Brilliant indeed

51OEUHk5YiLI was going to post about the delights of Amsterdam as a place to visit for a week — the cityscapes, the cafes, the restaurants, the museums large and small, the whole urban experience, all even better than we had hoped. But more or less as soon as we got back, I was felled by a nasty attack of a recurrent problem, about which all I will say is thank heavens for penicillin. Though industrial quantities of antibiotics do leave you feeling still pretty flattened, so it has been a few days of staggering from bed to sofa and back. But as I begin to feel more human I’ve had plenty of time to finish the book I’d just started before going away. Like Amsterdam, this too has been lauded to the skies by those who know it. It has been a delight, in both cases, to find that other people’s really warm recommendations are more than deserved (it doesn’t always happen!). And since you certainly don’t need me to tell you more about Amsterdam, but you might not have heard of Elena Ferrante’s My Brilliant Friend — I hadn’t until a couple of months ago, from the much better read Mrs Logic Matters — maybe I’ll just sing its praises instead.

It really is absolutely wonderful. But I’m not going even to try, in my limping way, to say why. Rather let me point you to this New York Review of Books review of Ferrante’s oeuvre by Rachel Donadio, and/or this review from the New Yorker by James Wood. If these don’t get you reading, nothing will!

Posted in Books, Italian matters | 3 Comments

Notes on Category Theory v.5

Here is an updated version of my on-going Notes on Category Theory, now 130 pp. long. I have done an amount of revision/clarification of earlier chapters, and added two new chapters — inserting a new Ch. 7 on categories of categories and issues of size (which much expands and improves some briefer remarks in earlier versions), and adding at the end Ch. 15 saying something about how functors can interact with limits. There’s quite a bit more that could be said in this last chapter, and I’ll have to decide in due course whether to expand the chapter, or return to the additional topics later, or indeed to only mention some of those topics in the end (I’m trying to keep things at a modestly introductory level). But for the moment I’ll leave things like this and move on to a block of chapters on adjoints and adjunctions. So here’s where we’ve got to:

  1. Categories defined
  2. Duality, kinds of arrows (epics, monics, isomorphisms …)
  3. Functors
  4. More about functors and categories
  5. Natural transformations (with rather more than usual on the motivation)
  6. Equivalence of categories (again with a section on motivation, why we want ‘equivalence’ rather than full isomorphism)
  7. [New] Categories of categories: issues of size
  8. The Yoneda embedding (shown to indeed be an embedding by using an easy restricted version of the Yoneda Lemma)
  9. An aside on Cayley’s Theorem
  10. The Yoneda Lemma (how to get to the full-dress version by two conceptually easy steps from the restricted version).
  11. Representables (definitions, examples, universal elements, the category of elements).
  12. First examples of limits (terminal objects, products, equalizers and their duals)
  13. Limits and colimits defined (cones, limit cones: pullbacks etc.)
  14. The existence of limits (in particular, having finite products and equalizers implies having all finite limits).
  15. [New] Functors and limits (preserving, reflecting, creating limits: hom-functors preserve limits, etc.)

Don’t hold your breath for the chapters on adjoints, though. After a very busy time for various reasons, I’ve a couple of family holidays coming up!

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Books published in the first quarter of 2015?

What books on logic (mathematical or philosophical logic) and/or philosophy of maths have been published in the first quarter of 2015? There’s John Burgess’s Rigor and Structure which I have started blogging about here. But what else has appeared so far this year?

Not that I’m lacking things to read! But I’d like to know what I’m missing, and I’m probably not alone. So maybe you would like to share any recommendations of recent titles which might be of interest to readers of Logic Matters?

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Mathematical depth

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.

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The Chiaroscuro Quartet: Mozart and Mendelssohn

ap092-cover-20140508-1=360x360This 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.

You can listen to excerpts on the Quartet’s website here. Very warmly recommended.

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Blue Monica

monica-vitti-152338
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.

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A little problem in high school geometry

7FLJl 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!

Continue reading

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Category theory: status report

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.

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893 posts later …

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.

Onwards!

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David Fray plays Schubert: Fantaisie

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.

71E82Cp9UWL._SL1500_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, for whatever reason, my favourite Schubert piano music: but again surely it could not be played better than it is here.

A wonderful disc then, most warmly recommended.  (And there is, by the way, a video about this CD on David Fray’s website.)

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