hockney exhibition which is still continuing). Mostly minor works, to be sure, but the cumulative affect a delight, with some gems you would be more than happy to live with. Like these gnarled olive trees.

A list of forthcoming autumn books in one of the weekend papers. At the end of this month, we get Maggie O’Farrell’s The Marriage Portrait. “Winter, 1561. Lucrezia, Duchess of Ferrara, is taken on an unexpected visit to a country villa by her husband, Alfonso. As they sit down to dinner it occurs to Lucrezia that Alfonso has a sinister purpose in bringing her here. He intends to kill her.” OK, you’ve got me! I thought Maggie O’Farrell’s Hamnet was wonderful: so can’t wait for this. Then Elisabeth Strout has another novel, fast on the heels of the marvellous Oh William!: in October we get a sequel, Lucy By The Sea. Then, not least, there is a new Kate Atkinson coming, Shrines of Gaiety. Enough said! Some happy autumn evenings ahead. Three reasons to be cheerful.

I was struck that the ones that stood out for me in that list of autumn books were all by women. And looking back at the list I keep, I see that most of the two dozen novels I’ve read this year so far have been by women. At least of the recently published ones, only one was by a man — Julian Barnes’s Elisabeth Finch. I must be missing out on something: but what?

A book taken down from the shelves one recent evening, The Faber Book of Landscape Poetry (a serendipitous as-new Oxfam purchase a while back). A real pleasure to dip into — some very engaging poetry, some familiar, some not at all. Perhaps not very challenging. Indeed, to be honest, perhaps a rather conservative selection. But then it was edited by a conservative, indeed a Conservative politician, the one-time Education Secretary Kenneth Baker. The thought strikes: which of the last six or seven Conservative Education Secretaries might have even had any literary interests, let alone sensibly edited some such book? Anyone?

Another forthcoming book: A. E. Stallings This Afterlife: Selected Poems. Something else to really look forward to. Pleasures of the reading kind will be plentiful, then. Outside books, the world isn’t doing so well, is it?

I have recently been re-reading Wuthering Heights, for the first time in however many decades. I’m not sure what prompted me to reach the book down from the shelves, and I wasn’t sure either how I would take to it in my dotage. Isn’t it a book for the young and over-romantic? Yet I am engrossed and carried away. What passionately driven writing! Utterly extraordinary. But you knew that.

Should I be surprised that Emily Brontë’s teacher in Brussels wrote of her “She had a head for logic, and a capability of argument unusual in a man and rarer indeed in a woman …”?

Back to logic for me. And so here is the revised chapter on primitive recursive functions from GWT2, just 11 quick pages. Some from a more computer science background complained a bit about what I wrote in the previous edition (and should indeed be similarly dissatisfied with the similar treatment in IGT2). I have, in particular, tried to make the passing remarks about “for” loops and “do while” loops less misleading. Have I succeeded?

I was about to post another rather critical book note on two more offerings in the Cambridge Elements series. I’ll still get round to that. But I wouldn’t want you to think that I’m always a relentlessly negative reviewer, would I? — perish the thought! So before I fall back into my more usual grumpy groove, let me pause for a positive recommendation of a fairly recent addition to the Springer series ‘Problem Books in Mathematics’.

Laszlo Csirmaz and Zalán Gyenis have put together a fairly challenging collection Mathematical Logic: Exercises and Solutions (2022). From the Preface:

Problems in this volume have been collected over more than 30 years of teaching undergraduate students Mathematical Logic at Eötvös Loránd University, Budapest. The problems come in great variety: routine applications of a newly introduced technique, checking whether the conditions of a particular theorem are really necessary, extending or ﬁnding the limitations of various methods, to amusing puzzles and interesting applications of established results. They range from easy questions and riddles to proving hard theorems when all the necessary ingredients are—hopefully—available.

After preliminary chapters on sets, strategies in games, and formal languages, the main chapters are on recursion theory, propositional calculus, first-order logic, some model theory (Ehrenfeucht–Fraïssé games, quantiﬁer elimination, ultraproducts), and formal arithmetic.

The problems are set within the context of reminders of key definitions and theorems, with the occasional hints for solutions: these take 128 pages. Then there about 200 pages of solutions. That page ratio will tell you that the solutions are typically not going to be fully-worked-through answers developed in the sort of detail that (e.g.) a student might be expected to turn in, but rather they are headline indications of the main ideas needed to get a solution (occasionally calling too on background mathematical knowledge). So this is a book, I’d say, better suited for a mathematically moderately strong reader, whether someone taking a taught course or someone self-studying an area of logic.

Of course, a book like this is going to reflect the idiosyncrasies and special interests of the authors (so for example, the propositional/predicate logic topics are almost entirely semantically driven — proof theory doesn’t get much of a look in). But such idiosyncrasies are no bad thing at all — it’s always illuminating to be coming at perhaps familiar topics from different angles. Dipping through this book, I have found it very interestingly put together, with some of the exercises requiring real thought; all but the most expert are surely going to learn from sampling it. So your university library should certainly get a copy.

Of course, there is no point in banging on about the absurdity of Springer publishing such a student-oriented book as a hardback/e-book way out of their price range. Maybe an eventual paperback is planned. (But in the meantime, you needn’t feel too sorry for the impoverished seeker after knowledge, as I’m sure that a PDF will have already found its way to the usual repositories — an eventuality which respectable readers of this blog must of course entirely deplore.)

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 Beginners: 15 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.