In our Mind review of Penelope Maddy’s Defending the Axioms, Luca Incurvati and I were rather skeptical about whether she could really rely on the notion of mathematical depth to do as much work as she wants it to do in that book. But we did add “We agree that there is depth to the phenomenon of mathematical depth: all credit to Maddy for inviting philosophers of mathematics to think hard about its role in mathematical practice.”
Since then, there has been a workshop on mathematical depth at UC Irvine co-organized by Maddy, and now versions of the papers there have been made available as a virtual issue of Philosophia Mathematica which will remain freely available until November this year. Looks interesting.
This is currently my favourite late-evening listening among recent releases — it’s the third CD by the Chiaroscuro Quartet. Each CD couples one of the Mozart Haydn quartets with another work: this time it is Mozart’s Qt 15, K. 421 with Mendelssohn’s Qt 2, op. 13. The performances are extraordinarily fine.
The Chiaroscuro are friends with other musical careers, who come together for the pleasure of playing together — and oh, how it shows! There’s a sense of listening in to private music making of exploratory intensity. The leader is Alina Ibragimova whose solo work is stella beyond words, but here Ibragimova in no way overshadows Pablo Hernan Benedi, Emilie Hörnlund and Claire Thirion: the togetherness, the shared style and understanding, is astonishing indeed.
If you haven’t heard their previous CDs then initially their sound is a shock: they are playing on gut strings, almost without vibrato. So the timbre is spare, the period sound unadorned: it can take a couple of hearings to get used to it. And if — like me — you already know the Mozart well and the Mendelssohn hardly at all, then another surprise is how the Chiaroscuro bring the works much closer in their worlds than you have previously heard them. The Mozart is more troubled, the 18-year-old Mendelssohn more austere: but this makes for a revelatory and satisfying programme.
Monica Vitti — for those who still end up at Logic Matters because of that little flurry of pictures of that icon of the Italian films of my youth which I posted a few years ago.
Let E be the midpoint of the side AD of a square ABCD.
Problem: Determine which has the greater perimeter, the square ABCD or the circle through E, B, C? (You can assume you know the value of π. Otherwise try to use elementary methods.)
The neat solution is, I think, rather satisfying, even though it doesn’t require any special ingenuity to find it. Metaproblem: Why is this solution aesthetically pleasing?
The answer — to the problem, if not the metaproblem — is below the fold: but try blowing the dust of your school geometry before looking at the cheat sheet!
It is an irksome habit, and I wish I could break it and plough on regardless. But I always find when writing anything lengthy that there comes a point by which I’ve accumulated enough little niggling worries about things I’ve already said that I just have to go back to page one and work through again from the beginning, ironing things out. So that’s what I’m doing at the moment with my category theory notes. The exercise is — surprise, surprise! — taking longer than expected, but I hope has made for some interesting expository changes, about which more in due course, when the revised/expanded notes are ready for prime time.
Meanwhile, here’s an update/expansion of the page on Category Theory giving links to (broadly speaking) student-orientated materials available online.
The Logic Matters blog started, back in the olden days, on March 9, 2006. It’s our ninth birthday. Will there be cake?
So here we are, at post number 894. The two stats packages which monitor Logic Matters give very disparate results, but a thousand or two people come to the site each day, and if there is a mention on reddit, for example, that number can jump well into the tens of thousands. And the stats agree that in the last five years, visits to the site in general and the blog in particular have gone up four-fold. So I seem to get to corrupt the minds of the youth spread the logical word to many more here than I ever did in my teaching career. Which makes it fun and all worth the effort.
I buy too many CDs and we go to a fair number of concerts, and so I usually blog only about some of the ‘five star’ discs or concerts which bowl me over. Which does mean that when I do offer reviews, they tend to be consistently full of superlatives. It is certainly not that I’m an uncritical listener: very far from it. Still, I don’t want to be carping or tediously negative here (I’ll keep that for the philosophy!). I prefer to write about music, when I do, from heartfelt enthusiasm. And this time, the enthusiasm is for David Fray’s latest CD.
