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.

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.

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 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 are quite likely to know what’s going on. But if you were quite 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. So I certainly wouldn’t recommend Agore’s pages 70-77 as my go-to presentation of Yoneda.

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 of a commuting diagram, 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 3 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 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.

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.

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.

Another categorial update

It’s been a month since I last posted about the category theory project, so a quick update  — and the end really is in sight!

  • I’ve just put online another revised version of Category Theory I. Little has changed except for some more corrections of typos (with particular thanks to Georg Meyer) and a few small changes for added clarity (with particular thanks to John Zajac). I’ve also made a few very small changes to better fit with what happens in Category Theory II as I steadily revise that.
  • More significantly, there is another version of Category Theory II linked on the category theory page. The old chapters on adjunctions are now in a much better state. I don’t think I found any horrendous errors, but the story is (I certainly hope!) a lot clearer in a number of key places.
  • In fact, the bit of recent work on this that I’m most pleased with is probably the proof of ‘RAPL’ (Right Adoints Preserve Limits). Tom Leinster and Steve Awodey offer fancy-but-unilluminating proofs. I spell out the sort of bread-and-butter proof idea I got from Peter Johnstone’s lectures (and my version is perhaps a little clearer for a first encounter than Emily Riehl’s?).
  • It’s a judgement call where to stop. For example, I still reckon (as I did before) that the Adjoint Functor Theorems are just over the boundary, as far as what is really appropriate for an entry-level introduction. However, I do now say just a very little about monads (so at least you know what the idea is), though I might yet add another example or two.
  • I still need to revise the last three chapters of Category Theory II. They should be in a reasonable basic state as these are the same final chapters that — in the previous arrangement — appeared as the last chapters in the 2023 published version of Category Theory I. However, in the somewhat more advanced context of Category Theory II it might be appropriate to expand the discussions a bit.

I’d hoped that Category Theory II would be paperback-ready by the end of this month. There have been unforeseen distractions. But I’m not far off. Watch this space.

More Cambridge Elements: Mark Wilson on maths, David Liggins on abstract objects

I overlooked Mark Wilson’s Innovation and Certainty when it was published in 2020. But I didn’t miss much. The topic is an excellent one — just what is going on when we make innovations like adding points and lines at infinity in geometry (to take a reasonably comfortable but still instructive example), and just how are these extensions justified? How can we be sure they don’t lead us astray? But heavens, Wilson’s discussion is arm-wavingly pretentious and tediously obscurantist. It is just a dreadful piece of writing, and it baffles me that the series editor let it pass. Don’t waste time on this.

By contrast, David Liggins’s brand new mini-book on Abstract Objects  is the very model for how Elements should surely be.  It is admirably lucid and plain-speaking, approachable by an undergraduate student, yet the way Liggins organizes the material (evidently reflecting a good deal of thought) ought to be useful too for readers with rather more background who, for example, want to revisit the area and perhaps return to thinking about it.

I do have quibbles. Well, more than quibbles. I suspect that we have pretty different views on Hale/Wright abstractionism (which is touched on), and on the value of the ideas in e.g. Charles Parsons’s Mathematical Thought and Its Objects or Øystein Linnebo’s Thin Objects, (neither of which is mentioned). And if I weren’t telling myself that I must concentrate on other projects, I’d certainly be moved to engage properly here. But disagreement is only to be expected with a fifty-page essay. And I’d still happily put this into the hands of a student. Wilson’s effort, not so much.

(Abstract Objects is still free to download from CUP for another 48 hours or so.)

Two new Cambridge Elements on Phil. Maths

Just briefly to note that there are two new short contributions in the Cambridge Elements series in the Philosophy of Mathematics, both free to download for another week. The Euclidean Programme by Alex Paseau and Wesley Wrigley critically examines the traditional idea that mathematical knowledge is obtained by deduction from self-evident axioms or first principles. How much of that idea can be rescued?

And Number Concepts by Richard Samuels and Eric Snyder takes an interdisciplinary approach to reviewing and critically assessing work on number concepts in developmental psychology and cognitive science. (And after all, shouldn’t philosophers of arithmetic be interested in the concepts deployed by folk arithmeticians?)

So far, contributions in this series have been, it seems to me, a rather mixed bunch. So naive induction is little guide, in this case, as to how worthwhile these new efforts will prove to be. But let’s live in hope. When I’ve had a chance to take a proper look, I’ll let you know what I think. But I thought I would post a quick note straight away, while these two Elements are still freely downloadable, and you can judge them for yourself.


These days I rarely visit the philosophy news website Daily Nous. But my eye was caught by a recent post (or in fact, a re-post) inviting readers to report markedly good experiences with journals — to counterbalance the frequent complaints about various  journals for the slowness of getting any decision, and worse.  And in the replies, the journal Ergo comes in for a lot of praise as outstanding in handling submissions. That had to be a surprise to me, because (ok, I’m obviously way off the pace here!) the journal had previously never crossed my radar.

So I took a look, here. And at one level I’m hugely impressed. It’s a genuinely open-access journal; its procedures seem quite exemplary in principle, and by all accounts work excellently well in practice. The online reading experience is terrific, with a well-designed look’n’feel. And if you download a PDF of an article, it is also very decently designed. (Someone with a good eye was involved in tweaking the under-the-bonnet engines driving the site.) As you will probably know, I’m all for open-access, and Ergo seems a splendid model for journals. All credit to those involved.


But ….

When I looked at the abstracts of the forty pieces published last year how many did I actually want to read?

Pretty much zero. I did try dipping into a few on topics that I could perhaps have mustered some interest in but (no names, no pack drill) I found them laboured and unexciting, and I just wasn’t drawn in at all. What did I overlook?

Now I’m well aware that this could indeed reflect much more on my increasing distance from the fray than on the quality of the papers. But equally, I really had little sense that I was missing out on a scene of bubbling intellectual ferment. I’m almost tempted to add: not like the good old days, eh?

(Oh, and I did notice that Analysis still comes in for praise in the Daily Nous comments. That’s good to hear.)

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