Back, at last, to category theory

I am underway, at last, with the project of  improving and updating my notes on category theory. So, here are the first four chapters of Category Theory I: Notes towards a gentle introduction.

The ‘I’ in the new title signals that I am carving the old notes into Part I and Part II, and I am planning to work up Part I into a decent shape, while quite putting aside Part II for a good while. (Lichtenberg: “Just as certain writers, after first dealing their material a rough blow, say it naturally falls into two parts”.) And ‘Notes’ is a frank admission that the material still won’t be very smoothed out, and I’ll not be aiming for a polished book-style finish. ‘Gentle introduction’ means that it goes no doubt far too slowly for some.

I’d be really interested in comments on Chapter 3, which has given me a lot of grief.

In concert: Peter Jablonski and Elisabeth Brauss

Not my usual kind of music! But, this week, here is a performance of Grażyna Bacewicz’s Double Piano Concerto of 1966. The soloists are Peter Jablonski, who last year released a very well received CD of Bacewicz piano works, and Elisabeth Brauss. And since this concert in December, they have recorded the Concerto together for another CD to be released in the spring. Visit the concert page here, press Katso (play), and go to about 22.30 for the Concerto (which last less than twenty minutes). I’m not sure what to make of the music, but I much enjoyed watching them play!

Peter Jablonski and Elisabeth Brauss also play a very short and delightful encore by Ligeti, at 42.10. And then, if you want something a lot calmer, here is another very short piece, this time by Hindemith, quite beautifully played by Elisabeth. Hopefully a trailer for a DG solo disk by her.

Logical Methods — on propositional logic

I have now had a chance to read the first part of Greg Restall and Shawn Sandefer’s Logical Methods, some 113 pages on propositional logic.

I enjoyed this well enough but I am, to be frank, a bit puzzled about the intended readership. The book’s Preface starts “Welcome to Logical Methods, an introduction to logic for philosophy students …”. And the text does indeed seem to start right from scratch. But Restall’s web-page for the book says “The text was developed through years of teaching intermediate (second-year) logic at the University of Melbourne.” While their Amazon blurb says “suitable for undergraduate courses and above.” Which suggests a rather unstable focus. And indeed, a significant amount of the material here, as we’ll see in a moment, is at what strikes me as a decidedly non-introductory level.

Certainly, things that can (and often should!) give pause to a philosophy student encountering formal logic for the first time are often skated over at speed. For example, when we do propositional logic, just what is the relation between the formal systems and our everyday inferences using the ordinary-language connectives? So, exactly what are these dratted “p”s and “q”s doing? On p.8 we are told that “declarative sentences express propositions”, and that we are going to be looking at propositional languages “where there are declarative sentences”. But then are also immediately told that our formal language is just designed “to express the forms of propositions combined with [the connectives]” (my emphasis). So do the “p”s and “q”s get interpretations as expressing propositions or not?

On p. 9 we are baldly told that “disjunctions will always be inclusive in this text” without a moment’s discussion of how things might or might not stand in ordinary language. And later, the much more vexed question of how the logician’s conditional might be related to the ordinary language conditional is relegated to a “challenge question” on p. 32. I wonder: if we don’t say rather more about the ordinary-language logical apparatus, how do we rack up a persuasive score sheet of the costs and benefits of various alternative formal choices? (Teachers using this book with real beginners might well be adding quite a bit of appropriate classroom chat on such matters as they go along — but I’m thinking here of a student reader taking the book “neat”.)

Again, the beginning reader is given just one worked example of a truth-table test for validity in action. And nothing is said e.g. about standard heuristics to speed things up (as in “you don’t need to work further on a line where the conclusion is true because that can’t give us a counterexample”) Yes, yes, of course truth-table testing complicated examples is as boring as heck. But surely(?) we do want our beginning students to be just a bit more au fait with how things can work out in practice.

So already, I’m not sure how well this is going to work with real beginners. But there are more serious worries. Restall and Sandefer advertise their book as presenting “proof construction on equal footing with model building” — but in fact that briskness over truth-tables is just one sign that their presentation is really skewed to emphasize proof-theoretic ideas. And so, long before we ever hear about the classical truth-functional interpretation of the connectives, we are tangling with why we might want detour-free proofs in a Gentzen-style natural deduction system. (By the way, much as though I like the elegance of Gentzen trees, I’m yet to be really persuaded that they trump Fitch-style proofs for introducing ND to students.)

