Peter Smith's Blog, page 100

Notes on Basic Category Theory, v.8a

It’s a balancing act. On the one hand, I don’t want to annoy readers with over-frequent announcements of minor revisions. On the other hand, I don’t want to keep propagating flawed versions when I have an improved offering in hand!

Anyway, I’ve been reading through the first 11 chapters of the Notes making some minor corrections and other changes. I’ve also had some much appreciated corrections of a few mistakes in later chapters from Alessandro Stecchina. Since I think there will be something of a pause before I can press on to re-read the rest of the Notes, here’s an interim update, to version 8a of the Notes.

Three quartets

PHQ in Zagreb

In some sense, the Allegri Quartet have been going for 60 years. But after 30 years, none of the original four were still playing; and now after another 30 years none of those four replacements is still in the quartet … I suppose it is a minor philosophical question, the persistence conditions for string quartets.

But let’s not puzzle about that! The question is: are they still worth listening to? On the evidence of a concert in the chapel of Trinity a couple of nights ago here in Cambridge, very much so. We heard them give a really fine performance of Schubert’s “Rosamunde” Quartet, with nuanced feeling and great togetherness from the four players. Compelling playing.

The second piece in the concert was Dvorak’s “American” Quartet, played by the Wihan Quartet (a new name to me: they are a long-established Czech string quartet currently in residence at the Trinity College of Music, London). This was performed with marked intensity and evident love for the piece, though (we agreed) sometimes there was a very slight sense of rush, particularly in the haunting Lento. The Wihan Quartet CD performance of the Dvorak — available to stream via Apple Music as I discovered afterwards — is extremely good, and the tempi there are a little more spacious. But their live performance was still rather fine.

After the interval, the two quartets combined to play the Mendelssohn Octet, with  verve and much evident enjoyment (and to the huge enjoyment of the audience). A terrific evening.

It is strange, though, that magical alchemy that makes the difference between rather good quartet performances, as these were, and truly great ones. After sampling the Wihan Quartet’s CD performance of the “American” Quartet I listened again to the Pavel Haas Quartet’s CD (which was the overall Recording of the Year for the Gramophone in 2011, and the top recommendation of BBC Radio 3’s “Building a Library”). The playing is just extraordinary in so many ways.  Difficult to listen to entirely dry-eyed.

The same goes for the PHQ’s latest CD, of the Smetana quartets. Again the Gramophone reviewer rightly reached for the superlatives. The second quartet is not so immediately appealing, and I’d never really appreciated it till we heard the PHQ play it at the Wigmore Hall: this recording however is again totally compelling. But it is their performance of the first quartet that I’ve been listening to time and again since it was released three months ago.  Extraordinarily affecting.

But don’t take my word (or the Gramophone‘s) for it: PHQ’s Dvorak and Smetana discs, like their other Supraphon recordings, are all available to stream on Apple Music. Do listen!

Notes on Category Theory, v.8

At long last — more or less exactly nine months since I started intermittently writing them — there is a first complete version of my Notes on Category theory (as they are now called). Or at least, the Notes are complete in the sense that I don’t intend to press on to add further chapters on significant new topics like monads or abelian categories. Rather the current plan is to leave these notes in more or less their present form, for all their shortcomings (and I hope in due course to start writing a differently organized, more discursive, bigger and better version).  Still, I would very much like to hear about errors of one kind or another. And I’ll no doubt issue occasional “maintenance upgrades” when I hear about mistakes or spot passages which really won’t do —  and perhaps I might add more illustrative examples or even new sections here or there to round out the treatment of existing topics where the coverage in retrospect seems too skimpy.

Since the previous version, I have expanded the chapter on some general results about adjunction, and  added a chapter on adjunctions and limits. This has entailed quite a bit of going back to earlier chapters, adding material to smooth the route to later theorems. I finish up by waving my hands at, though not elucidating the content of, the Adjoint Functor Theorems, General and Special.  But it is a non-trivial expositional task to explain these (the technical proofs aren’t hard; what isn’t so easy is to see is the motivation for the various new concepts — like the ‘solution set condition’ — which they involve). I’m not sure I yet have a sufficiently good grip on the place of these theorems in the scheme of things to give an illuminating account of the motivations. So I’m at the moment shirking the task of trying to explain more.

