Peter Smith's Blog, page 12

Jonathan Bennett (1930-2024), Michael Tanner (1934-2024)

The deaths have been announced of two Cambridge philosophers.

Jonathan Bennett was here from 1956–1968, and I was and remain a huge admirer. His early little book Rationality is a masterpiece, and for a good few years I was much intrigued by his defence of a Gricean programme in the philosophy of language which comes to fruition in his Linguistic Behaviour. And when it came to my retirement and I had to radically downsize my ludicrously big library, one of the few history of philosophy books that I couldn’t bear to let go was his Kant’s Analytic which still strikes me as the very paradigm of how to make the Great Dead Philosophers live as exciting interlocutors with something to say which is still worth grappling with. His energetic, direct, straight-talking style as a philosopher I found inspirational over the years.

Bennett was a prolific publisher, not so Michael Tanner, who spent his whole career in Cambridge, first as an undergraduate and then from 1961 until his retirement in 2002 as a lecturer. But he had a great cultural influence on many students over the years, in the way that dons of a different era could do. His Wagner evenings were legendary (and his eventual short book Wagner is a terrific read, even for those of us who never quite caught the bug). And he became a wonderful reviewer of CDs for the BBC Music Magazine, and of opera for the Spectator. He was passionately engaged, opinionated, insightful because — as a loyal Leavisite — he thought such things really mattered to life.

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Category Theory I — now in paperback

Sound the trumpets! Or at least, let me give a small toot …

Category Theory I: A gentle prologue  is now available as a print-on-demand paperback. This is Amazon only (sorry, but that’s easiest for me and cheapest for you), with ISBN 1916906389, at £5.99, $8.25, €7.50. Those prices are only trivially rounded up from the minimum possible (e.g. from £5.90), so I’m not taking any significant royalties.

I’m thinking of the paperback as still a beta version of the text. Functional, and I hope with no horrible mistakes, but surely not bug-free. I’ll still be very happy, then, to get corrections and comments and suggestions for improvement. The Amazon print-on-demand system makes future updates of the file for printing very straightforward and cost free.

The PDF is of course still freely downloadable from the category theory page; but many prefer to work from a printed copy. So, since the paperback is the price of — what? — just three coffees, and is actually quite nicely produced, why not treat yourself!

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Three things to read …

The death was announced a few days ago of W.W. Tait, whose work I have much admired over the years. His early technical work was on proof theory, and then he wrote with great knowledge and insight on the philosophy of mathematics and its history. His collection of papers The Provenance of Pure Reason is a must-read, and more excellent papers can be found on his otherwise minimalist website here. One piece of Tait’s which I can’t remember reading before and which will surely be of interest to anyone reading this blog is his piece contributing to Philosophy of Mathematics: 5 Questions: you can read it here.

Brian Leiter’s blog linked yesterday to a piece by a one-time colleague of mine, Leif Wenar, lambasting the pretentions of the “effective altruism” cult. I laughed out loud at his (surely just) comment that “To anyone who knows even a little about aid, it’s like [Will] MacAskill has tattooed “Not Serious” on his forehead.”. But this is serious stuff, and well worth a read here.

For some hard-core logical reading, here is another one-time colleague in action. Tim Button has just posted on the arXiv a forthcoming JSL paper “Wand/Set Theories: A realization of Conway’s mathematicians’ liberation movement, with an application to Church’s set theory with a universal set”. Tim describes a template for introducing mathematical objects which prima facie is much more liberal than standard set theory provides. Indeed it seems to very nicely encapsulate Conway’s liberation movement, allowing that (in Conway’s words)

(i) “Objects may be created from earlier objects in any reasonably constructive fashion.

(ii) Equality among the created objects can be any desired equivalence relation.”

Note, though, that Conway expected that any theory whose objects are so created in such a “reasonably constructive” fashion can be embedded within (some extension of) ZF. Tim aims to prove a stronger theorem: all loosely constructive implementations of the Wand/Set Template are not merely embeddable in (some extension of) ZF, but synonymous with a ZF-like theory. Which seems a surprise.

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Markdown joy

A minor improvement here …

You can now use simple Markdown syntax (you know the kind of thing, *italics* for italics, **bold** for bold) in comment boxes.

This is explicitly signalled in comment boxes on the blog; but it is also the case with comment boxes on static pages.

(Nerdy trivia: this is enabled by using the multi-purpose Jetpack plugin — which is wild overkill: but I’ve turned off more or less every other Jetpack option so I hope that nothing is inadvertently broken.)

