That nothing is certaine

I’ve been trying to put together better notes on elementary category theory; which is an engaging exercise, but that doesn’t lead to anything very interesting to report here. Another thing occupying me on and off has been writing quite extensive comments on a draft book by an ex-colleague: but although that does raise very interesting issues, it mostly wouldn’t be appropriate to rehearse them here.

Then there have been time-consuming domestic chores — including tackling the Book Problem once again. This time round, it’s been the non-work books. Piles around the house acquired over lockdown (and I’m not the only guilty party) were beginning to totter. So it’s been the occasion for an overdue major sort-out, with re-shelvings, donations of bags of books to Oxfam, and all those discussions “are either of us ever going to read that/read that again?”. Yes, I would for example, have liked to have read the  massive biography of Darwin: but life is too short (a phrase that becomes ever more telling as the birthdays clock by). That’s a precious three inches of shelf-space reclaimed. And so on it goes … But hardly the topic for a riveting essay here!

But perhaps the main reason for the lack of many posts recently has been low spirits as much as anything: try as you might, the state of the world just gets you down, no? The grim uncertainty of it all. But then,

… these things every one doe enwrap and entangle silly mortall men, void of all forecast and true understanding: so as this only point among the rest remains sure and certain, namely: That nothing is certaine …

Thus Pliny, “done into English” by Philemon Holland.

Philosophical Uses of Categoricity Arguments

Peter F. has written:

I wanted to let you know about a paper that was recently put on the arXiv which I think you and many readers of this site will find very interesting, in case it hasn’t been noticed yet:

Penelope Maddy & Jouko Väänänen “Philosophical Uses of Categoricity Arguments” https://arxiv.org/abs/2204.13754

This does indeed on a quick skim look a seriously interesting and thought-provoking paper (though not an easy read: there is a lot of detailed argument here). Thanks for the pointer!

Beginning Category Theory: Chs 1 to 13 (etc.)

I have now re-revised Chapters 1 to 13 of Beginning Category Theory. So here they are again, as before together in one long PDF with the remaining unrevised chapters from the 2015/2018 Gentle Intro. [You may need to force a reload to get the latest version, dated April 27.]

There are significant changes in the rhetoric of Chapter 3, though the intended general position hasn’t really changed. Elsewhere there are scattered, mostly minor, changes to improve clarity and readability. I’m still far from happy with the overall tone/style: but I hope I’m edging slowly, slowly in the right direction!

OK, the next major task is to tidy up the chapter on equalizers and co-equalizers. Now, I motivated the categorial treatment of products at some length by talking more informally, and pre-categorially, about what we want from pairing schemes. But at the moment, like too many every elementary presentations, I just plonk the definitions of equalizers and co-equalizers on the table without motivational pre-amble, and then pull the rabbit out of the hat and say “oh look, quotients of equivalence relations are a special case of co-equalizers!”.

That’s not at all satisfying, and I’d like to do better (as I see Awodey does)!

Like it or not, again

I experimented with a ‘like’ button for blog posts for ten days. I thought it probably wouldn’t be used much, and it turns out I was right (for I know that each post is actually read hundreds of times). So I’ve decluttered and removed the button again.

In fact, everything about “user engagement”, as they say, remains a bit of a mystery to me. It is very nice to know, for example, that the Beginning Math Logic study guide has already been downloaded almost a thousand times this month. But how the word gets around, who the readers are, what they make of the guide, is all pretty unfathomable.

Never mind. The overall site statistics (whatever they mean in absolute terms) continue to look perfectly healthy. So as long as I’m not entirely talking to myself, on we go …

Juliette Kennedy, Gödel’s Incompleteness Theorems

I was in the CUP Bookshop the other day, and saw physical copies of the Elements series for the first time. I have to say that the books are suprisingly poorly produced, and very expensive for what they are. I suspect that the Elements are primarily designed for online reading; and I certainly won’t be buying physical copies.

I’ve now read Juliette Kennedy’s contribution on Gödel’s Incompleteness Theorems. Who knows who the reader is supposed to be? It is apparently someone who needs the notion of a primitive recursive function explained on p. 11, while on p. 24 we get a hard-core forcing argument to prove that “There is no Borel function F(s) from infinite sequences of reals to reals such that if ran(s) = ran(s’), then F(s) = F(s’), and moreover F(s) is always outside ran(s)” (‘ran’ isn’t explained). This is just bizarre. What were the editors of this particular series thinking?

