Two new books — the published-today hardback of Jeremy Avigad’s new book from CUP, and the just-arrived, published-next-week, hardback of my GWT. Both excellent books, it goes without saying! But they have more in common than that. These copies are both produced in just the same way, as far as I can see to exactly the same standard, by Lightning Source, one of the main printers of print-on-demand books for the UK. So both copies have the same paper quality (reasonable, though with more see-through than we’d ideally like) and the same sort of case binding (a flat spine, with the pages not gathered in signatures, but glued: how robust will this binding be in the long term?).

Now, I’m perfectly happy with the hardback of my book. It is serviceable and looks professional enough (maybe one day I’ll get round to redesigning the covers of the Big Red Logic Books; but I quite like their simplicity). I’m just noting that, although Avigad’s book will cost you £59.99 (if you don’t happen to be able to call on the very large discount for press authors), it is another print-on-demand book produced to no higher standard. Once upon a time CUP’s books were often rather beautifully printed and produced. Now, I’m afraid, not quite so much.

“To thee, thou glorious son of worth”, from the new CD of music by Handel with the quite stellar Lea Desandre, Iestyn Davies, and Thomas Dunford’s Jupiter ensemble. Just wonderful.

I noted before that Jeremy Avigad’s new book Mathematical Logic and Computation has already been published by CUP on the Cambridge Core system, and the hardback is due any day now. The headline news is that this looks to me the most interesting and worthwhile advanced-student-orientated book that has been published recently.

I’m inspired, then, to blog about some of the discussions in the book that interest me for one reason or another, either because I might be inclined to do things differently, or because they are on topics that I’m not very familiar with, or (who knows?) maybe for some other reason. I’m not planning a judicious systematic review, then: these will be scattered comments shaped by the contingencies of my own interests!

Chapter 1 of MLC is on “Fundamentals”, aiming to “develop a foundation for reasoning about syntax”. So we get the usual kinds of definitions of inductively defined sets, structural recursion, definitions of trees, etc. and applications of the abstract machinery to defining the terms and formulas of FOL languages, proving unique parsing, etc.

This is done in a quite hard-core way (particularly on trees), and I think you’d ideally need to have already done a mid-level logic course to really get the point of various definitions and constructions. But A [Avigad, the Author, of course!] notes that he is here underwriting patterns of reasoning that are intuitively clear enough, so the reader can at this point skim, returning later to nail down various details on a need-to-know basis.

But there is one stand-out decision that is worth pausing over. Take the two expressions and The choice of bound variable is of course arbitrary. It seems we have two choices here:

Just live with the arbitrariness. Allow such expressions as distinct formulas, but prove that formulas like these which are can be turned into each other by the renaming of bound variables (formulas which are -equivalent, as they say) are always interderivable, are logically equivalent too.

Say that formulas proper are what we get by quotienting expressions by -equivalence, and lift our first-shot definitions of e.g. wellformedness for expressions of FOL to become definitions of wellformedness for the more abstract formulas proper of FOL.

Now, as A says, there is in the end not much difference between these two options; but he plumps for the second option, and for a reason. The thought is this. If we work at expression level, we will need a story about allowable substitutions of terms for variables that blocks unwanted variable-capture. And A suggests there are three ways of doing this, none of which is entirely free from trouble.

Distinguish free from bound occurrences of variables, define what it is for a term to be free for a variable, and only allow a term to be substituted when it is free to be substituted. Trouble: “involves inserting qualiﬁcations everywhere and checking that they are maintained.”

Modify the definition of substitution so that bound variables first get renamed as needed — so that the result of substituting for in is something like . Trouble: “Even though we can ﬁx a recipe for executing the renaming, the choice is somewhat arbitrary. Moreover, because of the renamings, statements we make about substitutions will generally hold only up to -equivalence, cluttering up our statements.”

Maintain separate stocks of free and bound variables, so that the problem never arises. Trouble: “Requires us to rename a variable whenever we wish to apply a binder.”

I’m not quite sure how we weigh the complications of the first two options against the complications involved in going abstract and defining formulas proper by quotienting expressions by – equivalence. But be that as it may. The supposed trouble counting against the third option is, by my lights, no trouble at all. In fact A is arguably quite misdescribing what is going on in that case.

Taking the Gentzen line, we distinguish constants with their fixed interpretations, parameters or temporary names whose interpretation can vary, and bound variables which are undetachable parts of a quantifier-former we might represent ‘’. And when we quantify to get we are not “renaming a variable” (a trivial synactic change) but replacing the parameter which has one semantic role with an expression which is part of a composite expression with a quite different semantic role. There’s a good Fregean principle, use different bits of syntax to mark different semantic roles: and that’s what is happening here when we replace the ‘’ by the ‘’, and at the same time bind with the quantifier (all in one go, so to speak).

