Peter Smith's Blog, page 3

Introducing Category Theory again

With a bit of useful proof-reading help from ChatGPT, and rather less from Claude, there’s a new version (2.8b) of ICT downloadable from the categories page.

Between them, my friendly AI assistants found perhaps half a dozen straight-up typos per chapter (misspelt words, omitted words, singular verbs with plural subjects and the like), and also caught the occasional symbolic foul-up, or carelessness with a technical term. So ChatGPT and Claude did earn their keep.

I asked only for reports of definite mistakes. Even so, most of the feedback I got was stylistic, complaining about informal locutions, the use of contractions like doesn’t and can’t, or the use of what they judged to be a less-than-academic tone. I mostly ignored such complaints.

As the chapter-by-chapter feedback went on, I got less impressed. Evidently, neither assistant was particularly reliable at even the most basic typo-spotting (as often enough a plain error found by one was entirely missed by the other). The style and organization of feedback would vary significantly from day to day (at least until I learnt to tell them to exactly copy the format of the previous session). ChatGPT could say that it had read all of a portion of a chapter when evidently it had stopped when it got to the tenth mistake (and then lied about doing that). It could also skip right over a large chunk of a chapter.

We all know by now about the way LLMs hallucinate — and you’ve probably seen the interesting research by Apple about the shortcomings of LLMs augmented, supposedly, with reasoning abilities. But the sheer flakiness of both ChatGPT and Claude when it comes to such a low-level task as proof-reading for plain errors remains quite surprising.  Even here, they don’t (as the phrase has it) “just work”. Which perhaps makes it a bit easier to understand why Apple are having such trouble getting an AI-assisted Siri to function at the standards they’d be looking for.

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Clara, Noa, Elisabeth

As the world continues to go to hell in a handcart, let’s just pause for ten minutes to remind ourselves that there is beauty and tenderness … So from a couple of months ago, here are Noa Wildschut and Elisabeth Brauß playing the Three Romances for Violin and Piano by Clara Schumann. (Click the Watch on Youtube button.)

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Introducing Category Theory (an almost final version)

Not the paperback (yet). But there is now a full version of Introducing Category Theory that I don’t plan to do anything much more with than get my AI friends to continue proof-reading onwards from about half way through. I’ll cast my eye over the Index, edit the cover, and sigh with relief.

You can freely download the PDF here. Do let me know if you spot any typos or thinkos, however small. Am I pleased with the result? Moderately. But it will have to do! As I say, at the very end “I am leaving issues about the interpretation and significance of categorial ‘structural set theory’ hanging – tantalizingly! – in the air, with major themes yet to be explored. But that must be for another day.”

The Amazon print-on-demand service, to be fair, doesn’t price-gouge. By taking minimal royalties, I’ll be able send the printed version out into the world for about £12.50, €15, $17. Not bad, surely, for a 500 page, larger-format, book. Depending on my limited patience with proof-reading, that should be around the end of this month. As always with the Big Red Logic Books, the PDF will remain open access.

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Schubert on Sunday 12: Fantasy for Violin and Piano, D. 934

Charlotte Saluste-Bridoux — the prize-winning young violinist, who has recently joined her teacher Alina Ibragimova as the new second violinist for the spectacularly good Chiaroscuro Quartet — has a fine Schubert CD newly out with her regular duo partner, the pianist Joseph Havlat.

Here is a video of them playing the D. 934 Fantasia at Wigmore Hall a couple of years ago — rightly so warmly received by the audience. Enjoy!

(Want more? You can browse through the previous eleven Schubert on Sunday videos via this link!)

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Categorial progress …

Slowly, very slowly, …

I would be really annoyed to paperback Introducing Category Theory and then immediately find silly typos or thinkos. So (as I’ve reported before) I have been getting ChatGPT and Claude Sonnet to proofread chapter by chapter. This is a pretty tedious and surprisingly time-consuming business. However, they are finding enough plain typos to make the exercise worthwhile. And — rather impressively — they are also catching some thinkos too. There is little overlap between the feedback from my friendly AI assistants, so it is worth using both. And although the prompts I use ask for reports of definite mistakes, I also get back a lot of more stylistic comments, a sprinkling of which are also well worth taking note of.

