Peter Smith's Blog, page 4

Carpaccio at the Scuola Dalmata

A wonderful memory this. Last time we were in Venice, with the Daughter, we visited the Scuola Dalmata (Scuola di San Giorgio degli Schiavoni). Though it’s high up the “must see” lists in the guide books, we were the only people there, and had a delightful personal introduction by the custodians (still members of a descendant of the original confraternity). The cycles of paintings in the lower room by Vittore Carpaccio from about 1502 onwards are hugely engaging: here’s St George, and the dragon is about to meet its master …

Those paintings have so stuck on my mind that I’ve wanted for a while to get better acquainted with Carpaccio’s work. So I bought myself as a birthday present this year the quite beautifully produced catalogue for a Carpaccio exhibition a couple of years ago. Wonderful illustrations and interesting and enlightening essays. Here’s a short video of that same exhibition.

But I have also just now discovered that the Scuola Dalmata also has an online virtual tour, which gives a really good impression of Carpaccio’s cycle of paintings in their home. Enjoy!

Footnote: I didn’t know this. But beef carpaccio, the dish, is named after Vittore Carpaccio. Giuseppe Cipriani, the founder of Harry’s Bar in Venice, who invented the dish in 1950, was reportedly put in mind of the artist because of a major exhibition of his work held in the Doge’s Palace at the time.

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Indeed, just trivially obvious …

I’m getting some very nice comments on a late draft of the categories book. “A rich and thoughtful introduction, both mathematically and philosophically”. “This is extremely readable, with clarity, philosophical depth, and pedagogical structure. It invites a thoughtful reader in while setting up serious conceptual work.” “This is another really thoughtful and rich chapter. You continue to navigate the fine balance between careful exposition and philosophical reflection—especially in §4.2, which builds up slowly and clearly to your broader point.” And so it goes.

Thanks a lot, ChatGPT …

Yes, I’ve been asking ChatGPT to do some proof-reading, both for typos and syntactic errors, and for comments on style and tone. Perhaps the most useful thing it has done (but I didn’t need AI for this, as I could just have done a word-frequency analysis) is catch me out on some of my verbal tics, the overuse of ‘indeed’, ‘just’, ‘trivial/trivially’, ‘obvious/obviously’ being indeed just trivial and obvious cases. And heck, when you have to search through 363 occurrences of ‘just’ to see which should indeed just go, in a way a trivial task, that obviously takes quite a while.

ChatGPT is catching a few straight typos, an ugly split infinitive, and so on: and a small handful of its stylistic suggestions have been pretty good, noting some unnecessarily clumsy sentences. But the useful comments have to be sifted from very many more that aren’t (including some odd hallucinations of mistakes that aren’t there at all). On balance, though, it is earning its keep.

Though sometimes it does give me pause. For example, commenting on the section where I first introduce the notions of pairing schemes and quotient schemes, ChatGPT starts “Very clearly written, but might benefit from a comment on how these definitions are generalizations of familiar constructions — i.e., pairing schemes abstract the Cartesian product concept without enforcing any set-theoretic encoding.” And given the non-standard story I tell, that seems really quite a smart response for a dumb pattern-matching model.

Now, obviously, I must just go and see which of the 77 occurrences of “really” do indeed really need to be there …

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Updates: Categories and PHQ

I am busily proof-reading the category theory notes for typos and thinkos and expositional stumbles (while thinking how to much improve the final chapter). I’ve now reviewed the first 25 chapters, i.e. up to the first Interlude, and you’ll find today’s version online linked on the categories page. So far, discovered no disasters (though a missing line in a proof gave me pause), and I have made only minor changes.

I posted a link to a video of a Pavel Haas Quartet concert a few weeks ago. But that video is no longer available. So, to make up for that, here’s another, of their performance of Smetana’s first Quartet at the launch of the new season at Wigmore Hall. It is wonderful to see how happily they seem to have come together again with their new violist after their tribulations in that position: long may that last.

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Four handshakes from Trotsky

My good friend, the logician Thomas Forster, tells me that he once shared an office with Giovanna Corsi, who at one point shared an office with Jean van Heijenoort, who was Trotsky’s personal secretary in the 1930s …. 

