There have been considerable distractions over the last month, but I hope to get back to a few more logical blog posts this month.
For those interested, there’s now another small update for Category Theory II, in which I bring some considerations of size to the beginning of Part II in a tidier way, and then later I make what I say at different points about (locally) small categories more consistent. I adopt, for reasons explained, an idea in Roy Crole’s Categories for Types that defines e.g. a small category not as have a set of objects but as having objects that are indexed by a set. I’m also a bit clearer about the differences between different ways of defining categories (something that didn’t really affect the discussions in Part I). Slow progress, but this required some actual thought and not just juggling technicalities!
Chamber music playing doesn’t get better than this. The Pavel Haas Quartet have made two stunning, award-winning, CDs of Quintets by Dvořák and by Brahms with Boris Giltburg as pianist. Now half the Quartet get together with Giltburg again to play all four Dvořák trios, the final Dumky Trio of course, and perhaps you already know the third trio — but the first two were quite new to me.
This is just a marvellous double CD. Here’s Katherine Cooper, writing for Presto Music:
As on the earlier recording of the [Dvořák] Piano Quintet, there are moments when the collective sound is so massive that it hardly seems credible that so few musicians are involved – notably in the final athletic stretches of this first trio (just before the music ebbs away like a mechanical toy running out of battery), in the near-symphonic first movement of the Piano Trio No. 3, and as the three hurtle towards the finishing-line in the first of the six ‘Dumky’ which opens No. 4. (Giltburg switches from his ’beloved’ rather soft-focus Fazioli to a bigger-boned instrument for these two later works, and both string-players match its brawn head-on).
The more introspective stretches are also beautifully done, not least the slow movements of the second and third trios (both composed in the aftermath of bereavement): the muted colours of that Fazioli really come into their own in the lovely elegy for Dvořák’s daughter in No. 2, and Jarůšek and Jarůšková respond in kind with a fragile lyricism that’s enormously touching.
A triumphant debut for this new super-trio, then, and one which whets the appetite for whatever they might choose to record next.
The reviewer at Europadisc is equally enthusiastic, concluding about the Dumky Trio:
Here, above all, these performers are in their element, and they deliver performances that are richly satisfying, exploring the full spectrum of tone colours, and combining penetrating introspection with infectious high spirits. The fast passages of the second Dumka are enough to lift anyone from despondency, while the strings’ ethereal response to the piano’s opening phrase in no.3 is a thing of wonder, as is the rounded tone of the piano’s single-line continuation. The fourth Dumka has a deliciously deliberate tread; the exultant opening of the fifth rings out, its glorious cello line punctuated by pizzicato violin chords; and the sixth and final Dumka is underpinned by wonderfully resonant cello fifths, giving way to spectacularly full-throttle tuttis. Make no mistake, this is Dvořák playing of the highest order and – even against some strong competition – these performances as a whole set a new benchmark in this marvellous quartet of works. Urgently recommended!
Well, we were supposed to have been in Copenhagen, visiting the Digital Nomad Daughter. But the day before we were intending to travel, she sent a message to say that they had just succumbed to Covid. We didn’t go. Perhaps we dodged a viral bullet anyway, since apparently Covid is rife there and we have yet to have the next round of vaccine booster shots. Not the week, then, that we were looking forward to. So we’ve been here in an increasingly autumnal Cambridge, misty first thing … and staying hazy this morning as we walked for the hundredth time up to the folly at Wimpole.
With unexpected time on my hands, I’ve been pressing on with Category Theory II and have already quietly uploaded two more heavily revised chapters, and added a third just yesterday. These chapters are I think much improved, with material re-arranged and expanded, and some proofs made rather more transparent. Not epoch-making stuff, true. But I confess I’ve rather enjoyed the process of trying to explain things better to myself, and hope that some others might eventually find the results useful.
