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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Burgess, Rigor and Structure — 2

9780198722229_450In his Preface, Burgess says that in putting together his book it became clear that he

would need to explain not only what rigour is, but also how set theory came to occupy the position … of being in some sense the accepted foundation or starting point for rigorously building up the rest of mathematics. Such … explanations are given in Chapter 2.

However, the chapter begins with additional observations on the development of modern mathematics which could have as easily been in the first chapter. In particular, Burgess touches on the way that a familiar space or number system may get expanded by the addition of ideal elements (e.g. points at infinity, complex numbers). Since we can’t carry over habits of thought acquired when working with the old familiar systems to the augmented systems, we’ll have to pay careful attention to what does and doesn’t follow from our assumptions about the new ideal elements — another driver towards increased rigour.

But, as Burgess goes on to note, other developments involve learning more about the familiar not by augmenting our original system with ideal elements but rather by making connections between seemingly very different bits of mathematics  — e.g. as in Galois’s theory connecting solutions of polynomials (an old topic) which a theory of permutations (part of the theory of groups).

And it is here, in the fact of the interconnectness of different branches of mathematics, that Burgess locates “crucial implications for the project of rigorization”. Making the connections again requires us to get more precise about the systems we are connecting. But local precision isn’t enough. More  tellingly (for present purposes)

To guarantee that rigor is not compromised in the presence of transferring material from one branch of mathematics to another, it is essential that the starting points of the branches being connected should at least be compatible.

Burgess gives a nice example of what he means: there’s evidently a problem about using analytic methods in Euclidean geometry if your geometry assumes the Archimidean axiom (given any two lengths, there is an integer n such that times the shorter length exceeds the greater length) yet your analysis essentially depends on infinitesimals (which don’t obey the  Archimidean principle).

But what does this show? Burgess writes:

The only obvious way to ensure compatibility of the starting points of different branches is ultimately to derive all branches from a common, unified starting point. The material unity of mathematics, constituted by the interaction of its various branches at their higher levels, virtually imposes a requirement of formal unity, of development within the framework of a common list of primitives and postulates, if the rigorization project is to be carried to completion.

But what exactly is the claim here? Suppose we do adopt Burgess’s favoured standard set theoretic framework. Within that framework, we can develop various forms of geometry, Archimidean and non-Archimidean, and various forms of analysis with and without infinitesimals. All can be “derived from a common, unified starting point”: but that’s plainly not enough to warrant transporting results from a version of analysis to a version of geometry, to stick for a moment to Burgess’s prime example. We need to make the right compatible pairings. But how is compatibility established? In this case, perhaps, by interpreting the geometrical theory into analysis. Now, interpreting both in a background set theory may be an aid to this: but to repeat, it can’t be enough just to interpret both ‘vertically downwards’ into set theory — the interpretations must make the right interpretative links ‘horizontally’ between a particular axiomatized geometry and the axiomatized analysis. And it is the horizontal links that matter if our theories are to be compatible: yet surely such interpretative links can in this case, and in some other cases too, be made without dog-legging through a full-blown set theory in which all branches of mathematics can be derived. So rather more needs to be said if we are to justify the thought that the only way of ensuring compatibility of different areas of mathematics is development within an over-arching  theory.

What about the example of elementary Galois theory — another of Burgess’s main examples — where we attack a problem about polynomial equations with some elementary group theory? Here it isn’t like a situation where we have two theories — as was the case with a synthetic and an analytic geometry — purporting to be about the ‘same’ things, lines, conic sections and so on, and hence a situation where straightforward issue of consistency can arise. Rather we have some abstract algebraic apparatus about permutations (any permutations of any finitely many objects) which makes merely conditional claims — if anything satisfies such-and-such, it is so-and-so. And then we find we can apply this conditional apparatus, initially surprisingly, to a case where it wasn’t at all immediately obvious that that there are significant permutations to get a grip on, investigation of which will deliver key results about the polynomial equations. If that, at an arm-waving level, is what is going on, then there is a question of the useful applicability of group theory in a certain domain. But it isn’t immediately clear what kind of issue of compatibility arises here in any sense which would press us to look for an all-encompassing unified starting point for all mathematics.

So Burgess surely has, to say the least, more work to do to make out the claim that the very project of rigorization requires a formal unification of mathematics.

But we have a lot more of Ch. 2 to come: to be continued …

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Burgess, Rigor and Structure — 1

9780198722229_450Modern pure mathematics is characterized by the rigour of its methods, and by its special subject matter, i.e. abstract structures. Or so the story goes. But exactly what is meant by rigour here? What, exactly, is meant by saying that modern mathematics is about structures? And what is the relationship between the drive to rigour and the drive to some sort of abstract structuralism (if that’s what it is)?

