The Language of Category Theory

I’m taking a week or so off from on working the d****d second edition of my logic text (it’s quite fun, if you like that sort of thing, most of the time: but it is good to take a break). I’m instead updating, just a little, my Gentle Intro to Category Theory, about which more when the revised version is ready for prime time (within the week, I hope). So I’ve now had an opportunity to take a quick look at Steven Roman’s An Introduction to the Language of Category Theory (Birkhäuser, 2017) which in fact has been out a whole year.

This book is advertised as one thing, but is actually something rather different. According to the blurb “This textbook provides an introduction to elementary category theory, with the aim of making what can be a confusing and sometimes overwhelming subject more accessible.” We might, then, expect something rather discursive, with a good amount of the kind of informal motivational classroom chat that is woven into a good lecture course and which can be missing from a conventionally structured textbook. But what we get is actually much closer to a brisk set of lecture notes. For the book travels a long way — through the usual introductory menu of categories, functors, natural transformations, universality, adjunctions (as far as Freyd’s Adjoint Functor Theorem) — and all in just 143 pages before we get to answers to exercises. Moreover, these pages are set rather spaciously, with relatively few lines to the typical page. So certainly there isn’t much room for discursive commentary.

And I would have thought that the sequencing of topics would leave floundering some of those who would appreciate a gentler introduction. So we get to the Yoneda Lemma long before we eventually meet e.g. products (and that as part of a general treatment of limit cones). Yet aren’t products a very nice topic to meet quite early on?  — in talking about them, we  explain why it is rather natural not to care about what product-objects are intrinsically (so to speak) but rather natural  to care instead about how the product gadgetry works in terms of maps to and from products. Here then is a rather nice example to meet early to motivate categorial ways of thinking. But not in this book.

Still, looked at for what it is rather than for what it purports to be — i.e. looked at as a  set of detailed lecture notes which someone could use as back-up reading for perhaps the first half of a hard-core course, to keep things on track by checking/reinforcing definitions and key ideas, with added exercises  (notes which could then later be useful for revision purposes) — Roman’s book does look pretty clearly done  and useful. But if you were wondering what the categorial fuss is about and wanted an introductory book to draw you in, I doubt that this is it.

[Two grumbles. The book is pretty pricey for its length. And why, oh why, in an otherwise nicely produced paperback have the category theory diagrams been drawn in such an ugly way, given the elegant standard LaTeX packages?]

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Georg Kreisel — partial bibliography

Clearing out an old box, I’ve come across a crumpled xerox of a bibliography of some two hundred papers and other pieces by Georg Kreisel, covering up to the early 1990s.

I believe this biblio was passed on to me by Dan Isaacson, though I cannot recall where he got it from. A quick internet search suggests that it isn’t readily available online. But it might well still be of interest to some, so I have scanned it and made it searchable, and here it is.

(Do let me know if there is a more complete biblio anywhere. I’ve always wondered what Kreisel’s reputation would now be had he had the expository facility — or at any rate, the desire to be understood — of e.g. a Putnam or a Feferman.)

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Pavel Haas Quartet play Schubert

Pavel Haas Quartet 2016    Photo: Marco Borggreve

For another couple of weeks you can listen via the BBC website to a characteristically intense performance of the Schubert G major Quartet D887 by the Pavel Haas Quartet, from the Schubertiade last June, recorded at the Angelika Kauffmann Saal, Schwarzenberg. Catch it while you can.

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Symbol for assignment of a truth-value?

Here’s an odd thing. There seems, browsing along my shelves, to be no really standard symbolic metalinguistic shorthand used in elementary books for assigning a truth-value to a wff (say, in the propositional calculus). You would have expected there to be some.

In the first edition of my Introduction to Formal Logic, I borrowed the symbol ‘\Rightarrow
to abbreviate ‘has the value … [on some given valuation]’ and wrote the likes of e.g.