But in any case, the Adjoint Functor Theorems arguably sit on one of the boundaries  between basic category theory and the beginnings of more serious stuff. So given the intended limited remit of the Notes (now highlighted by calling them notes on Basic Category Theory), the Theorems mark a reasonable point at which to stop for now.

So, with that by way of preamble, here is the new version of the Notes (190 pages). Enjoy!

Apple Music — some Day One impressions

So here I am, someone who — in phases — buys quite a lot of classical music. I have never really got into buying the music in the form of digital downloads. Much of the classical back catalogue can be acquired cheaply as second-hand CDs (and there’s fun to be had, searching the charity shops). While the small difference in price between buying a new release as a physical CD and as a digital download is usually balanced by what is still (for me, given our various players) the convenience of the CD — and anyway, it is good to have something of value to pass on to Oxfam if I decide that the recording isn’t one I want to keep.

But storage space has increasingly become an issue. So now, in the nick of time, along comes Apple Music.  This looks very tempting. For the price of one mid-range CD a month, here’s the prospect of  renting access to the vast iTunes library. (Yes, yes, of course I know that there were streaming services before! But old dogs, new tricks, etc. )

So today I’ve updated the software on various bits of Apple kit and signed up for the three months free trial.  How does it look on Day One, at least for a classical music listener? For to be frank, the whole set-up is surely not really designed with us much in mind (indeed Apple don’t really seem to give a fig about us — or else ages ago iTunes would have had the trivial tweak that would tell us that CDs which are assigned the genre “classical” comprise tracks rather than always, idiotically, songs).

The first and crucial question is: what is available to stream via Apple Music? Obviously a huge back catalogue, and many new releases. But there are also significant gaps. For example, you can’t get three of the last six of Gramophone’s ‘Recordings of the Month’ (Rachel Podger’s wonderful L’Estro Armonico, Andras Schiff’s revealing Schubert played on a fortepiano, and Alina Ibragimova’s new Ysaye Sonatas).  Hyperion seems to be one of the labels that has not yet signed up to Apple Music. Which explains not only Ibragimova’s absence, but also the fact that you can’t stream two of my other favourite recent buys, Marc-André Hamelin’s Mozart Piano Sonatas  and his Janecek/Schumann disc. And looking back, you can’t get e.g. any of the Hyperion Schubert Edition lieder discs. Now, these are in fact all available to buy on iTunes. But of course, existing gaps in what you could already purchase on iTunes also carry over to become gaps in what you can stream — so e.g. the coverage of historical recordings can be very patchy indeed.
What is the “user experience” like? Not bad, on any OS X or iOS platform at least, though it probably helps to already know what you are looking for. One big trouble (as The Daughter pointed out) is that Apple and others didn’t push for the standardization of metadata for classical recordings a million years ago. Presumably the record companies thought that it wasn’t going to be an issue, ‘cos we were all going to be reading CD inlays till the end of time. This means that searching can be a bit of a pain on iTunes and now on Apple Music. But — even more annoying — when you do find what you want, the information about tracks can be hopelessly inadequate. Here’s an extreme example of a common phenomenon. There’s a rather splendid L’Oiseau-Lyre 50(!) CD box called The Baroque Era full of interesting stuff. And heavens, you can stream the lot from Apple Music. Great.  So here are the 480 tracks nicely listed, with none of them assigned to their respective composers (even if you control-click to Get Info). Apple have no doubt just used the meta data as provided by Decca. But is no one at either end doing any quality control on this sort of thing? Another sign, perhaps, that Apple doesn’t really care about the classical listener enough to insist that everyone gets their act together.
A minor glitch when it comes to listening (or am I missing something?): you can’t adjust the gap between tracks in streaming an album. So movements of a sonata, say, can follow on each other with with a quite inappropriate rush. I suppose you could stream one track at a time to avoid this, but it is a (surely avoidable) annoyance.
Sound quality? No complaints at all. No doubt my ears are too old to catch the finest nuances, and so I wouldn’t be able tell the difference anyway between Apple’s encoding and the high quality streaming that is available more expensively from specialist providers. But I guess for most of us, listening on only moderately decent kit, in an averagely auditorily cluttered environment, Apple Music is just fine.