And now, not only is the — I agree! — reasonably intuitive idea of a detour-free proof canvassed, but we actually get a full-on, ten-page, proof of normalizability for intuitionistic propositional logic (starting as early as p. 53 in the book). I honestly can’t imagine too many thinking that this is where they want their beginning philosophy students to be concentrating, so early in their logical encounters!

Now, I don’t want to carp, so let’s now recalibrate our expectations, and think of this as in fact a second-level text with some brisk reminders of the more elementary stuff. Then, on positive side, it can be said that the normalization proof and other parts of the discussion of Gentzen style ND are very accessibly done. So I can e.g. well foresee the relevant sections getting into the next edition of the Study Guide as warmly recommended reading on entry-level proof theory. But yes, for me at least, that is where this material really belongs, a step or two up from a first introductory text for philosophers. Call me old-fashioned!


I note that the text was typeset by the authors (and some of their aesthetic choices are a bit wonky!). But that does raise a question. I do wonder why, in 2023, since they have a nice PDF to hand, they have gone done the route of conventional publication when they could have got the book into so many more students’ hands by going down the free-PDF-plus-cheapo-print-on-demand route? Just saying.

Jonathan Raban, 1942–2023

I have hugely admired Jonathan Raban ever since I first read Old Glory forty years ago, and have praised his books here before. For example, I wrote this a few years ago (editing a little):


Now, how did that happen? Wanting to re-read Jonathan Raban’s Coasting, his evocation of a journey of discovery sailing round Britain, first published over thirty years ago, I just found a couple of days ago that our copy has disappeared from the shelves. A mystery:  I wonder what happened to it. So it is very good to discover that the book has recently been republished, along with four other books by Raban, by Eland, in rather handsomely produced paperbacks. I remember the book as being extraordinarily well written. The blurb on our new copy tells me that  Raban here “moves seamlessly between awkward memories of childhood as the son of a vicar, a vivid chronicle of the shape-shifting sea and incisive descriptions of the people and communities he encounters. As he faces his terror of racing water, eddies, offshore sandbars and ferries on a collision course, so he navigates the complex and turbulent waters of his own middle age. Coasting is a fearless attempt to discover the meaning of belonging and of his English homeland.” Which indeed is how I recall the book. I look forward to it!

I’ve been put in mind to read Coasting again because I have recently been re-reading with great  enjoyment two of Raban’s other books that are on our shelves (as it happens, another two of the five that have been republished by Eland). First there was For Love & Money, which is subtitled “Writing, Reading, Travelling: 1969–87”, and which reprints some early reviews and occasional pieces. The writing is consistently humane and insightful, but more than that, it is just so beautifully readable (the number of times I thought, “I wish I could write even half as well”). And then there was Raban’s early masterpiece, Old Glory from 1981, notionally recounting his voyage down the Mississippi in a small boat.

I say “notionally” as this complex work is lightly disguised as a straight travel book, a literal recounting of a journey taken. But the one-time English literature lecturer warns us clearly enough. One of the epigraphs of the book is from T. S. Eliot (writing of the  Mississippi), starting “I do not know much about gods; but I think that the river/Is a strong brown god …” The other epigraph is from Jean François Millet: “One man may paint a picture from a careful drawing made on the spot, and another may paint the same scene from memory, from a brief but strong impression; and the last may succeed better in giving the character, the physiognomy of the place, though all the details may be inexact.”

So we are set up for this to be a mythic tale (“the river/Is a strong brown god”)  and for the “Jonathan Raban” who features as the narrator and his adventures to be a very loose rendition of the author and his own journey (“all the details may be inexact”). And a mythic tale is indeed what we get, an ordeal by water, with auguries and signs, battles fought, even a princess won (but also lost, for this is a flawed epic, and the journey ends in emptiness as the river becomes the sea). But woven together with this are encounters with old American myths of frontiers and journeys. And, presciently — so striking, reading again now, nearly forty years later —  Raban notes the “deep, unsatisfied capacity for hero-worship” that makes many Americans long for a “strong” leader, a saviour. This is a many layered book, artful in the artlessness of its transparent prose. Wonderful.