Let me share a belated discovery. I should have known before about the pianist Cordelia Williams who won the piano section of the BBC Young Musician of the Year in 2006 and has since released four CDs to considerable acclaim, and fifth just a couple of days ago. (There is a Cambridge connection too: she gained a First in Theology at Clare College about the time for her early competition success.)
Her new disc is “Cascade”, a recording of miniatures, Beethoven’s Bagatelles, Op. 126, Schumann’s Waldszenen. and Prokofiev’s Visions Fugitives. She writes “When I play the middle section of Beethoven’s Fourth Bagatelle from Op 126, I instantly leave the everyday world for another existence: floating, hazy, waiting. Just as abruptly, this shining life disappears like a burst bubble, leaving only a sense that something magical has flickered by. I’m struck over and over by the contrast between the idea of a ‘Bagatelle’ — a small trifle, something of little import — and the musical reality of these pieces. Yes, they are short, but there is such concentration of creativity, focusing each vignette to bright intensity. The depth of transformation and modulation conveys us great distances in each ‘trifle’ — musical content and emotional weight in inverse relation to length. These pieces were written well after Beethoven’s final piano sonata and I sense a whole lifetime of invention condensed into these small forms. Each of these six miniatures casts a brief spell, seeming to end after a minute or two but nevertheless leaving us subtly changed by the experiencing of it. I’ve enjoyed playing these Bagatelles for 15 years now but over the last three I’ve become captivated by this momentary magic in music, by the quality of fleeting beauty. My perception of life seems to become less linear, more a tumbling collection of moments of passing vividness. The ephemerality does not diminish the meaning contained within each of those moments – maybe it even increases their splendour.” The other pieces too she offers as cascades of momentary magic.
The playing here immediately strikes me as very fine, the familiar Schumann immediately captivating, the Prokofiev (which I didn’t know) evanescent fleeting glimpses, the Beethoven impressively convincing. I will want to listen and re-listen. But for Schubert on Sunday, I’ll be returning to her first CD of 2013 of the Schubert Impromptus. The Guardian reviewer wrote at the time “Don’t be lulled by Cordelia Williams’s sweetly cool reading of the first two of Schubert’s Impromptus D899 in this recording; there’s romantic fire lurking under the surface. No 3 is almost Brahmsian in its achingly beautiful sweep, and there’s a delicious, dark brooding beneath the delicacy of No 1, D935. Crucially, [she] manages to suffuse each of these eight miniatures with just the right degree of regret …”. These aren’t performances to replace Brendel, Lupu, Pires, Uchida. But they are a delight to listen to.
But if Cordelia Williams’s recording of the Impromptus made a fine debut disc, her return to Schubert a couple of years ago in her fourth CD “Nightlight” is something else again. The centrepiece of this disc is Schubert’s C minor Sonata D958. The Gramophone reviewer wrote, more eloquently than I could, “For all the architectural grandeur of the opening Allegro, [Williams] brings subtlety and contrast to its urgent rhetoric. When the purity of the voices in the Adagio’s chorale devolves into the menacing triplets, it is a plausible psychological progression from calm self assurance to abject terror. Its sinister undertow notwithstanding, the Minuet maintains poise and grace. The finale’s flight from the furies, despite its driven desperation, remains an escape inerrantly proportionate and flawlessly planned. Throughout the sonata, her playing is always natural, unforced and supremely lyrical, yet alive to every tragic implication in Schubert’s drama. … For all this album’s many strengths, the Schubert sonata alone is worth the price. Williams unapologetically takes her place among the most eloquent exponents of this great work in recent years.” Yes! And from another review: “Her performance of the strange un-minuet-like minuet is a marvel. Even the great names amongst Schubertians seem a little perplexed by this fugitive music. Perhaps it is Williams’ night time theme that helps her unlock the way the uneven phrases interlock so convincingly as though an insomniac Schubert were facing his demons in the wee small hours. Whatever the prompt, Williams catches the mood peculiarly well both here and in the finale. In not straining to find high tragedy, she brings the music closer in character to the other two of Schubert’s last piano sonatas. There is the bitter-sweet tang. There are the fleeting moments of sensual delight and of joy mixed with deep sadness and nostalgic regret. It all passes under her attentive fingers. The first two movements are just as good; this is one of the great performances of this sonata.”
