New package for setting proof tableaux

Suppose you want to typeset (using LaTeX of course) a proof tree that looks something like this: Screengrab copy

Maybe that’s an answer to some exercise for students, which is why you want a tree fully annotated with line numbers, justifications on the right, and justifications too for closing off branches. Previously, getting in all that detail using available LaTeX resources would have been tricky. Now help is at hand. There is a very nice new package prooftrees.sty which will enable you to generate that output easily. And, of course, generate simpler results too, without line numbers and/or justifications, if that’s what you want. The package seems very flexible and to provide, at last, the kind of purpose-built-for-logicians tool that we have needed for a long time. So all credit to Clea Rees for her work on this.

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Beethoven for a Later Age

IMG_1135On a recent visit to our favourite bookshop, Topping’s in Ely, I picked up a copy of Edward Dusinberre’s Beethoven for a Later Age: The Journey of a String Quartet. (In fact, if you click on the photo, you’ll see my copy on a table in an upstairs window of Topping’s, with the stunning view of the cathedral beyond.)

Dusinberre is the leader of the Takacs Quartet, and his book weaves together a little of the history of the circumstances of the composition of Beethoven’s quartets, some reflections on the quartets and the problems they pose those who play them, and thoughts and anecdotes about the life of a string quartet (and  of the Takacs in particular). I found it an absorbing and illuminating read — and at times, very touching in a quietly understated way. Warmly recommended.

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Penelope Maddy on Set-Theoretic Foundations

Penelope Maddy has uploaded a recent paper, Set-Theoretic Foundations, to her page on academia.edu (I needed to download the paper to get a readable version).

The paper shows Maddy’s usual admirable virtues of great clarity. In the first main section, she says something about the various senses in which set theory might be said to “provide a foundation for classical mathematics” (and in which it can’t be said to do so). The second section considers whether, and in what senses, category theory can be said to provide an alternative/better foundation. The third main section — and this makes the paper apt for a volume of essays celebrating Hugh Woodin, for it is on a topic he has views on! — is about the idea that we should think in terms of a multiverse of set universes, rather than a single universe in which (in some sense) all mathematics can live. Her ambitions are modest, however. Maddy writes, sensibly enough,

I won’t pretend to sort out all these complex and contentious matters, but I do hope to compile a few relevant observations that might help bring illumination somewhat closer to hand.

And, yes, I did find a number of Maddy’s remarks clarifying and illuminating. So I’ll spread the word about her paper here.

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CD choice #4

71BEu88cowL._SL1300_Continuing my occasional series of CD recommendations, this is not at all my usual fare. But I happened to hear a slightly jazz-influenced arrangement of Purcell’s “A prince of glorious race descended” on internet radio (Otto’s Baroque Music, since you ask), and was really rather taken with it. So I sought out the disc it came from and here it is: Music for  While: Improvisations on Purcell, by L’Arpeggiata directed by Christina Pluhar. The disc has had mixed reviews: I suspect, then, that it won’t be everyone’s jar of marmite (and there is an oddly misplaced bonus track of a Leonard Cohen piece, sung as if by Margo Timmins!). But I do think Pluhar’s arrangements of Purcell mostly work very enjoyably, and some of the singing is really rather fine. If you have an Apple Music subscription or the like, well worth a listen. (There’s a short  promotional video here.)

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Hilary Putnam, 1926–2016

Such was Hilary Putnam’s very long winding path, his changes of direction, his jumpings across to traverse new fields, that different readers will surely be gripped by different stages of the journey.

For me, the really golden period was from about 1960 to 1975. It is difficult now to overestimate the impact of Mathematics, Matter and Method, and Mind, Language and Reality, the first two volumes of his Collected Papers, which CUP brought out in 1975. Of course, we knew some of those 41 papers, originally published in very widely scattered places, if published at all. But bringing them together revealed Putnam to be an extraordinarily fertile, imaginative philosopher, unpicking the legacies of verificationism and behaviourism in defence of sane realisms about science and the mind. It helped too that he wrote with such stylish clarity, carrying an enviable amount of background technical knowledge lightly. I thought at the time that those papers exemplify analytical philosophy at its best. And I’d still warmly recommend any student beginning the serious study of philosophy to read the Putnam of those years. Such a very fine philosopher.