As I carefully go through, not so surprisingly, I also find other infelicities, or slight mismatches between what I say three hundred pages apart, and so on. So there is still more work to be done. And I need to expand the second Interlude (to say just something about the emerging picture of category theory) and slightly expand the final chapter on ETCS. But, famous last words, the end is in sight.

I have already made more than enough corrections and improvements to the April version to make it worth uploading another interim version: so here it is. Last minute comments and corrections still very much appreciated. (If you do download the PDF, check the verso of the title page. You should get Version 2.7f. Clear your browser cache if an older version appears!)

And The View of Delft? Stunning, no? By itself, worth the trip to The Hague, and now for a while my desktop picture. Click on the image for a larger reproduction …

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McLarty on ‘The roles of set theories in mathematics’

A couple of years back, I commented here on most of the papers in Categories for the Working Philosopher, edited by Elaine Landry (OUP). I’ve had occasion to revisit the first piece in the book, ‘The roles of set theories in mathematics’ by Colin McLarty. This is not the sort of wide-ranging discussion that we get in e.g. Penelope Maddy or John Burgess writing about the role of set theory. McLarty focuses on one familiar claim — namely that ZFC radically overshoots as an account of what ‘ordinary mathematics’ requires by way of background assumptions about sets. I have somewhat revised my comments.

McLarty in particular takes a look at the set-theoretic preliminaries expounded in two classic texts, by Munkres on topology and Lang on algebra, and comments on their relatively modest character. Though I’m not sure that McLarty is a particularly reliable close-reader of their texts. For example, he writes “Of course in ZFC everything is a set, including that the elements of sets are sets. Munkres hardly denies that all objects are sets.” But Munkres does deny just that — his informal set theory explicitly allows urelements which aren’t sets:  “The objects belonging to a set may be of any sort. One can consider the set of all even integers, and the set of all blue-eyed people in Nebraska, and the set of all decks of playing cards in the world.” 

Let that pass.  Neither Munkres or Lang gives a regimented summary of his set-theoretic assumptions in axiomatic form: if they did so, what would it look like? Something like the original Zermelo set theory with urelements plus choice, I guess. But McLarty suggests it could look like ETCS, the ‘Elementary Theory of the Category of Sets’ developed by Lawvere. Thus he writes “[Lang’s] account simply says nothing that would distinguish between ZFC and ETCS, because nothing in his book depends on those differences.” 

But recall: 

In ETCS, no element is an element of two distinct sets.And a subset of X cannot be a subset of Y unless X = Y.In standard set theory a set can be an element of X and a subset of Y: in ETCS this can’t happen unless X = Y.In standard set theory any two sets X and Y have an intersection and union: not so in ETCS.In standard set theory an element of a set X can itself have elements: not so in ETCS.

And on it goes. Yes, nothing that Munkres or Lang says explicitly rejects the deviant ETCS line. But the plausible explanation is that it just doesn’t cross their minds as an option that needs to be countenanced in an introduction to mainstream set-theoretic ideas to beginning graduate students of topology or algebra. (Any more than they countenance NF-style stratification, say.) I’d bet that Munkres and Lang would vote the traditional ticket at least when it comes to those sorts of issues on which ETCS and standard set theories disagree.

Now true enough, ETCS can be neatly re-packaged in the manner of Tom Leinster in his well-known ‘Rethinking set theory’. Leinster’s aim is, as he puts it, to show that “simply by writing down a few mundane, uncontroversial statements about sets and functions, we arrive at an axiomatization that reflects how sets are used in everyday mathematics.” And Leinster is at pains to point out that the axioms of this theory, in his presentation, do not overtly involve any essentially categorial notions — they just talk about “sets” and “functions” (without reducing the “functions” to “sets”, by the way). He is indeed emphatic that this ETCS-style story about “sets” is, unqualifiedly, a theory about sets as ordinarily thought of by mathematicians. But given the discrepancies such as those noted above between ETCS and routine assumptions about sets, that is tendentious to say the least.

