Peter Smith's Blog, page 45

Roman Kossak’s Mathematical Logic

What do you make of this?

Think of a number, say 123. What is 123? It is a sequence of digits. To know what this sequence represents, we need to understand the decimal system. The symbols 1, 2, and 3 are digits. Digits represent the first ten counting numbers (starting with zero). The number corresponding to 123 is 1 . 100 + 2 . 10 + 3 . 1. In this representation, the number has been split into groups: three ones (units), two tens, and one hundred.

I’d worry that someone who wrote that is hopelessly confused between numerals (expressions which reprsesent) and numbers (what the numerals represent). And just how can a number (that very thing which is represented) be “split into groups”?

Or what about this, following some examples of equinumerous collections?

All those equinumerous collections have their individual features, but there is one thing that they all have in common. That is this one thing that we call the size. This common feature is the size of the collection and of all other collections that are equinumerous with it. Now we can introduce the following, more formal definition: a counting number is the size of a finite collection.

So numbers are “features”, i.e. properties? A moment ago they were things that can be split into groups! (Great-uncle Frege is not resting quietly …)

Let’s put this sort of thing down to a certain arm-waving carelessness rather than confusion: still, it doesn’t exactly inspire confidence in the more conceptual/philosophical remarks in Roman Kossak’s Mathematical Logic (Springer 2018).

Actually, this book is mis-titled. There is little core logic here. The early chapter entitled ‘First-Order Logic’ is a fleeting introduction to first-order languages, too fast for real newbies, and the idea of a formal deductive system is only mentioned, and then without elaboration, at p.132 (and the book has just 155 pages before the final summary chapter and the appendices start). What the book is in fact centrally about is signalled by its subtitle: “On Numbers, Sets, Structures, and Symmetry”.

As Kossak says in his final summary, his “aim in this book was to explain the concept of mathematical structure, and to show examples of techniques that are used to study them. It would be hard to do it honestly without introducing some elements of logic and set theory.” The examples of structures are near all numerical ones. So Part I is an introduction to the construction of the integers, rationals and reals from the naturals, and a lightning tour of some of the presupposed set theory. This is done in with a fair amount of motivational chat, so some credit for that. But I still think the beginner would be notably better off reading one of the usual introductions to these ideas in elementary set theory books like Enderton’s or Goldrei’s. Charitably, the author is rushing on to get to what really does interest him, the book’s distinctive content in the seventy-odd pages of Part II.

And this Part, to get much more positive, is a very approachable introduction to some simple model theoretic ideas, but taking a rather different route in that some familiar texts.  So Kossack explores some of the first-order definable features of various structures defined over the natural numbers, the integers, the rationals, the reals, and the complex numbers, and he nicely brings out some of the perhaps unexpected complications. He helps himself to the compactness theorem and e.g. the Tarski–Seidenberg theorem (which are not proved) to give partial demonstrations of various results. And along the way, the reader is introduced not only to basic notions like that of an elementary extension but also somewhat more sophisticated ideas like being a minimal structure. This is done with a light touch, helpful examples, and again a good amount of motivational chat. I’m not sure that some of the sketched proofs are quite as clear as they could be, and there can be philosophical wobbles in the commentaries. But this Part of Kossack’s book does, I think, get across some basic model theoretic ideas without too many tears, in an unusually accessible way, making connections that aren’t often brought out; and (those wobbles apart) I enjoyed it and learnt from it.

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PHQ — Prague spring

Six months after their last live concert, with Covid restrictions easing in the Czech Republic, the Pavel Haas Quartet have been able to start playing again, streaming two concerts over the last weekend. It was wonderful to hear them.

One highlight, for me, was getting to know Martinů’s 7th quartet, which they played so engagingly in the first of the concerts, recorded for a Duke University series. John Gilhooly at Wigmore Hall had persuaded the PHQ to play a Martinů cycle starting last year  — but of course, like so many other musical plans, all that was thwarted by Covid. Hopefully the cycle (and a recording or two?) will still happen sooner rather than later: if this sample was anything to go by, PHQ will make Martinů their own in the same way that they give such compelling performances of the other Czech greats.

