This and that

One in twenty

One way or another, a newly-acquired book or two must come into the house most weeks. So in a year, that’s much more than a shelf’s worth. After a few years, even with occasional donations of books to Oxfam, there are far too many books in piles around the place, on furniture, on the floor, slotted in randomly here and then. So there comes that time again when Something Must Be Done. Perhaps one in twenty really needs to go. And this time, everything ought to be taken off shelves and dusted down …

So that was a week’s worth of mornings, just for the books at one end of the living room. “Will we ever read this again?”, “Good lord, I’d quite forgotten we had a copy of that”, “Do you remember finding that book in …?”, “Oh, it is more than time to reread …”, “Oops, we seem to have bought two copies of …”, “Hold on a moment, let me just remind myself what this is about …”, “Do you remember that exhibition …?”. And so it goes. It all takes time. But job done, and the shelves look a lot tidier. Bags of books line the hall waiting to be distributed to charity shops. Not one in twenty of course. But enough.

A recent report of research from Imperial College indicated that one in twenty are affected by, as it were, Very Long Covid, with symptoms lasting even a year. I seem to be in the unlucky group. Over six months on from catching Covid on a lovely trip to Perugia, I’m still getting chest discomfort and fatigue, of a pretty annoying kind. So, some days, a morning’s book clearing is stupidly tiring. Not much writing then gets done on matters categorial …

But I can read. What has been particularly enjoyable in the last few weeks? I don’t think I can put it better than this, from the blurb: “In Thunderclap, Laura Cumming reveals her passion for the art of the Dutch Golden Age and her determination to lift up the reputation of Fabritius. She reveals the Netherlands, where – wandering the narrow streets of Amsterdam, driving across the flatlands, or pausing at a quiet waterfront – she encounters the rich reality behind the shining beauty of Vermeer and Rembrandt, Hals and de Hooch. She shares too her relationship with her father, the Scottish artist James Cumming, who had his own deep connection to Dutch painting, and who taught her about colour, light and the rewards of looking deeply. This is a book about what a picture may come to mean: how it can enter your life and change your thinking in a thunderclap, a sudden clarity of sight. This is also a book about the precariousness of human life – the way it may be snatched from us in an instant. What can art do to sustain us? The work that survives tells its own compelling story in these pages.” I thought in advance that that must be over-selling the book! But I did find it hugely engaging. Especially when read with iPad to hand, to find reproductions of more of the paintings Laura Cumming mentions. Warmly recommended. Thunderclap stands out among books I’ve read recently — one in twenty, perhaps.

Bewildered of Cambridge writes …

A few posts back, I quoted Tom Leinster: “The level of abstraction in the Yoneda Lemma means that many people find it quite bewildering.” I can relate to that. Just a couple of days ago I managed to bewilder myself for a long afternoon, and it took me an embarrassing amount of time to locate where I’d composed arrows the wrong way around (contravariance bites back). I have a rule when writing this sort of expository stuff not to look things up, on the principle that working things out again for myself from first principles much increases the chances of being able to explain the topic lucidly to others. But the rule has its downside sometimes!

Anyway …

There is now a fifteen page chapter on Yoneda (the theorem, not its implications which I plan to tackle next). I take things in three stages, proving what I (non-standardly) call the Restricted Yoneda Lemma, and then the Intermediate Yoneda Lemma, before getting to the fully caffeinated Yoneda Lemma. The hope is that, chunked up like this,  each of the three stages almost writes itself …

For enthusiasts, the latest version of Category Theory II is here. anyone?

I have long, long since unsubscribed from all e-mails from, and I’ve very largely ignored it for years. I‘ve certainly never dabbled with the “pro” version. I did upload copies of IFL and IGT when I had recovered the copyright, and later replaced the old TYL Guide.  But otherwise I’ve kept my presence there pretty minimal and not used the site at all.

However, I was alerted recently to the fact that a (very) old version of my notes of category theory were still being downloaded from there. So I thought I should either replace those notes or perhaps just remove myself from (as I had got the impression that it really isn’t in favour).

I was a bit staggered to find, then, that despite the non-activity on my part and despite still having only eight live pieces there, the site reports 175K “views” over the years. I’m pretty sceptical. But I did find hundreds of recent messages from downloaders that I’d not acknowledged (apologies, then, to those who thought I was just being rude!).

So I’ve decided to keep my profile there for the moment. But is really much used these days?  What do people use instead, if anything?


I put myself on the waiting list for a Bluesky invite code more or less at the beginning, and then again after a few months, but have never got one. Is this other people’s experience too?

Updated: Thanks for the various kind offers of codes … I’m now signed up!

Not Copenhagen (but Schubert on Sunday 5)

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).


Typos, bad links, functorial frustration, and mercury makeup

It’s been one of those days.

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.

63 book notes and counting …

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.

Another related update. The ARCHIVE menu item now targets a new page which points onwards to four different archive pages (that’s to keep individual page-size sensible). This new Archive page points to (1) the same Book Notes page (which is now reasonably complete); (2) a page linking to other logical blog posts and a few papers/handouts on logic/phil. math (in progress); (3) a page with links to a variety of other old blog posts possibly worth revisiting (only just started). And there will also eventually be — for me, if not for many others! — (4) a page listing some of the music videos I have posted and which are still available.

Parsons on Mathematical Thought (revisited?)

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!!

New book notes

I have added three more short book notes to the Appendix to the Beginning Mathematical Logic Study Guide, discussing

These and other entries are also linked in the growing table of Book Notes.

Added: That’s very annoying! URLs pointing to “named destinations” inside a PDF work perfectly in Safari and Firefox on my MacBook but, it turns out, not in the iOS versions of Safari and Firefox on my iPad. I’ll have to rethink. For the moment, I’ve redone the PDF of the Appendix so that when the links don’t work as intended, you’ll now be taken straight to the contents page of the Appendix and can click on the internal link inside the PDF to take you to the book note you want ….

The Study Guide’s Appendix, a new version

I’m interleaving two projects at the moment, spending time on one or the other as the mood takes me. One project is to update the messy draft notes Category Theory II. The other is to revise and improve the Beginning Mathematical Logic Study Guide.

The Guide gives topic-by-topic recommendations for reading on various areas of math logic. Early versions had an Appendix which also looked, book-by-book, at some of the Big Texts that covered more than one area. That Appendix has stayed online, but has been left untouched for the better part of a decade. Yet it gets downloaded pretty steadily, about a fifth as often as the main Guide (so between three and four hundred times a month, if we can believe the stats counter, which I don’t).

At long last, there’s a new version of the Appendix, now in the same format as the Guide. I’ve updated the entry on Ebbinghaus, Flum and Thomas, and added entries on Mileti and Avigad (basically from the blog posts here). I’ll be adding/revising more over the coming three months or so as I do more (re)reading as background homework for revising the Guide.

What other good books published in, say, the last twenty years and not yet mentioned in the Appendix should I take another look at? Rautenberg, for sure. Kaye’s The Mathematics of Logic and Kunen’s The Foundations of Mathematics are certainly worth a revisit. But what other suggestions of wider ranging, near-entry-level books are there?

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