Not perfect then. But still, there is a lifetime’s listening of extraordinary recordings old and new available to stream, and reasonably easy to find if you know what to look for  — and I guess other labels will sign up if Apple Music takes off as you would expect. I’ve already spotted, oh, a dozen recent CDs I had it mind to buy which I can now stream. So at least modified rapture.

And rapture at less than $10/£10 a month seems rather a bargain to me.

Notes on Category Theory, v.7

Progress seems to have been a bit slow for various reasons, but I have now added two short-ish chapters to the Notes on Category Theory. One is a chapter on Exponentials added between the chapters on limits and the start of the chapters on Galois connections and adjunctions. The other is second chapter on adjunctions, showing how to generalize certain results we saw in the special case of Galois connections to apply to adjunctions more generally.  I have also added some material earlier to make some of the new later proofs work more smoothly.  Here then is the latest version of the Notes, some 170+ pages.

As I have stressed before, I’m myself learning as I go along from the project of writing these Notes. So as more light dawns, I of course can see how I could have arranged/explained earlier material rather better (OK, a LOT better). But for the moment, the plan remains to add a few more chapters, before eventually starting over and trying to get everything into a more ideal shape.

Notes on Category Theory v.6b

Illness (not so great) followed by fortnight’s holiday (really excellent) stopped work on category theory for a while. Very slowly getting back to it. But in the meantime, a number of people have very kindly been sending corrections to the last version of the Notes. I have also tinkered in minor ways, improving the last chapter in particular. There are just about enough changes to warrant another “maintenance upgrade”, making some of the needed repairs and improvements.

I hope to have a couple more chapters on adjoints ready for prime time later in the month — but finding a neat expository path through the material is a challenge, so don’t hold your breath!

The plan at the moment is for another five or six more chapters in total to round off Part I of these Notes, on basic category theory (I’m not sure yet whether I also need to say anything about monads for what is going to follow). And then — having got the bit between my teeth — I’d like to continue, by discussing some logic and set theory in a categorial way in Part II of the Notes. Promises, promises.

Peter Cropper, 1945–2015

It is very sad to hear of the untimely death of Peter Cropper, for forty years the inspirational and charismatic leader of the Lindsay String Quartet.

When we lived in Sheffield, we often went to hear the Lindsays play at the Sheffield Crucible Theatre (not the big hall, but the very intimate studio theatre where the players sit in  a central space surrounded by tiered seats, a few hundred people just yards away). These could be extraordinary occasions, which always involved Peter Cropper talking to the audience about what the Quartet was about to play. His enthusiasm and passionate involvement made for memorable evenings.  The playing wasn’t always immaculate — but “Who wants perfection? Perfection is sterile. We’re human beings.” Peter Cropper was also instrumental in setting up quite exceptional series of concerts over the years in  Sheffield’s ‘Music in the Round’ (I guess using his personal warmth and contacts to entice world-renowned musicians to this small venue).

The Lindsays were at the height of their musical achievement in the period when we were able to hear them so often, and the CDs that came out at this time — often, like the ‘Bohemians’ series, after series of performances in Sheffield and the Wigmore Hall — are consistently wonderful and are among the very best quartet performances we have. Peter Cropper’s playing on their Haydn disks shows a warmth and a delight in Haydn’s endless invention that is absolutely captivating. The quartet’s Beethoven is unsurpassed.

But I also have more personal memories. When at school, The Daughter was taught the violin by Nina Martin, Peter Cropper’s wife. And Nina would a couple of times a year arrange concerts of her pupils (and pupils of local viola and cello teachers, so quartets and baroque concertos could be played). After the concerts, some pupils and parents would often go back to Peter and Nina’s house, and crammed into the sitting room there would be more music. I can remember now a scratch, sight-read, performance of the first half of Mendelssohn’s Octet with six assorted school kids, including The Daughter on fourth violin, plus Nina busking on second viola, and Peter attacking the first violin part with his usual passion, and inspiring  a moment of magic from his impromptu ensemble.

Logical snippets (again)

[Updating a post from fifteen months back.] For almost three years now, I’ve been a contributor to the useful question-and-answer site, math.stackexchange.com. This is a student-orientated forum, not to be confused with the truly wonderful mathoverflow.net which is its research-level counterpart. Think of my efforts as (hopefully) constructive procrastination on my part.