Another highlight was the performance of the Brahms Piano Quintet, joined by Boris Giltburg in the second of the concerts, recorded for a Spivey Hall series. In particular, the gentle second movement was simply magical (with Giltburg making the pianists in a couple of discs I know seem positively flatfooted). Another recording, please, of this and Shostakovich Piano Quintet! — I’ve also heard Giltburg play that with the PHQ a couple of times quite outstandingly.

These concerts, perhaps, had more significance for the PHQ than just restarting playing; for they were joined for the first time by their new violist Luosha Fang. They had suddenly parted company with the violist, composer and conductor Jiří Kabát at the beginning of 2020, asked the prize-winning Luosha Fang to join them, and (again) plans were blown up by Covid. Now, I don’t have a good enough ear to be the best judge, but from these concerts she is surely an inspired choice for them. Her playing seems wonderful (for example in the exposed viola part in the movement  of the second quartet by Pavel Haas that PHQ played as an encore for the Duke concert), and very much in keeping with the style of the quartet. We can only hope this transatlantic marriage works out for them all.

The links I gave for the concerts in a post last week have now expired, as each was only available for three days. Hopefully the recordings will eventually be made available more widely.

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Self publishings …

The three Big Red Logic Books continue to be downloaded at a very steady rate, about 1500 times a month between them — and over 200 paperback copies sold a month too. Now I know that there are enough sales to justify any additional effort, I might explore the costs and benefits of moving away from the easiest default option of using the Am*z*n self-publishing system. But plainly this self-publishing malarky is a pretty good way of getting what you write into people’s hands or at least onto their screens.

Of course, I realize full well that with things as they are, many people will need the imprimatur of a university press for their book to get brownie points for tenure/promotion purposes. But equally, there are a lot of cases where that doesn’t apply. For example, I chanced upon this notice of a Festschrift for Keith Hossack to be published next year by Bloomsbury. For £85. What exactly is the point of publishing like that? Wouldn’t everyone involved much  rather have had their work read and thereby have Hossack’s work more widely promoted? As it is, how many university libraries will be forking out in these straitened times?

There are some downsides to totally independent self-publishing — so as I’ve said before, what we need is probably some more projects for book-publishing like the planned BJPS Open series. It will be interesting to see how that develops. Open access journal publishing, like the Philosophers’ Imprint, seems to be doing well and gaining standing.  With enough support, something similar could and should happen for book publishing.

I’ve noted before that by far the most frequent download from the whole Logic Matters site  is (the successor of) the Teach Yourself Logic Study Guide — downloaded more than 1700 times in April, for example. And encouraged by that, I have found myself over the few weeks working hard on an update, in a slightly different format which will now probably lead to another Big Red Logic Book. But it will be some weeks before a new version is ready for prime time.

It is actually a rather enjoyable project (sad but true!), revisiting some familiar old texts and taking another look at some less familiar ones, and writing/revising mini-overviews of the various areas the Guide covers. Should I be dispirited (‘it has taken this long for the penny to really drop?!’) to still be finding out new things along the way? Or should I just enjoy the continuing journey? The latter, I think!

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Sabine Devieilhe sings Ravel

Sabine Devieilhe sings Ravel’s Cinq mélodies populaires grecques at Musée d’Orsay. Eight minutes of delight. Start the video linked at the foot of the review here at 49.38.

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Feferman’s virtual book: Logic, Mathematics, and Conceptual Structuralism

In the 2018 collection of articles Feferman on Foundations (about which more in a follow-up post), we learn that shortly before his death Sol Feferman proposed to OUP a sequel to his terrific volume of papers In the Light of Logic. He wanted to collect together some more of his later papers of broader philosophical interest, under the suggested title Logic, Mathematics, and Conceptual Structuralism.

Sadly the proposal seems not to have been taken up. However, but apart from any introduction he might have written, the book does exist in a virtual state, as all the papers are on Feferman’s website. So here is his proposed Table of Contents, linked to the papers which are divided into five groups (hopefully I’ve linked to the right targets!) :

I. My Route

Philosophy of mathematics: 5 questions.

II. The Mathematical Mind

Mathematical intuition vs. mathematical monsters.And so on… Reasoning with infinite diagrams.Gödel’s incompleteness theorems, free will, and mathematical thought.

