## More Dvorak

If you loved the Pavel Haas Quartet’s recording of the Dvorak Piano Quintet No. 2, Op. 81 and String Quintet op. 97, then you  should also love this new Dvorak recording by the terrific Jerusalem Quartet (with Veronika Hagen and Gary Hoffman). They here play the String Sextet Op 48 and then the String Quintet op. 97 again. The Sextet was (I confess!) new to me, and is quite delightful: the performance is as good as you would expect.

The Quintet is again beautifully played. The Jerusalem’s playing is slightly gentler, slightly more restrained (I suppose) than the Pavel Haas’s: but warmly recommended too.

(You can listen on Apple Music: other streaming services are available …)

## Dry January

For my Dry January, I tried to quit reading stuff about Brexit (after all, surely nothing much was going to happen for a month).

Well, I miserably failed to quit outright. But I did cut down a bit, and I did get more Dickens read. But hell’s bloody bells, what an appalling mess on so many levels.

(That’s what you come here for, no? Incisive political analysis like this!)

## Category Theory: A Gentle Introduction

At long last, I have updated my  notes Category Theory: A Gentle Introduction (now some x + 291 pages).

A good while ago, I received lists of corrections from a number of people, and just recently I’ve had another tranche of corrections, making over a hundred in all. Mostly these corrections noted typos. But there were also enough mislabelled diagrams, fumbled notation mid-proof,  etc., to have no doubt caused some head-scratching. So I can only apologize for the delay in making the corrections.

I have also added a new early chapter and restored a couple of sections that were in a rather earlier version but got lost in the last one (thereby breaking some cross-references and no doubt producing more head-scratching).

These notes were originally written for my own satisfaction, trying to get some basics clear. But I know some people have found them useful (despite their very obvious shortcomings, unevenness,  and half-finished character). So I hope some others will find the update helpful: you can download it from the category theory page here.

## 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, look at this for what it is rather than for what it purports to be. In other words,  look at this 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). Then Roman’s book does seem to be  pretty clearly done  and likely to be useful for some students. 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 available elegant standard LaTeX packages?]

## 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.)

## 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.

## 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??

Posted in This and that | 5 Comments

## 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

## 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.