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Elisabeth Brauss at Wigmore Hall

A wonderful lunchtime concert by Elisabeth Brauss at Wigmore Hall this Monday. She played Prokofiev (8 Pieces from Op 12) and Beethoven (Sonata in E Flat Major, Op.31 No.30). Such technical control combined with joie de vivre and great sensitivity too when called for. Really as good as it gets. The audience loved her as usual, and she responded wreathed in smiles. A delight. You can listen on BBC Sounds for another four weeks.

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Core logic again

I have mentioned Neil Tennant’s system(s) of what he calls Core Logic once or twice before on this blog, in friendly terms. For the very shortest of introductions to the core idea of his brand of relevant logic, see my post here on the occasion of the publication of his book on the topic. (And there is a bit more info here in a short note in which I respond to some criticisms on Neil’s behalf — unnecessarily, it turned out, as he published his own rejoinder.)

I notice that Neil has now written a piece outlining his developed ideas on Core Logic for Philosophia Mathematica. If you want to know more, this might be a good place to start. You can download this paper here.

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The blog is eighteen!

Post #1 was back on March 9th, 2006. And here we are again, with post #1715, and with the blog yet another year older, if not a year wiser. I’ll raise a glass.

Since its last birthday, I’ve started to put together archive pages to make it easier to find those old posts which may (or then again, may not) be still worth reading. I really must make the effort to complete that job.

But I’m afraid that just recently there hasn’t been so very much happening here. However, the end of the category theory project really, really, is in sight now, and when it at last gets off my desk, I do hope to have a bit more time and energy to devote to posting more often. For a start, there’s a pile of books — and not just logic books — I’d like to say something about.

Today though, my self-denying ordinance stands: it’s back to concentrating on reworking the final few pages of Category Theory II … Wish me luck.

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Ana Agore, A First Course in Category Theory

The most recommended introductory books on category theory (at least for pure mathematicians) are probably those by Steve Awodey, Tom Leinster, and Emily Riehl. All three have very considerable virtues. But for differing reasons, each presents quite steep challenges to the beginner (especially for self-study). Having, back in the day, worked through Awodey’s book with a reading-group of super-smart Cambridge Part III (i.e. graduate) students, I can only report that we found it engaging but a much bumpier ride than the author surely intended. Leinster’s shorter book, although my favourite, is often quite compressed and I’m told that students can again find it quite tough for that reason. Riehl’s book is full of good things — her title Category Theory in Context points up that she is particularly seeking to make multiple connections across mathematics. But she goes at pace and the connections made can be distractingly/dauntingly sophisticated.

So there is certainly room on the shelf for another introductory book, especially one advertised as being “unlike traditional category theory books, which can often be overwhelming for beginners. …[It] has been carefully crafted to offer a clear and concise introduction to the subject. … the book is perfectly suited for classroom use in a first introductory course in category theory. Its clear and concise style, coupled with its detailed coverage of key concepts, makes it equally suited for self-study.” So: does Ana Agore’s recently published A First Course in Category Theory (Springer, Dec. 2023) live up to the blurb?

Here’s the very first sentence of Chapter 1: “We start by setting very briefly the set theory model that will be assumed to hold throughout.” Which is garbled English. Quite unsurprising, I’m afraid, from Springer who don’t seem to proof-read their books properly these days. And I do wonder whether Agora has run her text past enough readers including a native speaker or two. For in fact there are quite a few unEnglish sentences. Fortunately, the intended message is only occasionally obscure, at least to this reader who has the advantage of knowing what Agore should be saying. I suspect, however, that some — especially if English is not their first language — may sometimes stumble.

The Preface tells us that the book is based on lecture notes from a graduate course. And that’s how it reads. We get action-packed notes, with a lot of detail given at a relentless pace, and with really very little added motivating classroom chat. The typical approach is to plonk on the table a categorial definition without preliminary scene-setting, and then give a long (sometimes very long) list of examples. And the level of discussion sometimes seems rather misplaced — is it really helpful for the introduction of categorial ideas to be interrupted, as early as p. 7, by an unobvious argument more than a page long to show that epimorphisms in Grp are surjective?

Again as early as p. 12, we are given the categorial definition of a subobject of C as an equivalence class of monics with codomain C. What could motivate pulling that strange-seeming rabbit out of the hat? We aren’t told. Rather, we quickly find ourselves in a discussion of how the definition applies in KHaus vs Top.

Another case: on p. 24 the definition of a functor is served up ‘cold’, followed by thirty-five examples. Or more accurately, we get thirty-five numbered items, but general points (e.g. that functors compose) are jumbled in with particular examples.

All in all, this does read rather like handout-style notes expanded with more proofs written out and with multiple extra examples, but served up without the connecting tissue of classroom remarks which can give life and direction to it all and which the self-studying reader is surely going to miss rather badly.

What does the book cover? How is it structured?