One of the very finest Schubert recordings of the last ten years, it is widely agreed, is Fray’s CD of the Op 92 Impromptus and the Moments Musicaux. He plays those pieces with luminous artistry and acute sensitivity — taking some notably slow tempi yet never seeming mannered or other than fully immersed in the complexities and ambiguities of Schubert’s music. There is, I have remarked elsewhere, something Richter-like in Fray’s intensity, and in his wonderful ability to impose his vision of the music.
Fray has now returned to Schubert, and indeed firstly to a piece indelibly associated with a quite extraordinary recording by Richter — the G major sonata D894. And the comparison in some ways is still very apt. For Fray too takes the first movement unusually slowly. Where, for example, Brendel in his later digital recording takes 17′.16″ and Paul Lewis 17′.28″ — both very fine performances — Fray takes 19′.06″. This still falls far short of Richter’s astonishing, bordering-on-the-perverse, 26′.51″ in the (in)famous 1979 recording. Yet here is the magical thing: from Fray’s way with his very slight holdings-back, the slightest hesitations, to his control of the architecture of the movement, everything gives his performance a seeming scale closer to Richter’s. (He takes little more than half a minute more than Mitsuko Uchida’s 18′.29″ and yet Fray’s first movement at crucial moments seems markedly more spacious.)
Another magical thing is the wonderfully nuanced clarity of Fray’s playing here — effortlessly cantabile passage work, forte passages which never become brashly declamatory, unending attention to detail with nothing exaggerated or out of place.
Now, there can be a problem — can’t there? — with performances of some of Schubert’s major works: how to make a satisfying whole of a piece that starts with one or two immense movements — immense both in scale and emotional weight. (One of the many things that I particularly admire about the Pavel Haas Quartet’s Schubert CD which I reviewed here is the extraordinarily balance they achieve across the four movements of Death and the Maiden and again of the String Quintet.) How does the rest of David Fray’s performance of the ‘Fantasie’ sonata hold up against this test?
Extraordinarily well, I would say. Partly this is because, although the first movement is played very expansively, it is never becomes heavy. And partly because of the compelling readings he gives of the other movements. I was rather surprised, when I checked, that Fray’s timings in the last three movements are all rather quicker than Brendel, Lewis and Uchida. Yet he plays with such grace and attention to texture and detail that Schubert’s music is given all the space it needs, and there are again quite magical touches. The Trio in the third movement catches with your breath. The final Allegretto dances through its episodes with a wonderful lightness of touch in building to its conclusion.
In short, I would say that of the dozen or more great performances of the G major sonata that I have on disc, this is at least as fine as any: it is worth getting Fray’s new CD for this alone.
But there is much more. Fray follows with a lovely performance of the haunting Hungarian Melody D817. And then for the last two major pieces on the CD he is joined by his one-time teacher Jacques Rouvier. First, they play the Fantasia in F minor D940. This, the incurable romantics among us will remember, was dedicated by Schubert to his young pupil the Countess Karoline Esterházy, often thought to have been the object of his hopeless love: and the piece certainly calls for yearning and passion. I have long loved the old recording from 1978 by Imogen Cooper and Anne Queffélec. But Fray and Rouvier are perhaps even better. Certainly, the yearning in the first dotted theme (with its Hungarian echoes) is as intense; the passion as the music then goes through its evolving moods — stormy, a burst of sunshine, clouds regathering interspersed with more moments of fleeting happiness — is as heartful; the moment when the first yearning theme returns is as affecting; the build-up through the fugato passage to the very final appearance of the initial theme as the music comes to its abrupt recognition that the yearning is indeed hopeless, all this is wonderfully well done.
The CD concludes with the ‘Lebensstürme’ Allegro D947. This is not, for whatever reason, my favourite Schubert piano music: but again surely it could not be played better than it is here.