You can stream the CDs from Apple Music etc. And here is a film of Cordelia Williams performing the first movement of the D958 Sonata (and I believe she is planning to be performing D959 and D960 over the next couple of years — so catch her concerts if you can).
If you have access to a library which subscribes to Springer Link, you should be able to download an e-copy of this very recent addition to the growing list of editions of Gödel’s various notebooks. (If you don’t have good library access, then tough — Springer are price-gouging at £111.50 for the PDF, and more for the print-on-demand version.)
The editors Maria Hämeen-Anttila and Jan von Plato write in their short Preface
If there is one “must” to be cleared in the enormous mass of the Kurt Gödel Papers kept at the Firestone Library of Princeton University, it is the series of four notebooks titled Resultate Grundlagen. Gödel wrote these 368 pages between 1940 and 1942, except for the ﬁrst 33 and last 12 pages. There is a continuous page numbering and the same goes for the theorems. It has been a great fortune for us to meet the task of transcribing, translating, and editing these notebooks.
And later, in their introductory essay
Resultate Grundlagen [RG] is a collection of results Gödel considered ﬁnished. … Close to two thirds of RG deal with set theory … Next to set theory, RG contains results on arithmetic and recursive functions. Type theory is one clearly separate topic, and so is what Gödel called “positive logic.” The latter relates to intuitionism which was one of Gödel’s permanent interests from the early 1930s on. This interest is clearly seen in [RG] with about one part in four devoted to intuitionistic logic and its interpretation.
So that tells us two things. First, about the topics of the RG notebooks themselves. And second, inadvertently, that the language of this edition is sometimes only an approximation to good English. Evidently, Springer’s contribution to the publishing of this book didn’t run to a native-speaker copy-editor. This matters, I think, for two reasons. First, readers for whom English is not their first language will stumble. Second, the editors have (oddly to my mind) not given their transcription of Gödel’s obsolete German shorthand in a parallel text (surely an achievement worth preserving for future researchers): so occasionally the reader might wonder whether seemingly odd or stuttering phrasing is in the original or is a result of rendering into clumsy English. In fact the editors write
RG is a polished shorthand text when compared with such sources of preliminary work as [other notebooks]. There are next to no cancellations, but there are additions that often result in awkward sentence structures. The question is to what extent such passages should get improved in translation.
Given this sort of issue, why indeed not pre-empt a reader’s questions with a parallel text, as in the canonical edition of the Works?
On the key set-theoretic content, the editors write
After the transcription and translation work was done, we were lucky to ﬁnd in Akihiro Kanamori a reader without comparison of Gödel’s results on foundations. … Aki took up the task and presented us with a splendid essay on The remarkable set theory in Gödel’s 1940–42 Resultate Grundlagen, an essay that explains how Gödel had arrived at numerous results independently discovered by others later, sometimes much later, in an anticipation of the development of set theory from 1942 on, the year Gödel left formal work in logic and foundations.
Which is good to know; but since Kanamori’s essay isn’t included in the book as an introduction (and isn’t yet available elsewhere), the rest of us will have to wait a little for a knowledgeable guide to Gödel’s achievement in RG. All that said, it remains astonishing to find how productive Gödel was in those years when he was publishing so little. Fascinating but frustrating to dip into.
That’s mostly because I went back and tinkered again with the first four chapters of Part II. And I’m still not super-happy with the fifth chapter, introducing hom-functors. But it is better than it was, and eliminates a foolish mistake or two, so I guess that’s progress.