And yes, I did indeed hugely enjoy re-reading Coasting. But Raban’s best book is surely Passage to Juneau (and this is saying something, for he is one of the very finest writers of non-fiction prose in English of recent decades). This is his 1999 book, notionally about sailing his 35ft ketch up the Inside Passage, from his home in Seattle to Juneau in Alaska. But like all great travel books, it is about so very much more — the voyage of Captain Vancouver that he is retracing, about the original Indian inhabitants and their relationship to the sea, about the idea of the sublime, about the death of his father, about his feelings for his young daughter, about other escapees he encounters at the edge of the world. And as ever “journeys hardly ever disclose their true meaning until after — and sometimes years after — they are over”. A masterpiece. Read it.


Jonathan Raban died a few days ago, aged 80. A stroke some years ago kept him confined to a wheelchair, an ironic fate, as the Guardian obituarist notes, for a writer who saw his journeys as “a means of escape, freedom and solitude, I could be happy … in a way I couldn’t be at home”. But he kept writing. And as another Guardian tribute tells us, “Raban’s memoir Father and Son is due to be published this autumn. … He is survived by his daughter, who said that on his last day he saw a bald eagle swooping and playing in the wind from his hospital window, before it flew off over Puget Sound, off Washington state.”

In concert: Chiaroscuro Quartet

Here is a video of the wonderful Chiaroscuro Quartet at a recent concert in Stockholm. They begin with Haydn’s String Quartet Op. 33, No. 4, followed (at 16:30) by Emilie Mayer’s String Quartet No. 1 (first performed in 1858). The concert concludes (from 47:30) with Beethoven’s String Quartet, Op. 59, No. 3.

I leave you to make your own mind up about the Mayer piece. But I’m really linking this video because of the performance of the third Rasumovsky. That quartet has been a quite special favourite of mine ever since I saw Godard’s One Femme Marieé at a very impressionable age — do you remember this scene? I have never found out which quartet’s performance was used in the film — though of the recordings from that era which I know, it is quite similar to that by the Amadeus.

The greatest live performance I’ve ever heard was by the Pavel Haas about three years ago, who were absolutely on fire at Wigmore Hall. I have never heard the transcendental second movement played more affectingly, or the final movement propelled with such passion. Astonishing (as the audience obviously felt at the time). But this performance by the Chiaroscuro, with their distinctive timbre on gut strings, is very fine indeed.

Two pages on slice categories

I am intermittently thinking about categories again, and I think I got something wrong in the current set of notes. I there defined slice categories in a way that doesn’t work, at least given my initial preferred definition of categories (which is the same as Awodey’s or Riehl’s).

OK: What is an arrow in a slice category (C/X) from the object (A, f \colon A \to X) to the object (B, g \colon B \to X) — where, of course,  A, B are C-objects, and f, g are C-arrows?

Since we are constructing (C/X) from data in C the natural thing to do is to use a C-arrow j \colon A \to B which interacts appropriately with f and g, giving us a commuting triangle with f = g \circ j.

But that still leaves two options. The simpler option (1) identifies the needed C/X-arrow with j by itself. A more complex option (2) takes the needed C/X-arrow to be the whole commuting triangle, or if you like, the triple (f, j, g). In the earlier set of notes I went for the simpler (1). But I don’t think this can be right for a reason I explain. And I now note that Leinster initially goes for (2) (though his language then wobbles).

So here are two improved(?)pages on slice categories which I do hope now get things right. But let me know if I have gone off-piste!

Restall & Standefer, Logical Methods

A new introductory logic textbook has just arrived, Greg Restall and Shawn Standefer’s Logical Methods (MIT).

This promises to be an intriguing read. It is announced as “a rigorous but accessible introduction to philosophical logic” — though, perhaps more accurately,  it could be said to be an introduction to some aspects of formal logic that are of particular philosophical interest.