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One future for journal publishing, revisited

Back in September, I noted that Tim Gowers announced a new journal, Discrete Analysis. As I said then, the content of the journal probably won’t be of much interest to most readers of Logic Matters. But the form of the journal is fascinating. It is to be an arXiv overlay journal. In the briefest headline terms, what this means, in Tim Gowers’s words, is that

rather than publishing, or even electronically hosting, papers, it will consist of a list of links to arXiv preprints. Other than that, the journal will be entirely conventional: authors will submit links to arXiv preprints, and then the editors of the journal will find referees, using their quick opinions and more detailed reports in the usual way in order to decide which papers will be accepted. … [So] The articles will be peer-reviewed in the traditional way.

The first issue is now online. And rather striking it looks too. There is more about the launch issue and the project in Tim Gowers’s latest blogpost. I wonder if any logicians out there will be spurred on to think about starting such an open-access journal? It seems a terrific idea.

By the way, for those of you with some little maths, if you click on the banner for the lead article, you get a short editorial intro. to Terence Tao’s paper, and — this is nice touch! — a link to a video of a talk that Tao gave about his proof, which (even if you don’t really follow the later parts of it) gives you a sense of what’s going on.

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Book Note: A little light algebra

Aluffi algebra Suppose, e.g. as a philosopher of mathematics, you want to get to know more mathematics (after all, it is always a jolly good idea for a philosopher of X to know more than a mere smidgin about X). One area which you have seen to be absolutely central to the modern mathematical curriculum is abstract algebra. Fine: but what to read if — quite a big ‘if”! — you do decide to get to know more about this?

It depends what base you are starting from, of course. At a really introductory level (first year undergraduate, perhaps), one really nice option which I can warmly recommend is Alan Beardon’s relatively short Algebra and Geometry (CUP, 2005), which is very well put together and indeed not-too-abstract. But perhaps this doesn’t take you far enough to get a more rounded sense of modern algebra.

So suppose you want more than you’ll get from Beardon; or perhaps you already have some — possibly fragmentary, possibly half-remembered — knowledge of algebra, and want to go up a level in sophistication and detail. What then? One option is Saunders Mac Lane and Garrett Birkhoff’s Algebra (3rd edition, AMS Chelsea, 1999). This is a distant and slightly more advanced descendant of their famous A Survey of Modern Algebra (which originally dates back to 1941), and is notable for the way it weaves into the story categorial ideas, with a fairly light hand and in an illuminating way. But the treatment is quite brisk, and I think there is now a better option, taking quite a similar approach, but in a rather more engaging way. This is Paolo Aluffi’s Algebra: Chapter 0 (American Mathematical Society, 2009). Its presentation of even quite complex ideas is typically exemplary, both in giving motivation, and in explaining official definitions, presenting the proofs etc, and all often written with a light touch. So this is admirable. and I think is particularly suitable for self-study.

The chapter titles indicate the coverage. I, Preliminaries: sets and categories (so yes there is early introduction to categorial ideas, but again done with a light touch, emphasising the notion of ‘universal properties’). II, Groups, first encounter (the basics done very clearly). III, Rings and Modules (getting as far as a first look at complexes and homology and exact sequences). IV, Groups, second encounter. V, Irreducibility and factorization in integral domains. VI, Linear algebra. VII, Fields. VIII, Linear algebra, reprise. IX, Homological algebra. So things eventually get pretty serious, and you can bail out well before the end but still with a good sense of the topics and approach of modern algebra.

These days, alternative standard recommendations for tackling algebra at this sort of level include e.g. Serge Lang’s Algebra (3rd revised edition, Springer, 2002), and David Dummit and Richard Foote’s Abstract Algebra (3rd edition, Wiley, 2004). But these are quite a bit longer than even Aluffi’s weighty volume, and really cover unnecessarily much (for our purposes). And, more to the current point, I would say neither is a particularly attractive read.

So headline summary: if you want to learn some algebra, dive into Aluffi’s Algebra: Chapter 0.

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And now we are ten …

Francesco Guardi: Forte S. Andrea Del Lido, Venice. Fitzwilliam Museum, Cambridge

Francesco Guardi: Forte S. Andrea Del Lido, Venice.
Fitzwilliam Museum, Cambridge

This blog started ten years ago today. We’ve survived. So let’s have a birthday card, a favourite picture, from down the road in the Fitzwilliam.