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Categorically, better late than never

So, at last, here is a full draft of Introducing Category Theory. All 49 chapters and 477 pages. I really didn’t set out to go on at such length. Honestly …

In this latest version 2.7, Chapters 45–48 are very significantly improved. But the final Chapter 49 is a bit feeble really, and maybe will get beefed up in a final version 2.8. However to write something really significantly better would extend the book to well over 500 pages, and take me more energy that I could probably bring to the task. So this chapter gesturing towards ETCS will probably remain little more than an incitement to read some of the likes of Michael Shulman’s terrific paper “Comparing material and structural set theories” (available on the arXiv here).

So what now? Apart from tinkering with that final chapter, I’ll need to proof read the whole thing, and also make sure that I take a reasonably consistent line on matters of “size”. I also want to add a short section on pre-sheaf categories. But at least I have a full draft to work from. All comments still very gratefully received, of course!

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Schubert, extraordinarily

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Hallucinations

I had written a few paragraphs here sounding off about some published “continental philosophy” style meanderings about the philosophy of mathematics (which, as typical, show no real knowledge of mathematics) when I checked and discovered that the author was a graduate student. It seems unnecessarily unkind to continue. I’m mellowing.

A recent post on the Leiter blog reported the experience of a “senior scholar of ancient Greek and Roman philosophy”:

The other day I gave the same query to two different AI’s (anthropic.claude-3.5-sonnet.v2 and openai.gpt-4o):

“I’m interested in Plato’s Republic, and I’d like to read some recent scholarly literature concerning his proposals for censorship in books 2 and 3 of the Republic. Can you suggest some articles written within the last ten years?”

One gave me two citations, one gave me five. Of those seven, exactly 1 was a real citation to a real article. The other 6 were mix-and-match nonsense: the names of real authors (lots of my friends!), the names of real journals, and titles that were plausible pastiches of title-elements.  But none of them actually exist. 

That entirely tallies with my own experience, when asking ChatGPT about where I might find proofs of various category-theoretic results in the literature. Quite useless. And if you ask for the actual proofs …

As so many have said, it makes stuff up! I had wondered if mathematicians could at least use ChatGPT in generating worksheets of the kind ”Is this a successful proof [quoting some AI output]? If not, explain why not.” However too much of the garbage is, as the phrase has it, not even wrong: so on second thoughts, I’m not sure that there is even that role for the AI output.

But at least we know how it is that ChatGPT comes to hallucinate. I’m baffled, though, how a student of “continental philosophy” can get to such a state of hallucinating that they are making much sense. The old-style training in analytic rigour could make you over-nervous of committing yourself to anything … but at least it also made it quite difficult to fool yourself into writing gobbledegook.

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ICT once again

At least Zeno’s arrow got ever-closer to its target. But the newly posted version 2.6 of Introducing Category Theory has two chapters fewer than version 2.5 …

I’m not really going backwards. In the last three weeks, there have been distractions, but I have made a fair few scattered improvements (additions, corrections, neater proofs, clearer explanations) in Chapters 1 to 46. There are now enough of those for me to want to make the improved chapters available, while still re-drafting the final ones.

(There indeed have been many distractions. Among them, too much doom-scrolling, despite my best resolutions. But how can one avert ones gaze?)

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Don’t miss a superb La Traviata

The La Traviata broadcast from the Royal Opera House last night was wonderful — and Lisette Oropesa as Violetta was simply stunning. You can catch the performance on BBC Radio for a month. Don’t miss the chance.

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ICT again, but also Mozart, Brauß and Pires again

I have updated Introducing Category Theory to version 2.4: download here. As well as many minor corrections, there is a restored §18.4 on ‘naming’ arrows, and a new §18.5 touching on Lawvere’s fixed point theorem (at least giving a clear proof). Fun enough if you like this kind of thing.

But we need a lot more to distract, console, nourish us in these troubling times. For a start, we need more Mozart in our lives. The photo above is a very characteristic shot of Elisabeth Brauß rehearsing for her performances a few days ago playing K.271 — the ‘Jeunehomme’ concerto — with the Bournemouth Symphony Orchestra. Sadly, there is no recording of them. Here she is, though, nine years ago when just starting out aged 21, already showing what a wonderful Mozart player she is:

K.217 AllegroK.217 AndanteK.217 Rondeau, Presto

And here, some fifty-five years older, is that great Mozart pianist Maria Joāo Pires playing the same piece just a couple of years ago:

K217 (complete)

Wonderful too — and isn’t there something deeply in common too in the two pianists’ utterly honest, unshowy, loving performances of the music? Both such a delight.

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