III. What is Classical Logicality?

Tarski’s conceptual analysis of semantical notions.Logic, logics, and logicism.Set-theoretical invariance criteria for logicality. Which quantifiers are logical? A combined semantical and inferential criterion

IV. Conceptual Structuralism

Logic, mathematics, and conceptual structuralism.Conceptions of the continuum.

V. New Axioms for Mathematics?

Does mathematics need new axioms?Is the Continuum Hypothesis a definite mathematical problem?

Some of these pieces are rightly very well known; others I haven’t come across before — so I’m now really looking forward to reading this virtual book.

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The Pavel Haas Quartet return for online concerts

The great Pavel Haas Quartet suffered a real blow at the beginning of 2020 when they suddenly parted company with their violist. And then Covid struck. Like so many musicians, the PHQ have been having a hard time of it since. They were able to play a few local concerts in or around the Czech Republic in the summer, joined again by their founding viola player Pavel Nikl (as in the photo above). But it is six months now since they last gave a concert. However, an online concert is announced for Friday May 7th, for a series organised by Duke University, with the video available for 72 hours. They will be with their new violist, Luosha Fang, playing Beethoven’s Quartet in F Minor, op. 95, no. 11 (“Serioso”) and Martinů’s Quartet no. 7, H. 314 (“Concerto da camera”). For details, and very inexpensive tickets, see here. (The concert isn’t that well advertised, so do spread the word.)

A second online concert, with the pianist Boris Giltburg, is available from May 8th, from Spivey Hall at Clayton State University — though the programme details don’t seem to be given. But (again cheap) ticket details are here. The PHQ are also filming in the Suk Hall at the Rudolfinum a concert with Boris Giltburg  for the Library of Congress series.

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More readings

I’ve been preoccupied for the last three or four weeks with a big domestic project which I’ve been putting off for years. But with help from two men with a van, and after multiple trips to the town dump, lots of shelf building, painting and the like, the job is done, and our rather large garage is looking positively civilized. At last. Rather satisfying. Tiring days, though, so not much energy left for logic.

Very enjoyable non-logical reads, however, have included Lorna Sage’s Bad Blood (which somehow I’d not read before), the ever-readable Jane Gardham’s Bilgewater, and Anne Tyler’s newly paperbacked and wonderfully humane Redhead by the Side of the Road.

Not so much enjoyable logic though. Various thoughts keeping nagging away at me about ways to improve my IFL; I think I’m just not going to be able to resist the urge to write a bigger and better third edition. So I’ve been dipping into various relevant books and articles. I’ve just reread the interesting but patchy long piece on the ‘History of Natural Deduction’ by Jeff Pelletier and Allen Hazen published in the final volume of The Handbook of the History of Logic, which has set me off worrying again about which particular choices in constructing an ND system are of some conceptual significance. Other contributions in that weighty volume (which is titled ‘Logic: A History  of its Central Concepts) look potentially interesting too. But the first piece on ‘History of the Consequence Relation’ by Conrad Rasmus and Greg Restall turned out to be sadly disappointing: it covers the terrain too superficially to be very illuminating. I’ll report back on some of the other contributions. I’ve also been revisiting Neil Tennant’s Core Logic; but although I’m certainly sympathetic to his project, I’m again finding Tennant’s mode of presentation unnecessarily off-putting (we need a short Core Core Logic).

However, I am enjoying getting to grips belatedly with Nils Kürbis’s Proof and Falsity, which has now been out a couple of years. The topic is going to be negation and it’s role in deduction in general and natural deduction systems in particular, and you can’t get more basic than that! The opening two chapters — which take us some 90 pages into the book — set the scene, giving a sympathetic elaboration of the Dummett/Prawitz theme that it is the rules of inference which give the logic operators their meanings. As is familiar, the distinctively classical double negation rule and its equivalents seem to be anomalous, standing apart from the meaning-conferring balanced pairings of introduction/elimination rules which according to the Dummett/Prawitz theme fix meanings (Kürbis has helpful things to say in his Chapter 2 about what makes for such a stably balanced pairing). So I really want to see how the story now develops over the coming chapters — can you cleave to what is defensible about proof-theoretic semantics for the logical operators while giving a classical treatment of negation?