If \mathsf{P} \Rightarrow \textrm{T} and \mathsf{Q} \Rightarrow \textrm{F} then \mathsf{(P \land Q)} \Rightarrow \textrm{F}.

But on reflection this was pretty silly, given that the symbol ‘\Rightarrow‘ is already overloaded (not in my book, but elsewhere — like on math.stackexchange! — where, for a start, some use it for the conditional, some use it in place of a turnstile, and some get in a tangle by using it ambiguously for both!). It seems wiser not to add to possible confusion, especially when readers might well simultaneously get to see the double arrow being used in one of these different ways.

A bit of notation that is used, not at all consistently but often enough, is square double-brackets, so ‘[\![\ldots]\!]‘ is used for ‘the value of …’, and we write the likes of ‘[\![\mathsf{P}]\!] = \textrm{T}‘. But this seems to me a bit cluttered for elementary purposes — I’m after readability, rather than portability to more sophisticated contexts. And it misses the dynamism(??) of some type of arrow.

So for the upcoming second edition, I’m tentatively minded to use the \mapsto symbol for value-assignment, and write instead

If \mathsf{P} \mapsto \textrm{T} and \mathsf{Q} \mapsto \textrm{F} then \mathsf{(P \land Q)} \mapsto \textrm{F}.

(I suppose a colon could be another possibility, but I’d rather have something more distinctive. And the likes of ‘T(\mathsf{P})’ isn’t so pretty/easy to read in bulk and is conventionally part of an augmented object language.)

Any objection to the revised arrow? Am I missing some sufficiently  established (or even just nicer) alternative??

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A Christmas card

Ghirlandaio, The Adoration of the Magi, Spedale degli Innocenti, Firenze

With every good wish for a happy Christmas and a peaceful New Year

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CD of the year 2017

Their recording history has been extraordinary. In 2010, the already much admired Pavel Haas Quartet released their fourth CD, Dvořák’s String Quartets Op. 96, the “American”, and Op. 106. It won the Gramophone Award for chamber music disc of the year, and the overall Award for Recording of the Year. Their next disc was in 2013, astonishing performances of Schubert’s “Death and the Maiden” Quartet, and the Quintet (with Danjulo Ishizaka as the second cellist). Again they won a Gramophone Award for Chamber music disc of the year.

Why the three year gap? In part, because the Quartet changed second violins, being joined by Marek Zwiebel who strikes me as a remarkable occupant of the role — his interaction with Veronika Jarůšková’s first violin seems both technically and musically superb.

The Quartet’s next disc was the 2015 recording of the two Smetana quartets. Again a triumph, again a Gramophone Award for Chamber music disc of the year, and other awards too. But we have had to wait two and a half years for the follow-up. Why another long gap? Sadly, their founder violist, Pavel Nikl felt he had to leave the Quartet in 2016 for reasons of family illness. His replacement is another fine viola player in the same Czech tradition, another pupil of the great Milan Škampa’s, Radim Sedmidubský (for a long time the violist in the Škampa Quartet). We were at a concert shortly after Radim Sedmidubský had joined, and he then seemed understandably a little reticent, not quite at home. But a year later, at another concert, he was just terrific and fully part of the Quartet, and the Pavel Haas were back to, if not surpassing, their very best.

So here is the first recording by the new line up, a generous CD of two Dvořák Quintets, the second Piano Quintet Op. 81, with their friend Boris Giltberg on piano, and the String Quintet Op. 97, with their more-than-friend Pavel Nikl as the second viola. And the warmth and understanding between friends comes through, time and again. “Another Pavel Haas Quartet disc, another triumph,” says the Gramophone magazine. “The playing is breathtakingly good, each performer maintaining their own personality and yet working together to conjure a special magic,” says the Guardian. “Something that always takes my breath away with this quartet is the range and breadth of dynamics and tone colours that they produce, as well as the perfect blend of sound that they make whilst still allowing individual members’ contributions to come to the fore when required. The very opening of the piano quintet is a case in point: with its gentle cello melody supported solely by a rocking piano accompaniment it makes for a beautifully hushed opening, and as played here by cellist Peter Jarůšek it is simply sublime. Take also the first movement of the string quintet, where the players move from digging in with such force that it sounds like their strings are about to snap, to the most delicately tender chords.” — so writes the reviewer on the Presto site. I agree with every word.