So its seems to me that option 1c is markedly more attractive than A has it (it handles issues about substitution nicely, and meshes with the elegant story about semantics which has true on an interpretation when is true however we extend that interpretation to give a referent to the temporary name ). The simplicity of 1c compared with option 2 in fact has the deciding vote for me.

Good news! The second edition of GWT is available as a (free) PDF download. This new edition is revised throughout, and is (I think!) a significant improvement on the first edition which I put together quite quickly as occupational therapy while the pandemic dragged on.

In fact, the PDF has been available for a week or so. But it is much nicer to read GWT as a physical book (surely!), and I held off making a splash about the finalised new edition until today, when it also becomes available as a large-format 154pp. paperback from Amazon. You can get it at the extortionate price of £4.50 UK, $6.00 US — and it should be €5 or so on various EU Amazons very shortly, and similar prices elsewhere. Obviously the royalties are going to make my fortune. ISBN 1916906354.

The paperback is Amazon-only, as they offer by far the most convenient for me and the cheapest for you print-on-demand service. A more widely distributed hardback for libraries (and for the discerning reader who wants a classier copy) will be published on 1 December and can already be ordered at £15.00, $17.50. ISBN: 1916906346. Do please remember to request a copy for your university library: since GWT is published by Logic Matters and not by a university press, your librarian won’t get to hear of it through the usual marketing routes.

Having recently been critical of not a few books here(!), let me mention a rather good one for a change. I’ve had on my desk for a while a copy of Topology: A Categorical Approach by Tai-Danae Bradley, Tyler Bryson and John Terilla (MIT 2020). But I have only just got round to reading it, making a first pass through with considerable enjoyment and enlightenment.

The cover says that the book “reintroduces basic point-set topology from a more modern, categorical perspective”, and that frank “reintroduces” rather matters: a reader who hasn’t already encountered at least some elementary topology would have a pretty hard time seeing what is going on. But actually I’d say more. A reader who is innocent of entry-level category theory will surely have quite a hard time too. For example, in the chapter of ‘Prelminaries’ we get from the definition of a category on p. 3 to the Yoneda Lemma on p. 12! To be sure, the usual definitions we need are laid out clearly enough in between; but I do suspect that no one for whom all these ideas are genuinely new is going to get much real understanding from so rushed an introduction.

But now take, however, a reader who already knows a bit of topology and who has read Awodey’s Category Theory (for example). Then they should find this book very illuminating — both deepening their understanding of topology but also rounding out their perhaps rather abstract view of category theory by providing a generous helping of illustrations of categorial ideas doing real work (particularly in the last three chapters). Moreover, this is all attractively written, very nicely organized, and (not least!) pleasingly short at under 150 pages before the end matter.

In short, then: warmly recommended. And all credit too to the authors and to MIT Press for making the book available open-access. So I need say no more here: take a look for yourself!

A break from logical matters, away for half-a-dozen busy days in Athens, followed by visiting family on the island of Rhodes for a week. Both most enjoyable in very different ways. Then we needed a holiday to recover …

But I’m back down to business. The first item on the agenda has been to deal with some very useful last comments on the draft second edition of Gödel Without (Too Many) Tears, and to make a start on a final proof-reading for residual typos, bad hyphenations, and the like. I hope the paperback will be available in about three weeks. If you happen to be reading this while thinking of buying the first edition, then save up your pennies. The second edition is worth the short wait (and will be again as cheap as I can make it).

I’m still finding the occasional slightly clumsy or potentially unclear sentence in GWT. I can’t claim to be a stylish writer, but I can usually in the end hit a decently serviceable level of straightforward and lucid prose. But it does take a lot of work. Still, it surely is the very least any author of logic books or the like owes their reader. I certainly find it irksome — and more so with the passing of the years — when authors don’t seem to put in the same level of effort and serve up laborious and uninviting texts. As with, for example, Gila Sher’s recent contribution to the Cambridge Elements series, on Logical Consequence.

Mind you, the more technical bits have to fight against CUP’s quite shamefully bad typesetting. But waiving that point, I really have to doubt that any student who needs to have the Tarskian formal stuff about truth and consequence explained is going to smoothly get a good grasp from the presentation here. And I found the ensuing philosophical discussion quite unnecessarily hard going. And if I did, I’m sure that will apply to the the intended student reader. So I’m pretty unimpressed, and suggest you can give this Element a miss unless you have a special reason for tackling it.

Another book which readers of this blog will probably want to give a miss to is Eugenia Cheng’s latest, The Joy of Abstraction:An Exploration of Math, Category Theory and Life (also CUP). This comes garlanded with a lot of praise. I suppose it might work for some readers.

But the remarks supposedly showing that abstract thought of a vaguely categorial kind is relevant to ‘Life’ are embarrassingly jejune. The general musings about mathematics will seem very thin gruel (and too often misleading to boot) to anyone who knows enough mathematics and a bit of philosophy of mathematics. Which leaves the second half of the book where Cheng is on much safer home ground “Doing Category Theory”.