There are three long chapters. Chapter 1 (82 pp.) is on Categories and Functors, taking us up to the Yoneda Lemma. Chapter 2 (70 pp.) is on Limits and Colimits. Chapter 3 (98 pp.) is on Adjoint Functors. There follows a welcome chapter (26 pp.) of solutions to selected exercises.

But note that although Agore tells us about subobjects early on, we don’t get round to subobject classifiers. We meet limits and colimits galore, but we don’t meet exponentials. And again as contrasted with e.g. Awodey, while of course we get to know about categories of groups and groups as categories, we don’t get to know about groups in categories, internal groups.

In a little more detail, Chapter 1 covers what you would expect, basic definitions and examples of categories, types of arrows and special objects (like initial/terminal objects), functors, natural isomorphisms and natural transformations more generally, hom-functors and representables, ending up with Yoneda. There are some oddities along the way — the idea of elements as arrows from 1 (like the idea of ‘generalized elements) is never mentioned, I think, while the idea of a universal property makes its first appearance on p. 16 but seems never to be given a categorial treatment.

Tom Leinster has written “The level of abstraction in the Yoneda Lemma means that many people find it quite bewildering.” It’s a good test for an introductory book how clear it makes the lemma (in its various forms) and now natural the relevant proofs seem. How does Agore do? Here’s her initial statement.

She then adds that the bijections here, for a start, form a natural transformation in C:

If you are reading this review you will know what’s going on. But if you are new to the material, I bet — for a start — that these notational choices won’t be maximally helpful, and the ensuing pages of proofs will look significantly messier and harder work than they need to be. I certainly wouldn’t recommend Agore’s pages 70-77 as my go-to presentation.

Chapter 2 on limits and colimits continues in the same style. So the first definition is of multiproducts (rather than softening us up with binary products first). There’s no initial motivation given: the definition is stated and some theorems proved before we get round to seeing examples of how the definition works out in practice in various categories. We then meet equalizers and pullbacks done in much the same spirit (I don’t suppose anyone will be led astray, by the way, but contrary to her initial definition, Agore now starts allowing fork diagrams with non-equal parallel arrows to count as commuting).

On the positive side, I do very much approve of the approach of first talking about limits over diagrams, where a diagram is initially thought of as a graph living in a category, before getting fancy and re-conceptualizing limits as being limits for functors. And if you have already met this material in a less action-packed presentation, this chapter would make useful consolidating material. But, I’d say, don’t start here.

And much the same goes for Chapter 2 on adjunctions, which gets as far as Freyd’s Adjoint Functor Theorem and the Special Adjoint Functor Theorem. This is another rather relentless chapter, but with more than the usual range of examples. Some proofs, such as the proof of RAPL, seem more opaque than they need to be. Again, I wouldn’t recommend anyone starting here: but treated as further reading it could well be a useful exercise to work through (depending on your interests and preferred mathematical style).

So the take-home verdict? The book advertises itself as a ‘first course’ and as suitable for self-study. However, I do find it pretty difficult to believe it would work well as both. Yes, I can imagine a long graduate lecture course, with this book on the reading list, as potentially useful back-up reading once the key ideas have been introduced in a more friendly way, with more motivating classroom chat. But for a first encounter with category theory, flying solo? Not so much.

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Miracle on St David’s Day.

St Davids Cathedral. Wales

I have just noted, with delight, that the poet Gillian Clarke — the much admired, much loved, one-time National Poet of Wales — has a new collection of poems coming out from Carcanet Press this month. We have read and reread and read her work again ever since we lived in Wales, and find it so deeply appealing. If by some ill chance, you haven’t really come across her poetry, try — yes, do try — her Selected Poems of 2016, or at least browse her website.

In this new book, we are told, “The poems in … The Silence begin during lockdown, to whose silences Clarke listens so attentively that other voices emerge. As the book progresses, that silence deepens, in the poems about her mother and childhood, about the Great War and its aftermaths, and in her continuing attention to Welsh places and names, and the rituals which make that world come in to focus. In these scrupulous, musical poems, Clarke finds consolation in how silence makes room for memory and for the company of the animal- and bird-life which surrounds us. These poems, compulsively returning to key images and formative moments, echo and bring back other ways of living to the book’s present moment.”

Since the poem is on her website, I hope that Gillian Clarke will forgive me if I reproduce here a particularly touching poem of hers, appropriate to the day, dating back to a real event in the 1970s.

Miracle on St David’s Day.

‘They flash upon that inward eye
which is the bliss of solitude’
(from ‘The Daffodils’ by William Wordsworth)

An afternoon yellow and open-mouthed
with daffodils. The sun treads the path
among cedars and enormous oaks.
It might be a country house, guests strolling,
the rumps of gardeners between nursery shrubs.