These are the topics of John Burgess’s new book Rigor and Structure (OUP, 2105). Very big topics for a relatively short, four chapter, book. Though Burgess in fact says that he originally intended to write even more briskly, planning just a long paper to make and defend his key novel claim, about which more in due course. But in the writing he found that for some audiences he needed to say something about how we got to where we are. So the defence of his main claim doesn’t come until Ch. 3, with a corollary then drawn  out in Ch. 4. First, though, we get Ch. 1 about how the ideal of rigour developed and how it is realised in present-day mathematics, and then Ch. 2 explores how set theory came to occupy the special position of being “in some sense the accepted foundation or starting point for rigorously building up the rest of mathematics”.

Burgess’s topics couldn’t be more central to the philosophy of mathematics, and he writes with an engaging clarity and directness. The book is intended to be, and mostly should be, accessible to philosophical readers without too much detailed background in mathematics and equally accessible to mathematical readers without too much background in philosophy.  But how far can we get, how deep can we go, in such a short book written for a dual audience and assuming so little background? Well,  let’s dive in and see! I plan to comment here as I read through.

If we want to understand what characterizes rigorous mathematics, Burgess suggests, it would be good to have a foil, significant examples of not-so-rigorous-mathematics: but  we will not find many of those looking round the contemporary scene in pure mathematics. So if we want throw modern standards of rigour into relief, it is natural to look to the history of mathematics to provide some contrasts. Burgess’s first chapter ‘Rigor and Rigorization’ aims to sketch in just enough history to do the job.

Now, I imagine that most readers of this blog won’t in fact need this element of scene-setting and so will be able to skim or skip through this chapter at considerable speed. There are, rather predictably, two main themes we can pick out from Burgess’s story: (a) the development of analysis, (b) the development of geometry, Euclidean and non-Euclidean.

(a) On analysis, Burgess says something about the roots of the calculus in the manipulation of infinitesimals and also about the early reliance  on “geometric intuition” (in something like a Kantian sense — not mere hunches, but e.g. the “perception” that if this curve tracing a function goes from a negative value to a positive value then it must cross the axis x = 0 and so for some value the function takes a zero value). He also describes what he calls the use of “generic reasoning” (extending the application of reasoning patterns known to have exceptions, without any clear story about what makes for favourable ‘safe’ cases) — though some more examples would have been good to have.

Now, in using infinitesimals, naively construed, we have to delicately alternate between treating them as strictly non-zero and allowing them to vanish: how are to make sense of this? We need better conceptual analysis of what is going on in taking differentials and integrating here (and the account that won out in fact eliminated the need to take infinitesimals seriously). Again, geometric intuition gets it wrong about e.g. the relation between continuity and differentiability (once we see that we can define an everywhere continuous nowhere differentiable curve): so how can we avoid relying on fallible intuition? By better analysis of concepts again and by closer analysis of reasoning. Failures of generic reasoning also force rigorization in the guise of exploring just what underpins various proof methods, again better analysis of modes of reasoning so we can understand their proper domain of application etc.

(b) As for Euclidean/non-Euclidean geometry, there’s a familiar story to be told about what happens in Euclid, its successes and failures in rigour. And then the development of non-Euclidean geometry (especially in the process of exploring what can and can’t be proved from Euclidean geometry minus the parallels axiom) forces what Burgess calls a “division of labor” between mathematical geometries and physical theorizing about which geometry is best suited to be recruited to describe the world. But then, if geometric/physical intuition can no longer play its role inside pure mathematical geometries, how else can we  conceive their exploration other than (in modern vein) the deductive elaboration, ideally with gap-free proofs, of various suites of axioms (where being an axiom now means only being a starting point, not being a basic truth about the world).

So familiar  themes emerge. The growing importance of sharp conceptual definitions, explicitness about what axioms and principles of reasoning we are working with (precision being essential, truth-to-the-world not), the emerging neo-Euclidean ideal of gap-free proofs from those explicitly acknowledged starting points (with no smuggled-in extras or reliance on “intuition”)  — these will be very familiar tropes to those philosophy or maths students who have gleaned just a very little history of mathematics.

Now, I’m not complaining in reporting that Burgess is here going over familiar ground. Those students who don’t already know the familiar stories could well find the opening fifty-page chapter useful, as it is all pretty lucidly done. Still, many other readers — surely including most readers of this blog — will be able to skip. So let’s move on …

To be continued.

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Blue Monica

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.


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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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Category theory again

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!

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The Pavel Haas Quartet at the Wigmore Hall


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).

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