The balance of the book is unusual. The first 113 pages are on propositional logic. There follow 70 pages on (propositional) modal logic — this, no doubt, because of its philosophical interest. Then there are just 44 pages on standard predicate logic, with the book ending with a short coda on quantified modal logic. To be honest, I can’t imagine too many agreeing that this reflects the balance they want in a first logic course.

Proofs are done in Gentzen natural deduction style, and proof-theoretic notions are highlighted early: so we meet e.g. ideas about reduction steps for eliminating detours as early as p. 22, so we hear about normalizing proofs before we get to encounter valuations and truth tables. Another choice that not everyone will want to follow.

However, let’s go with the flow and work with the general approach. Then, on a first browse-and-random-dipping, it does look (as you’d predict) that this is written very attractively, philosophically alert and enviably clear. So I really look forward to reading at least parts of Logical Methods more carefully soon. I’m turning over in my mind ideas for a third edition of IFL and it is always interesting and thought-provoking to see how good authors handle their introductory texts.

In concert: Pavel Kolesnikov plays Schubert D899

Since earlier in the pandemic, one cheering lifeline has been provided by filmed concerts. Here’s a plan. At least until the sunnier days of spring are here, I’m going to post a weekly link to share some performances, ones that you too might find really worth pausing over for a reflective moment one evening. They might be old or new, probably in fact just part of a concert, perhaps just half an hour more or less — and available online at least for the next couple of weeks, so you can find a chance to stop to watch and listen. Let’s see how it goes.

I’m starting with the opening of Pavel Kolesnikov’s Wigmore Hall concert just before Christmas. He began with a deeply felt, mesmeric, performance of Schubert’s D899 set of Impromptus. Kolesnikov astonished in the great G major sonata recently too; I find him a quite wonderful Schubert pianist. Enjoy!

(The impromptus last just over 30 mins: the rest of the concert — Bach, Adès, Schumann — is predictably excellent too!)

Self-publishing and the Big Red Logic Books

One way of increasing the chance of your books actually being read is to make them freely downloadable in some format, while offering inexpensive print-on-demand paperback versions for those who want them. Or at least, that’s a publication model which has worked rather well for me in the last couple of years. Here’s a short report of how things went during 2022, and then just a few general reflections which might (or might not) encourage one or two others to adopt the same model!


As I always say, the absolute download stats are very difficult to interpret, because if you open a PDF in your browser on different days, I assume that this counts as a new download — and I can’t begin to guess the typical number of downloads per individual reader (how many students download-and-save, how many keep revisiting the download page? who knows?). But here is the headline news:

PDF downloadsPaperback sales
Intro Formal Logic112211112
Intro Gödel’s Theorems7432627
Gödel Without Tears4394677
Beginning Mathematical Logic25863493

No doubt, the relative download figures, comparing books and comparing months, are more significant: and these have remained very stable over the year, with about a 10% increase over the previous year.

As for paperback sales of the first three books, these too remain very steady month-by-month, and the figures are very acceptable. So we have proof-of-concept: even if a text is made freely available, enough people prefer to work from a printed text to make it well worthwhile setting up an inexpensively priced paperback. (In addition there’s also a hardback of IFL which sold 150 copies over the year, and a hardback of the first edition of GWT sold 40 copies up to end of October, before being replaced by a new hardback edition.)

The BML Study Guide was newly paperbacked at the beginning of the year, not with any real expectation of significant sales given the rather particular nature of the book. Surprisingly, it is well on course to sell over 500 copies by its first anniversary.

Obviously, an author wants their books to conquer the world — why isn’t just everyone using IFL? —  but actually, I’m pretty content with these statistics.


To repeat what I said when giving an end-of-year report at the beginning of last January, I don’t know what general morals can be drawn from my experiences with these four books. Every book is what it is and not another book, and every author’s situation is what it is.

But providing an open-access PDF plus a very inexpensive but reasonably well produced paperback is obviously a fairly ideal publication model for getting stuff out there. I’d be delighted, and — much more importantly — potential readers will be delighted, if rather more people followed the model.