I am reading poetry to the insane.
An old woman, interrupting, offers
as many buckets of coal as I need.
A beautiful chestnut-haired boy listens
entirely absorbed. A schizophrenic

on a good day, they tell me later.
In a cage of first March sun a woman
sits not listening, not feeling.
In her neat clothes the woman is absent.
A big, mild man is tenderly led

to his chair. He has never spoken.
His labourer’s hands on his knees, he rocks
gently to the rhythms of the poems.
I read to their presences, absences,
to the big, dumb labouring man as he rocks.

He is suddenly standing, silently,
huge and mild, but I feel afraid. Like slow
movement of spring water or the first bird
of the year in the breaking darkness,
the labourer’s voice recites ‘The Daffodils’.

The nurses are frozen, alert; the patients
seem to listen. He is hoarse but word-perfect.
Outside the daffodils are still as wax,
a thousand, ten thousand, their syllables
unspoken, their creams and yellows still.

Forty years ago, in a Valleys school,
the class recited poetry by rote.
Since the dumbness of misery fell
he has remembered there was a music
of speech and that once he had something to say.

When he’s done, before the applause, we observe
the flowers’ silence. A thrush sings
and the daffodils are flame.

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Reasoning with Attitude

Here is a familiar thought, one that many of us find attractive:

For some classes of sentence, their primary semantic role is not to report a special class of facts but rather to express certain attitudes.

Moral claims are a paradigm case for this sort of treatment. The idea that such claims are referential in nature, aiming to track moral facts which are out there in the world independently of us (so to speak) is metaphysically puzzling, to say the least. A rival expressivist account looks prima facie attractive. If, in even the loosest sense, meaning is use, then the semantic story about moral discourse should surely be rooted in its use to express, share, and engender attitudes. Or so the story goes. And expressionism about other areas of discourse too can look attractive: think, for example, of Ramsey’s idea of attributions of probability (or at least some classes of these) as expressing degrees of beliefs. Perhaps modal judgments too can be handled in this way, without mystery-mongering: a judgment that P is necessary expresses something about P‘s special role as a fixed point in our web of belief, rather than magically latching onto facts regarding other possible worlds, whatever that can mean.

Here is another familiar thought, also one that many of us  find attractive:

For some classes of expression, their primary semantic role is to be explained by their role in inference.

For example, it is the introduction and elimination rules governing our inferential moves with the logical connectives that basically capture their meaning. Or so the story goes. And a more wide-ranging inferentialist semantics has its attractions. Maybe, unlike inferentialism about individual logical operators, the more general story will need to talk more holistically about the role of an expression in a whole wider inferential practice which mediates between experiencing the world and acting on it — think, for example, of Sellars. But again that chimes with a pragmatist, meaning-is-use, stance.

Given the separate attraction of these general ideas, it looks an obvious move to see what we can get by putting them together. And indeed, it could well be that they can help each other out in crucial ways. Think, for example, of the Frege-Geach problem for expressivism, which has it that naive expressivism can’t account for those unasserted uses of moral claims (as in the antecedents of conditionals) which aren’t expressing attitudes — we need an account of the inferential role of such claims. Or in the other direction, think how an inferentialism about logical connectives could perhaps be improved by appeal to reflections about the  role of negation in expressing an attitude of rejection or about the way that conditionals engage with expressing suppositional modes of thought.

It looks, then, as though there could very well be work to be done by an expressivist take on inferentialism or an inferentialist take on expressivism. So it is surprising then that no one has explicitly set out to put our two themes together like that. Until now. For this is the prospectus of the new book by Luca Incurvati and Julian Schlöder, Reasoning with Attitude: Foundations and Applications of Inferential Expressivism.

A central contention of this book is that, their differences notwithstanding, expressivism and inferentialism are best seen as opposing referentialism on the basis of the same pragmatist insight: that semantic explanations should not go beyond what is needed to explain the role of words in our practices. Expressivists focus on the attitudes that words are used to express; inferentialists focus on the inferences that words are used to draw. In this book, we lay the foundations for inferential expressivism, a theory of meaning which countenances both aspects of our linguistic practice and explains meaning in terms of the inferences we draw involving the attitudes we express.

This promises, then, to be a really engaging, ground-breaking book. I’m (with regrets!) not going to break my self-denying ordinance and start blogging about it right now instead of finishing the category theory books, though I have started reading Reasoning with Attitude with considerable enjoyment.

And you too might want to make a start, to see whether the book’s themes and approach appeals. For you can freely do so. I’m delighted to report that the authors’ research grant has enabled OUP to publish the book under their open-access scheme. You can download it here.

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