The Dvořák pieces are wonderful, and give great joy: the playing couldn’t be bettered. This is indeed the disc you want in your Christmas stocking.

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Bob Hale (1945-2017)

I am very sorry to hear that the philosopher Bob Hale has died. I have always admired his work, and when — back in the day — we used to bump into each other at conferences, he always struck me as an admirable man. Here is an interesting long interview with him from a year ago.

[Added] A memorial notice from the Sheffield philosophy department.

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CD of the year 2017 — runners up!

As I said, those lists of books-of-the-year only make me feel quite hopelessly out of touch, and leave me rather sadly wondering how to possibly find the time to read more. But lists of CD recommendations are much more cheering, and encourage enjoyable experimental listening on Apple Music (other streaming services are available …). So let me share some of my favourite new recordings from 2017. Here are my five runners-up for CD of the year, in no special order.

Glenn Gould recorded his 1955 Goldberg Variations a few months  before his 23rd birthday.  Now Beatrice Rana has recorded the same music when less than a year older. And this is another quite extraordinary performance. Utterly gripping from the beginning. From the Gramophone review: “The variations that in some hands become merely strong and affirmative are beguilingly multi-layered  …. Gentler numbers benefit from Rana’s ability to conjure the most translucent of textures  … In the famous so-called ‘Black Pearl’ (Var 25) she allows Bach’s tortured dissonances to speak for themselves …  the tension finally released by the joyously airborne Var 26. In some hands, these last variations, which build on that sense of joy, can seem rather forced …. Not here, though, where they range from the bucolic to the transcendental. After a Quodlibet that rejoices in its simple good humour, the return of the Aria is as emotionally multifaceted as you would expect – mysterious, quizzical, noble, resigned, hopeful – setting the seal on a life-affirming disc.”

Ivana Gavrić‘s Chopin disc groups some of Chopin’s earlier Mazurkas, seperated by a couple of Preludes, a Nocturne and the Berceuse. This makes for something like a concert programme (you should listen up to the Berceuse — which is quite hauntingly played, her left hand rocking the cradle in a way that somehow catches at the heart — and then take an interval!). Some of the Mazurkas are very familiar, but many were (as good as) new to me. Her unshowy, undeclamatory, playing seems just entirely appropriate to the scale and atmosphere of the pieces,  often tinged with melacholy as they are.  She is across the room from a group of you, friends and family perhaps, rather than performing to a concert hall. And repeated listenings reveal the subtle gestures and changes in tone she uses to shape the dances; these are wonderfully thought-through performances.

Haydn’s inexhaustible humanity can be a comfort and inspiration in these dark times, no? So I have returned again and again to the  Chiaroscuro Quartet‘s wonderful exploration of  the Opus 20 quartets, completed this year. Four friends, occasionally coming together to play concerts and record, perform with delight and bold inventiveness and warm insight. Their use of gut strings makes for wonderful timbres, now earthy, now confiding, now echoing a viol consort. This is extraordinary playing, and not just from Alina Ibragimova who leads the quartet: the sense of ensemble and the interplay of voices puts some full-time quartets to shame. Richly rewards that repeated listening.

And here is Alina Ibragimova again, this time continuing her long-standing partnership with Cédric Tiberghien. They have now recorded four double CDs of Mozart Sonatas —  in fact there are two sets from this year. The early pieces written by the very young Wolfgang are dispatched with affection and bring out the moments of musical magic that are scattered even there. The mature Mozart is played as well as I have ever heard.  As the BBC Music Magazine said of one of the discs, “Tiberghien’s limpid phrasing, radiant cantabile and velvety, cushioned tone provides a continual source of pleasure, complemented ideally by Ibragimova’s silvery-toned exploratory zeal, as she delights in Mozart’s gentle textual interplay, as though discovering its special qualities for the first time.” A constant delight.