I’m not sure how the book will strike readers who are not already well-versed in the topic and who are not at least intrigued by Dummett’s wranglings with the issues; but the project appeals to me. I confess I’ve taken a peek at the final chapter, and I’m doubtful that I’ll want to end up where Kürbis does. However, the writing is engagingly clear and direct, and there have already been some telling arguments. So I’m much enjoying the journey so far.

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The Chiaroscuro Quartet at Wigmore Hall

Wigmore Hall continue their stunning series of freely streamed concerts with the Chiaroscuro playing Haydn’s “Bird” and Schubert’s “Rosamunde”. One of my very favourite quartets playing, in particular, favourite Schubert — great! Wonderful performances (remarkably so, as the always impressive violist Hélène Clément was standing in at very short notice for Emilie Hörnlund).

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Recent readings

Logic/philosophy books have been arriving. A book aimed at students which is shallow and slapdash (evidently dashed off too fast, without taking enough advice from those who have actually thought hard about the topics). Another book aimed at students which I thought might be apt for mentioning in the Logic Guide: a bit better as far as it goes, but disappointingly thin — covers too much in far too short a compass, so the student reader again isn’t well served. A monograph on a topic that I’m much interested in, written in clotted prose and with a depressing penchant for obscure arm-waving generalities. And a straight logic text recommended to me, which I’ve again looked at with an eye to the Guide. It’s written for computer scientists and maybe later chapters on compsci topics are better; but so far I’m pretty unimpressed. All of these books could have done with an interventionist editor.

OK, I’m not going to name names. Because then I’d need to spend a lot more time making out the critical cases than I really feel inclined to do. Life is too short. But not very cheering or invigorating reading experiences.

Also recently arrived — a reprint of Abbott and Mansfield. If you are of a certain age and were properly brought up, those names will be so familiar, and you’ll probably have a vivid school-days recollection of the green cover of their grammar book. I’m trying to relearn Greek to keep the brain from rusting. And what is cheering is that I find that the core of Abbott and Mansfield looks remarkably familiar! For example, I still have much of the paradigm of λυω by heart, learnt at an age when these things stick. … I knew it would come in useful one day. Better late than never.

And a surprise birthday present from Mrs Logic Matters, Ross King’s new book The Bookseller of Florence. King’s story ranges widely over the intellectual life of renaissance Florence, and it all zips along most enjoyably. So this I can indeed warmly recommend as a fun and enlightening read.

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Slow progress …

There have been distractions. So it’s been painfully slow progress on various projects over the last few weeks. In particular, it has been very slow work trying to write the early (pre-category-theoretic, scene-setting) chapters I wanted to put together for the Gentle Introduction. Partly that’s because I have been trying to decide what I actually believe about set-theoretic reductionism/foundationalism/imperialism …

Here though is a fairly recent paper that I’ve found very helpful. And I am prompted to mention it here because I discovered (in talking to a small and very unrepresentative sample!) that word about it hadn’t got around. So — if you are interested in the topic — let  me recommend

Penelope Maddy, ‘Set-theoretic foundations’, in Caicedo et al, eds., Foundations of Mathematics, 2017.

Maddy begins, “It’s more or less standard orthodoxy these days that set theory – ZFC, extended by large cardinals – provides a foundation for classical mathematics. Oddly enough, it’s less clear what ‘providing a foundation’ comes to. Still, there are those who argue strenuously that category theory would do this job better than set theory does, or even that set theory can’t do it at all, and that category theory can. There are also those who insist that set theory should be understood, not as the study of a single universe, V, purportedly described by ZFC + LCs, but as the study of a so-called ‘multiverse’ of set-theoretic universes – while retaining its foundational role. I won’t pretend to sort out all these complex and contentious matters, but I do hope to compile a few relevant observations that might help bring illumination somewhat closer to hand.” And, particularly in the first part of the paper, it strikes me that she does an admirable and very  judicious job of distinguishing various things that might be meant by talking of foundations here. Though that still leaves me with much to think about.

In another neck of the woods, I am getting back to updating what was the TYL Study Guide, and I have just uploaded a slightly revised version of Chapters 1 to 8. I hope to finish the new Chapter 9 over the coming days.

About fifty years late, I’ve also been enjoying thinking a bit more carefully about the Prior Analytics, and (relatedly) about ways in which the opening chapters of IFL could be significantly improved.

And so it goes … But, as I say, slow progress all round.

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