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Encore #20: Keeping on keeping on …

I am getting ever-more buried in the expositional details of re-writing/expanding my Gentle Introduction to category theory. Which is fun, if you like that kind of thing, and I’m learning a great deal from the exercise:  but it is not really the sort of project liable to prompt snappy blog contributions. Techie details won’t be of interest to many; and more philosophical ruminations are difficult to keep focused.  So I suspect that contentful logical posts may become (even more) sporadic. You’ll just have to make do with the non-logical bloggings! — like this last pre-birthday encore.  This is a repost from only a few months ago, and the thoughts are still very much in my mind.

Until the lights go out (Aug 18, 2015)

“If you don’t know the exact moment when the lights will go out, you might as well read until they do.” I think I will adopt that as my motto. It’s from Clive James, in the Introduction to his Latest Readings which comes out next week, a little book that reflects on a delightfully idiosyncratic range of authors and books. James is still reading, and writing too with humanity and wit and insight and, as we say, with undying enthusiasm. Except our author is dying. Though he confesses to feel “the childish urge to understand everything”, an urge which “doesn’t necessarily fade when the time approaches for you to do the most adult thing of all: vanish”.

I’ve been living the last few weeks with Clive James’s previous book, Sentenced to Life. We share the same still-small town-within-a-city, share similarly eclectic tastes (or perhaps not so much eclectic as tastes that place us in the same generation and in somewhat overlapping worlds), and share some favourite authors: we also share rather more that I wish we didn’t. So I have been finding these last poems (if that’s what they prove to be) speaking particularly directly.  Poignant, regretful, looking death in the face without illusion. And very accessible too  – as if, by imposing such clear form, such unabashed rhyming structures, our poet is making his stand against the formlessness to come. There is no false consolation here.  But still (as Blake Morrison puts it in a review) “When in death, we’re in the midst of life – that’s the recurrent, bleakly hopeful theme.”

Not that it is all bleak. Far from it. There is still Jamesian fun to be had: his Compendium Catullianum, for example –

My girlfriend’s sparrow is dead. It is an ex-sparrow.

Though after the jests, we are back to a recurrent thread: “For I …

Miss you the way you miss that stupid bird:
Excruciating. Let’s live and let’s love.
Our brief light spent, night is an endless sleep.

And our poet, before the lights go out, is so vividly aware of the small particulars of his ever-narrowing world:

Once I would not have noticed; nor have known
The name for Japanese anemones,
So pale, so frail. But now I catch the tone
Of leaves. No birds can touch down in the trees
Without my seeing them. I count the bees.

I can recognise that. And so much else here.


I would quote whole poems from Sentenced to Life  if I could. But here is one of the last poems, one of my favourites, first published a year ago in the New Yorker.

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Encore #19: Schubert’s piano music

Is it something about the philosophical temperament? But I have known a number of philosophers for whom Schubert’s later piano works are absolutely central among the music which means the most to them. Certainly they are for me. I have blogged over the years about a number of recordings — more than I realised — including these posts: 

From 2009 Sometimes, in an idle moment, I jot down — be honest, don’t we all? — a list of the eight discs I would select as my Desert Island Discs. Impossibly difficult of course! But one constant choice is the last Schubert piano sonata, D960 (and if I had to save one of the eight discs from the waves, then this would probably be it). But which recording? Well, that’s almost impossible too. … I still think that that has to be one of the Brendel recordings: after listening to others, I always listen to him again with a sense of coming home. Perhaps I love his 1988 recording the most.

From 2010 The third double-CD has recently been released of Imogen Cooper’s stunning journey through late Schubert in her QE Hall recitals in 2008 and 2009. The first two pairs of disks got some extraordinarily warm reviews, and this too strikes me as just wonderful. Thoughtful, deep, utterly persuasive. And lyrically beautiful.

At the end of the second of the new disks she reaches the pinnacle, the end of her journey  — an amazing performance of D. 960.  As I said in an earlier post, I’ve accumulated over too many years recordings of this sonata played by Schnabel, Richter twice, Brendel three times, Schiff, Kovacevich, Perahaia, Uchida, and Lewis as well as the earlier Imogen Cooper; and yet here are new insights, a wonderful overall architecture, and an utterly compelling performance. This could well become my new “Desert Island” choice of  all those performances.