The Doric Quartet give us a driven, intense, performance of two of Schubert’s greatest works. From the Gramophone review: “Even in a work as familiar as the Quartettsatz the Doric lend character through elasticity of phrasing, which nicely counterbalances the piece’s inherent energy. … The main event, the G major quartet, is very impressive too, spacious without ever being ponderous. … The quartet build up their own kind of relentlessness, one that becomes more and more potent upon repeated hearings.” Convincing and emotionally gripping playing. (If you like the Pavel Haas’s take-no-prisoners Death and the Maiden, you should like this too.)

Others that almost made it: the very fine Schone Mullerin from Christian Gerhaher and Gerold Huber (but how often can you listen to that?), the second Scarlatti disc from Angela Hewitt, and for lighter relief, ‘The Italian Job’ from Adrian Chandler and La Serenissima.

Finally, what about the much hyped recording of the last two of Schubert’s sonatas from Krystian Zimerman? Try the opening of the first or second movements of D960 — I found the playing to be so affected, the hesitations and rushes forward so unnatural as to be simply unbearable.

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Logic books of the year?

It’s the season when the literary supplements are full of choices of books of the year. And I for one am made to feel I just Haven’t Been Keeping Up. Ah well …

In fact, the only 2017 published novel I seem to have read this year — though with great enjoyment and at the warm recommendation of Mrs Logic Matters — has been John Banville’s Mrs Osmond (his sort-of-sequel to Portrait of a Lady).

What about new logic/philosophy of maths books? It seems to have been a relatively thin year (or again, have I not been keeping up?). I have mentioned over the year a couple of new books by Jan von Plato. First, his introduction to and translation of Gentzen’s shorthand notebooks (which seems a major achievement — and of considerable interest even if the history of logic is not your primary concern). And second, his partial and opinionated (but therefore interesting and instructive) history of theories of deduction and computation. As I said before, the book retains the flavour of a thought-provoking and engaging lecture course, which makes for readability.

The other book I have highlighted here is Neil Tennant’s Core Logic, the result of some forty years of wrestling with entailment, the transitivity of entailment, the avoidance of explosion, and related matters. Some (many?) will think we should just keep things simple, allow that a contradiction entails anything, and not fuss. Neil has much to say about the gains in being fussy.

Just in the last few days I have got two newly published books, Cezary Cieśliński’s The Epistemic Lightness of Truth, and Elaine Landry’s edited collection Categories for the Working Philosopher (both, by the way, outrageously expensive, even after discounts). The latter seems to be a very mixed bag (of the three papers I’ve read, one bad, one no real news at all, and one very helpful). The former, however, looks good. If, like me, you are (a) interested in formal theories of truth, and (b) are inclined to some deflationist/minimalist view about truth according to which ‘It is true that p‘ shouldn’t get you much further than plain ‘p‘, then you will be very interested in  Cieśliński’s project: and the opening chapters are promisingly crisp and clear and accessible (though probably presuppose a bit more from the reader than the author thinks).

So what have I forgotten/overlooked? What are your logic/phil maths picks of 2017?

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When I were a lad …

When I were a lad, a conference on logic/phil maths was rare indeed, and got you a wet weekend in Keele, if you were lucky. Now there are more conferences than you can shake a stick at, and in delightful places too. Just in the last couple of days, these have caught my eye …

Foundations in Mathematics: Modern Views (Munich, April 2018)

Philosophy of mathematics: objects, structures, and logics (Mussomeli, Sicily, May 2018)

Logic Colloquium 2018  (Udine, Italy, July 2018)

My conference-going days have to be over. But I’d have relished any of those!

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