So I tried to approach this part of the book with fresh eyes and without prejudice, shelving what has gone before. But, to my surprise, I found the level of exposition to rather less good than I was expecting (knowing, e.g., Cheng’s Catsters videos). She is aiming to get some of the Big Ideas across in an amount of detail, and I was hoping for some illuminating “look at it like this” contributions — the sort of helpful classroom chat which tends to get edited out of the more conventional textbooks. But I’m not sure that what she does offer works particularly well. Try the chapter on products, for example, and ask: if you haven’t met the categorial treatment of products before, would this give you a good enough feel for what is going on and why it so compellingly natural? Or later, try the chapter on the Yoneda Lemma and ask: would this give someone a good understanding of why it might be of significance? I’m frankly a bit dubious.

So that’s a couple of recent CUP books that I did acquire, electronically or physically, and am sadly not enthused by. But in their bookshop there is another new publications which looks wonderful and extremely covetable, a large format volume on The Villa Farnesina. On the one hand, acquiring this would of course be quite disgracefully self-indulgent. On the other hand …

I confess to have given little thought in the past to questions of just when objects of problematic provenance in our museums should be repatriated. But, better late than never, I realize I can’t conjure any cogent reason why the “Elgin Marbles”, the Parthenon Frieze and the rest, shouldn’t now be returned by the British Museum and displayed in the beautiful Acropolis Museum. That museum, as we found last week, is already worth a trip to Athens in itself, and the huge gallery waiting for the originals of the rest of the frieze is just stunning. Time the marbles went home.

A last call for comments/corrections (please!!) for the draft second edition of Gödel Without Tears — I plan to finalize and publish a paperback version around the end of the month. You can download the draft here (though I imagine that anyone interested will have done so by now).

I’ve mentioned before that my three hundred pages of introductory notes on category theory are downloaded surprisingly often — frequently enough for it to be rather embarrassing, given their current ramshackle state. So, with GWT2 simmering on the back burner while I wait to see if there are any last minute suggestions to deal with, I’m getting back to thinking a bit about categories.

I was, for a while, stumbling over two things when thinking about how to revise/develop the notes. Firstly, I didn’t have a clear enough conception of where I wanted to get to. Secondly, I’ve become increasingly unhappy with the way the very opening chapters are handled (with those distracting sermons about set theory!). But I think that things are now falling into place rather better.

On the question of scope, of where to finish, I’m lowering my sights a bit. I had occasion, the other day, to be looking at the classic book on topos theory by Mac Lane and Moerdijk. It starts with a scene-setting fourteen pages of “Categorical Preliminaries” — a glorified checklist of what you need to bring to the party if you are planning to dive into the book. And that checklist more or less exactly corresponds to the topics sort-of covered (in rushed way towards the end) in the existing notes. So that’s persuaded me that maybe, after all, the notes do get to a sensible enough stopping point (and perhaps only need be rounded out with some brief pointers to routes onwards).

And on the question of how to start, I’ve decided that fussing at the outset about such issues as whether we should identify functions with their graphs just doesn’t make for a happy beginning. That’s largely got to go! But this makes for quite a bit of fiddly re-writing over the initial chapters.

I’m having a family break for a couple of weeks, so the new version of the first seventeen chapters or so won’t be ready for a few weeks. But I’m feeling decidedly cheerier about the project of improving those notes, at least enough for me to rest fairly content with the unambitious result.

The wider world continues to go mad and/or bad in various depressing ways. The most distractingly enjoyable novel I have read just recently? Perhaps Elspeth Barker’s atmospherically gothic O Caledonia. I dived in because of an enthusiastic recommendation by Maggie O’Farrell. I enthusiastically pass on the recommendation!

I have also been much distracted by Edmund de Waal’s The White Road, swept along by his obsession with porcelain and its origins (with walk on parts for Spinoza and Leibniz by the way). Strangely gripping I find!

A heads up, as they say. Jeremy Avigad’s new book Mathematical Logic and Computation has now been published by CUP (or at least, an e-version is already available on the Cambridge Core system if you have access — with the hardback due soon). Here’s a link to the front matter of the book, which gives you the Table of Contents and the Preface. Between them, they give you a fair idea of the coverage of the book.

As you’d expect from this author, this book is very worth having, an excellent addition to the literature, with plenty more than enough divergences and side-steps from the more well-trodden paths through the material to be consistently interesting. Having quickly read a few chapters, and dipped into a few more, I’d say that the treatments of topics, though very clear, are often rather on the challenging side (Avigad’s Carnegie Mellon gets very high-flying students in this area!). For example, the chapters on FOL would probably be best tackled by someone who has already done a course based on something like Enderton’s classic text. But that’s not a complaint, just an indication of the level of approach.

When the physical version becomes available — so much easier to navigate! — I’m going to enjoy settling down to a careful read through, and maybe will comment in more detail here. Meanwhile, this is most certainly a book to make sure your library gets.

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.