I’ve updated the corrections list for Beginning Mathematical Logic, which is getting to be long enough to raise the issue of whether to update the paperback before the next edition planned for next year. On the one hand almost all the typos are quite trivial — repeated words or obvious omissions (the sort of thing the eye so easily skips over). On the other hand it increasingly bugs me to know that they are all there. Annoying.
I’ve also had to correct some of the links on the Book Notes page, as I unaccountably didn’t check the page properly before putting it online a couple of days back. Annoying again.
But I’ve mostly been chewing away again at the early pages on functors in Category Theory II finding it frustratingly difficult to make things run smoothly. More annoyance (to add to the considerable irritation of finding I’d made a careless slip in Category Theory I).
Ah well. It’s very nice, then, to be able to hit a cheering note. Here’s a book recommendation. I’ve just finished the very enjoyable How to be a Renaissance Woman by the art-historian Jill Burke. As the blurb says “The Renaissance was an era obsessed with appearances. And beauty culture from the time has left traces that give us a window into an overlooked realm of history — revealing everything from sixteenth-century women’s body anxieties to their sophisticated botanical and chemical knowledge. [Burke] allows us to glimpse the world of the female artists, artisans and businesswomen carving out space for themselves, as well as those who gained power and influence in the cut-throat world of the court.” And as a reviewer puts it “Terrific … Drawing on early published beauty pamphlets, letters, poems, songs, diaries and recipe books, not to mention treatises by both men and women and the rich material of Renaissance art, [Burke] has emerged with enough knowledge to open her own Renaissance Body Shop …” (for the book ends with some recipes for lotions and potions ….: but not the ones for makeup with mercury or arsenic, you’ll be glad to know). A fun read, and illuminating too — it will change the way you see some Renaissance portraits.
Just spotted an elementary thinko in Category TheoryI.
I carelessly define a finite limit as a limit over a diagram with a finite number of objects. But standardly, it should of course be “with a finite number of objects and finite number of arrows”. And I define a small limit as a limit over a diagram with no more than set-many objects: it should of course be “with no more than set-many objects and set-many arrows”.
No great harm done as the definitions do little work in Category TheoryI but were really for future use in Part II. Very annoying all the same. And the twin mistakes have been there in however many earlier iterations, and no-one has pointed them out.
This website and blog has been going seventeen years. So even if I comment on just four logic/phil. math. books a year in a way worth preserving, that’s going to result in over sixty book notes of one kind of another. And indeed, here they are. I’ve replaced a partial table with a new webpage which will be easier to maintain. This page links to shorter or longer comments on (i) the 23 books also covered in PDF form in the Appendix to the Study Guide, plus (ii) another 40 books.
Also, you might have noticed, I have updated the old TYL (“Teach Yourself Logic”) menu to GUIDE (“The Study Guide and Book Notes” on the front page), and its target page now makes the link onwards to this new page of Book Notes quite a bit more prominent.
For a long time, back before CDs, Schubert’s piano music for me meant Alfred Brendel. Early on, I had very extravagantly bought his box of Schubert LPs, which I played and played for years. Still the performances which I find myself comparing with all others, and returning to often. So this week let’s have Brendel in his prime, playing the first set of impromptus.
Back in 2008 and into 2009 I blogged at considerable length about Charles Parsons’s book Mathematical Thought and Its Objects. Then, in February 2009, under the title “Parsons, the whole story, at last” I wrote “I’ve now had a chance to put together a revised (sometimes considerably revised) version of all those posts into a single document — over 50 pages, I’m afraid. You can download it here.” But the link points to defunct storage on the university site, and I don’t seem to have the original LaTeX file or the PDF. Drat. Not a great tragedy, but I put a lot of work into it at the time. And I wanted to revisit part of it … Ah well.
The very longest of extremely long shots: did anyone out there happen both to download and to keep a copy? I think the file was called simply “Parsons1.pdf”.
Later: Search over! Shawn Standefer and Felix Mühlhölzer have both kindly sent me a copy!!