From 2011  Here’s some other recordings to recommend — the new two CD set of Schubert from Paul Lewis, the D. 850 Sonata, the great D. 894 G major, the unfinished ‘Reliquie’ D. 840, the D.899 Impromptus — and last but very certainly not least the D. 946 Klavierstücke (try the second of those last pieces for something magical again).  By my lights, simply wonderful. I’ve a lot of recordings of this repertoire, but these performances are revelatory. Which is a rather feebly inarticulate response, I do realize — sorry! But if you love Schubert’s piano music then I promise that this is just unmissable.

From 2011 Thanks to Askonas Holt, her agents, three videos of Imogen Cooper playing at a concert in 2009 have just been posted on YouTube (the video isn’t HD, but the sound is just fine). There is a nice performance of Schubert’s Hungarian Melody D817, and a lovely short piece of Janacek, ‘Good Night’ from On an Overgrown Path, which I haven’t heard for ages. But then, on a quite different scale of length and emotional intensity, she is joined by Paul Lewis for a stunning performance of the Schubert Fantasie D940. And this is surely as good as it gets: two of the greatest Schubert pianists seemingly as one in their shared vision of the piece. Just wonderful.

From 2013  Suppose, just suppose, that one of the very greatest and most loved actors of his (or her) generation was tackling one of the high peaks of the repertoire, returning perhaps to a previous near-triumph with a quarter of a century’s more experience. Every broadsheet would have extensive reviews, though telling most readers about a performance they will never see (even if in reach of London and in funds to make the journey), because the tickets have long since sold out. There will be interviews with the actor, even colour-supplement spreads. You know how it goes.

Now here is one of the very greatest  and most admired pianists of her generation, still at the very height of her powers, returning to Schubert’s last piano sonata, a quarter of a century after a very fine earlier recording. You would have thought that the broadsheets with arts pages would at least notice this major event. After all, nearly all their readers can afford the CD, which is so accessible that it will arrive at the click of a button. So here surely is something worth reviewing. Here too perhaps is even an occasion for a retrospective pieces on a remarkable artist. But no. As far as I’ve seen, nothing. Which is not at all unusual these days. It is difficult  not to feel (as you do when look round your fellow concert-goers, noting all the grey heads) that slowly but inexorably a deep engagement with classical music is becoming less and less central to our cultural life.

Well here, let it be said, is the most extraordinary music-making, indeed inviting the most personal engagement. Maria João Pires offers us a performance of the A Minor Sonata D845  played with intense intimacy. There is nothing declamatory here: she is sitting across the room, playing for us listening, close around her. There’s a care to every phrase which isn’t mannered, wonderful lightness of touch when called for, and moments when time is slowed to a pause (her magical way with the Trio of the third movement). For this sonata alone you will want the disc.

But then there is D960. What is to be said? For many, this is very high on the list of  the music that matters the most, that has to be returned to time and again. We have a most wonderful inheritance of recordings from Schnabel onwards of this “music which … is better than it can be performed”. Brendel, Richter and Imogen Cooper, all more than once, Lupu, Kovecevich and Uchida — all are stunning in their different ways, all their recordings are to be listened to repeatedly.  And Pires herself recorded the sonata 25 years ago.

It would be absurd (or at least absurd for me) to try to make comparisons. Let me just say that this new recording is surely a more than a worthy addition to that great inheritance. This is not one of those more brooding performances where we are made conscious from the beginning that this is the work of a dying man (performances which give the first movement such ominous weight as to unbalance the whole sonata). There is an intimate directness to her undeclamatory opening: again, we are sitting with Pires — she is not addressing us across a concert hall. And it is only slowly that the intensity is ratcheted up in the first movement (especially about 11 minutes in), and then we are gripped by a new tension as the opening theme and development return once more. The following Andante sostenuto is unsentimental, played with a rhythmic delicacy that becomes magical. The Scherzo is played vivace con delicatezza as Schubert asks: but Pires deconstructs the Trio with bass emphases which I can’t recall being made quite so aware of before,  harking back to previous movements. The final Allegro has moments of lightness again but also a certain weight and drive giving a more-than-satisfying balance to the whole sonata, the tumbling final chords finishing in a sudden silence.

This won’t replace your other recordings of D960, how could it? But you will want to listen and listen again, and you will hear more from Pires each time. Wonderful.

From 2015 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 not, 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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