Whatever the author’s strengths, they don’t include the knack of attractive exposition. So I can’t recommend this for reading as a book. But if you already know your way around the Gödelian themes, you could perhaps treat this Element as an occasionally useful scrapbook to dip into, to follow up various references (indeed, some new to me). And I’ll leave it at that.

Birthday treats

Another birthday yesterday, a quite daft number, but much better than the alternative. A few treats, and two of them anyone else can enjoy too for a relatively modest outlay. One was the book accompanying the new Raphael exhibition at the National Gallery. Will we brave the covid-ridden crowds to go up to the exhibition ourselves? As asking the question that way suggests, I rather doubt it. But the catalogue book, as nearly always for the National  Gallery’s major events, is pretty terrific, and will be a lasting pleasure in itself. And the DVD of the 2020 Salzburg Festival Così fan Tutte which we watched last night is just a delight. Hugely enjoyable, very engagingly acted, with some wonderful singing, in particular from Elsa Dreisig. So both the book and the DVD are very warmly recommended; we all need cheering.

Beginning Category Theory: Chs 1 to 11 (and more)

To avoid readers having to juggle two PDFs, and to keep at least some cross-references between new and old material functioning, I have decided to put the newly revised chapters together with the old unrevised chapters from the Gentle Intro into one long document. So here is Beginning Category Theory which starts with eleven revised chapters, followed by all the remaining old chapters [with prominent headline warnings about their unrevised status].

The two newly revised chapters are

  1. Pairs and products, pre-categorially [Motivational background]
  2. Categorial products introduced [Definitions, examples, and coproducts too]

Note: these early revised chapters are not final versions. Revised chapters get  incorporated when I think that they are at least better than what they replace, not when I think they are as good as they could be. So, needless to say, all comments and corrections will be very gratefully received. Onwards!

Greg Restall, Proofs and Models in Philosophical Logic

There does seem little consistency in the level/intended audience of the various books in the Cambridge Elements series. John Bell’s book on type theory is pretty hard-core graduate level, and mathematical in style and approach. John Burgess’s book on set theory I found to be a bit of a mixed bag: the earlier sections are nicely approachable at an introductory level; but the later overview of topics in higher set theory — though indeed interesting and well done — seems written for a different, significantly more mathematically sophisticated, audience. It is good to report, then, that Greg Restall — as his title promises — does keep philosophers and philosophical issues firmly in mind; he writes with great clarity at a level that should be pretty consistently accessible to someone who has done a first formal logic course.

After a short scene-setting introduction to the context, there are three main sections, titled ‘Proofs’, ‘Models’ and ‘Connections’. So, the first section is predictably on proof-styles — Frege-Hilbert proofs, Gentzen natural deduction, single-conclusion sequent calculi, multi-conclusion sequent calculi — with, along the way, discussions of ‘tonk’, of the role of contraction in deriving certain paradoxes, and more. I enjoyed reading this, and it strikes me as extremely well done (a definite recommendation for motivational reading in the proof-theory chapter of the Beginning Math Logic guide).

I can’t myself muster quite the same enthusiasm for the ‘Models’ section — though it is written with the same enviable clarity and zest. For what we get here is a discussion of variant models (at the level of propositional logic) with three values, with truth-value  gaps, and truth-value gluts, and with (re)-definitions of logical consequence to match, discussed with an eye on the treatment of various paradoxes (the Liar, the Curry paradox, the Sorites). I know there are many philosophers who get really excited by this sort of thing. Not me. However, if you are one, then you’ll find Restall’s discussion a very nicely organized introductory overview.

The shorter ‘Connections’ section, as you’d expect, says something technical about soundness and completeness proofs; but it also makes interesting remarks about the philosophical significance of such proofs, depending on whether you take a truth-first or inferentialist approach to semantics. (And then this is related back to the discussion of the paradoxes.)

If you aren’t a paradox-monger and think that truth-value gluts and the like are the work of the devil, you can skim some bits and still get a lot out of reading Restall’s book. For it is always good to stand back and see an area — even one you know quite well — being organised by an insightful and eminently clear logician. Overall, then, an excellent and very welcome Element.

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