Yes, to produce a book this way, you need to be able to replicate in-house some of the services provided e.g. by a university press. But volunteer readers — friends, colleagues and students — giving comments and helping you to spot typos will (if there is a reasonable handful of them) probably do at least as good a job as paid publisher’s readers, in my experience. Writers of logic-related books, at any rate, should be familiar enough with LaTeX to be able to do a decent typographical job (various presses make their LaTeX templates freely available — you can start from one of those if you don’t feel like wrangling with the memoir class to design a book from scratch). Setting up Amazon print-on-demand is a doddle. You’ll need somehow to do your own publicity. But none of these should be beyond the wit of most of us!

The major downside of do-it-yourself publishing, of course, is that you don’t get the very significant reputational brownie points that accrue from publication by a good university press. And we can’t get away from it: job-prospects and promotions can turn on such things. So they will matter a great deal in early or mid career.

But for those who are well established and nearer the end of their careers, or for the idle retired among us … well, you might well pause to wonder a moment about the point of publishing a monograph with OUP or CUP (say) for £80, when you could spread the word to very many more readers by self-publishing. It seems even more pointless to publish a student-orientated book of one kind or another at an unaffordable price. So I can only warmly encourage you to explore the do-it-yourself route. (I’m always happy to respond to e-mailed queries about the process.)


Finally, I can somewhat shamefacedly add a last row to the table above, about work in (stuttering) progress towards an announced but as yet far from finished paperback:

PDF downloadsPaperback sales
Beginning Category Theory7482N/A

This download figure is embarrassing because, as I’ve said before, I know full well these notes are in a really rackety state. But I can’t bring myself to abandon them. So my logical New Year’s resolution is to spend the first six weeks of the year getting at least Part I of these notes (about what happens inside categories) into a much better shape. I just need to really settle at last to the task and not allow myself so many distractions. Promises, promises. Watch this space.

Music for the end of the year

Somewhat to my surprise, I have posted here over a hundred times in the last year. But very many of the posts were of (at best!) pretty ephemeral interest — for example, giving links to then current drafts of the Beginning Mathematical Logic Study Guide, to updated chapters for Gödel Without (Too Many) Tears (lots of those), and updated chapters for the stuttering notes on Category Theory (lots of those too). Other posts were logic/maths booknotes, not all exactly friendly. But I wasn’t always mean: there was warmer praise for a number of books, including the following very mixed bag:

I’ve also been intrigued by the opening chapters of

but I’m not at all sure what to make of Tennant’s deviant form of logicism and his handling of logical objects more generally. And as you’ve noticed, I’m still wrestling with and learning from

More, no doubt, about this very substantial book in the new year.


There have also been a dozen and a half posts on particular musical enthusiasms. So, since you may have a little more time over this holiday season, let me repost links to three wonderful filmed performances which are still available to watch. First, the wondrous Pavel Haas Quartet, recently at Wigmore Hall:

As I wrote before, it makes for a rather dramatic stage presence, Veronika Jarůšková with her mass of golden hair and a golden yellow dress catching the stage lights,  the rest of the quartet in the most subdued of subfusc. And there’s a lot of drama in the performances too. But in one respect, the way the quartet play couldn’t be further from what is visually suggested — the equal balance, the closeness of the ensemble, the intense way they listen to each other, is as ever remarkable. So here they are, playing Haydn’s Op. 76 No. 1, Prokofiev’s second String Quartet No. 2, and then Pavel Haas’s String Quartet No. 2 (that’s the one with percussion in the final movement). On this occasion, I thought, the Prokofiev was especially fine: it is difficult to imagine the deeply affecting Adagio being played better.

Next, here is Elisabeth Brauss, also performing at Wigmore Hall to the warmest of receptions:

The recital started Beethoven’s Op. 109 Sonata, which inspired Elisabeth to quite mesmerising playing with heart-stopping moments: transcendental music, and a performance to more than stand comparison with the very best I’ve heard. Sadly, this part of the recital is no longer available online. But in the rest of an engagingly varied programme she offered us some rarely performed Hindemith, Brahms’ late four Klavierstücke, and Schumann’s Faschingsschwank aus Wien, all done with such verve and then wonderful delicacy, as variously called for — just a delight. You can watch here.

Thirdly, let’s revisit the extraordinarily stellar Lea Desandre, filmed by candlelight, from Rouen … escaping our mad world for an hour. Sheer delight again, and just wonderful singing and playing.

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