The category theory page here has been much expanded with links to (i) some online lecture notes, and (ii) some books which are freely (and legitimately!) available online in one form or another.
I am not at all aiming to include everything that is available out there: but on the other hand, if I have missed something good, do please let me know!
To the Wigmore Hall again last week to see hear the Pavel Haas Quartet playing Czech music old and new. The most engaging concert, we agreed, that we’d been to for months. Superlatives are in order!
To begin, Dvorak’s four Miniatures Op. 75a for two violins and viola (not so much a trio as pieces for violin accompanied by the other two strings). These were played wonderfully by Veronika Jaruskova with enough lyrical beauty in the final movement to make something really rather special out of these charming miniatures. Then into the first Janacek String Quartet, ‘Kreutzer Sonata’, with this often jaggedly urgent and restless music played with the PHQ’s characteristic controlled passion and commitment. The quartet’s ensemble, their unity in tone, is quite remarkable.
The PHQ were then joined by Colin Currie for the UK première of a piece commissioned for them, Jiri Gemrot’s Quintet for two violins, viola, cello and marimba. The sound world of the marimba sometimes interweaving with the first violin, sometimes set against strong chordal writing for the quartet, was captivating and often very exciting. The atomspheric slow movement in particular produced quite some magical sounds: “the first violin sings with an eerie, disembodied voice to a hypnotic accompaniment from the marimba”, as the FT reviewer puts it. We weren’t entirely convinced by the piece, although it somehow sounded so very Czech — occasionally the writing for the strings seemed too blocky, when more imaginative textures could perhaps have been developed. But the playing was just superb, and we’d certainly return to hear the Quintet performed again.
After the interval, the Pavel Haas played Pavel Haas, the second quartet ‘From the Monkey Mountains’, the one which features percussion in the last movement. This was, as we would hope and expect, a matchless performance of a very fine piece (a few years ago we might have said ‘neglected masterpiece’, so we owe PHQ much for championing Pavel Haas’s quartets). If you don’t know it, get their recording!
A five star concert, then. And the audience very rightly gave the PHQ the warmest, cheering, applause at the end. I hope they felt properly appreciated. For we only half filled the Hall, albeit in the middle of the week on a chill night. Which is a bit depressing when it is one of the world’s very greatest quartets, surely approaching the height of their considerable powers, and offering a programme which was engaging rather than challenging.
I just regret that we couldn’t go back to hear them again in another Bohemian programme on Saturday, though we’ve already pencilled in their next two concerts at the Wigmore: for the moment, we’ll just have to look forward to the PHQ’s Smetena CD (which is due out in late April).
I wasn’t able to do much work in January for family reasons, but levels of concentration and energy are returning. So, much later than I’d hoped, here at last is an updated version of the Notes on Category Theory (still very partial though now 109 pp.). There are three newly added chapters, and some minor tinkering earlier. We now cover:
Categories defined
Duality, kinds of arrows (epics, monics, isomorphisms …)
Functors
More about functors and categories (and the category of categories!)
Natural transformations (with rather more than usual on the motivation)
Equivalence of categories (again with a section on motivation, why we want ‘equivalence’ rather than full isomorphism)
The Yoneda embedding (shown to indeed be an embedding by using an easy restricted version of the Yoneda Lemma)
An aside on Cayley’s Theorem
The Yoneda Lemma (how to get to the full-dress version by two conceptually easy steps from the restricted version).
Representables (definitions, examples, universal elements, the category of elements).
First examples of limits (terminal objects, products, equalizers and their duals)
Limits and colimits defined (cones, limit cones: pullbacks and upshots)
The existence of limits (in particular, having finite products and equalizers implies having all finite limits).
Coming soon: a further chapter on limits (on functors and the preservation of limits) and then a chapter on Galois connections as a nice gentle lead-in to the chapters on adjoints.