Of course, many of the questions on the site, including many I’ve found myself answering, are very ephemeral or very localized or based on very specific confusions. But a proportion of the exchanges to which I’ve contributed might, for one reason or another, be of some interest/use to some beginners and near beginners in logic.

So a while back, I put together a page of links to these logical scraps, morsels, excerpts, … snippets, shall we say. The links are grouped by level and/or topic.

This snippets page isn’t the most visited area of Logic Matters, but it gets more than enough hits to make it worth keeping alive. So — although I’ve been contributing to math.stackexchange rather less of late — I’ve just added links to some more recent postings there. Spread the word to students as/when appropriate.

Postcard from Cornwall

 

We have been in Cornwall for a fortnight. Saint Mawes, since you asked. Much to be recommended for riposo totale. We have already booked to return to the same place next year.

Now I’m back, this blog will splutter into life again. Though I’ve just been deleting a little rather than adding. I had been posting initial discussions of the opening chapters of John Burgess’s Rigor and Structure. I was, however, beginning to find the book surprisingly thin and unhelpful, and didn’t have anything useful to say: so rather than continue carping I’ve decided to remove those posts. I did read on further, to get to what were advertised as the main novel claims. But I must be missing the point as what I found seemed banal. I seem to be too out of sympathy with Burgess style and approach. Your mileage may, of course, very well vary.

For a sharply contrasting book, at least in the level of depth and care, can I instead recommend (well, I’m only a couple of chapters in, but it is going terrifically) Ian Rumfitt’s The Boundary Stones of Thought (OUP). This is a predictably serious discussion of the nature of logic in which Rumfitt defends classical logic against a variety of broadly anti-realist attacks. Exemplary and inspiring stuff.

Burgess, Rigor and Structure — 4

[After a delay, let’s continue …!] In the second half of Chapter 2 of his book, Burgess first turns to discuss some opponents of the project of rigorization when conceived as the project of regimenting mathematics into standard set theory.

We get a mixed bag of comments. For example, Burgess has a short section on opponents whose beef is with the distinctively classical nature of (post-)Cantorian set theory. It is perhaps enough to point to the obscurity of Brouwer’s intuitionistic critique to explain its early failure to win many adherents. Burgess then adds that “whatever the merits of [Dummett’s later defence of intuitionism], it came far too late, long after the mathematical community had made up its mind” (p. 82). Likewise, “it has transpired that much more of classical mathematics can be salvaged constructivistically [particularly, Bishop-style] than it originally appeared … [but this]  came to be appreciated only too late, after the struggle was essentially over” (p. 85).

Now, these remarks gesture in a rather arm-waving way to something of the history leading to the de facto hegemony of classical ideas. But of course they hardly justify that hegemony. Indeed, the point that we can salvage a surprising amount of applicable mathematics in a weaker, non-classical, framework might be thought rather to undermine the force of some of the earlier considerations that were supposed to weigh in favour of ZFC as a canonical framework for regimenting mathematics.

Similarly, Burgess has a brief section on predicativism as a different line of attack on Cantorian set theory. And Burgess is of course right that, back in the day, Weyl won even fewer converts than Brouwer. But we now know that (starting indeed from constructions already proposed by Weyl) even more mathematics can be salvaged in a weak classical but predicative framework where the only sets countenanced are sets of natural numbers. So again, exactly where does this leave the supposed justification for taking a theory as strong as ZFC as canonical?  Burgess doesn’t say.

Going off in a different direction, Burgess then returns to consider more general issues related to the idea of rigour (that would arise whatever our preferred foundational framework, if any). He has a short but routine section on deduction-as-regimented-in-logic-texts versus deduction-as-practiced-by-mathematicians, and briefly discusses the role of computer-aided proofs. And in between, there is a probably unnecessary excursus on what he calls the “Theoretical Mathematics” controversy about rigour and proof in arising from a paper twenty years ago by Jaffe and Quinn (I hadn’t come across this before, and Burgess’s discussion certainly doesn’t encourage me to follow it up).

So, again, I have to report that readers of this blog really won’t be missing much if they skip this chapter (while students who have e.g. never heard of intuitionism or predicativism will probably be left rather unclear what is going on, and really ought to be pointed to rather more substantial discussions, e.g. in the relevant articles in SEP). My interim recommendation is: open the book at p. 106, at the beginning of Ch. 3, and start reading from there …