McLarty himself goes on to highlight some of the differences between ETCS and standard ZFC-style theories (with or without urlements): but unlike e.g. Mike Shulman, this doesn’t make him hesitate to call ETCS a set theory. Maybe that’s impolitic. Rather than that the distracting implication (in Lawvere too) that we’ve being doing our set-theory-for-ordinary-applications wrong, I’d say it is better to spin a positive message that a significantly different way of handling pluralities may serve some purposes better (more economically, with less redundancy).  

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Postcard from Amsterdam

A week in Amsterdam, mostly to meet up with the digital nomad Daughter, which was of course hugely enjoyable.

We have been staying a few hundred yards behind the Concertgebouw, in a quietly so-civilised area, with a great scattering of cafes and restaurants on almost every corner, and a tram-stop five minutes away. Need I say that the Amsterdam tram system is amazing? And as for the much-used bike lanes … It all makes the public transport and the provision for cyclists back in Cambridge seem rather depressingly shabby and second-rate.

Between family meals and wandering round the canals of the inner city, we have seen a lot of art, old and new!

It is a bargain to become friends of the Rijksmuseum, so we have been there a number of times (much better than trying to take in too much at once). I think we’d smuggle out Vermeer’s Little Street if we could only take home one painting.

The museum has been busy but mostly not so crowded as to spoil the experience. Except for all the idiots who push to the front, stand there fiddling around focusing their cameras or phones, snap and wander off, never actually looking at the “must see” painting in front of them, just the image on their camera screen …. Irritating and distracting or what? The rooms with earlier paintings (and carved wooden statues and altarpieces) are much quieter: here’s a rather wonderful Mary Magdalene by Crivelli.

We were also quite bowled over by the Anselm Kiefer exhibition split between the Van Gogh Museum and the Stedelijk Museum next door. Extraordinarily powerful. We had tickets for the last possible entrance time of the day, which still gave us two hours, and this meant that we had the galleries almost to ourselves by the end. I had to restrain myself from getting art books in other museums but, as Mrs Logic Matters agreed, the book accompanying this exhibition is hugely worth getting.

And last, but certainly not least, we have made a pilgrimage to The Mauritshuis in The Hague (taking the train, more fantastic public transport). We especially wanted to see three paintings which are so very familiar from reproductions, but which we had never encountered face-to-face before, Fabritius’s The Goldfinch, Vermeer’s View of Delft, and The Girl with a Pearl Earring.

The View of Delft was a revelation — in real life, much larger, more luminous than on any page. But of course. Can we smuggle that home too? And as for “The most famous girl in the world” as the museum call her, that tronie is just magical. As their notes remark, “Vermeer painted the girl without any hard lines. We call this technique sfumato … This means that all kinds of details are missing, such as the hook for the earring – Vermeer did not paint that at all. And where does her nose turn into her cheek? So you can see more than Vermeer actually painted: your brain fills in the details so subtly that you don’t even notice it happening. But this also means that everyone sees something different and that we each have our own Girl.” Which is a rather striking thought.

And then we met Rembrandt van Rijn, in perhaps the very last of his self portraits. So much life has been lived, such triumphs and sadness.

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Lea Desandre, Claude, Categories …

We have just booked tickets to see the wondrous Lea Desandre at Wigmore Hall on Tuesday 16 September, singing Dowland and Purcell. It should be an amazing concert. If you haven’t ever seen her perform then this candlelit concert from Rouen three years ago has recently re-appeared on YouTube: watch it while you can. (Louise Ayrton, the terrific first violin for Ensemble Jupiter in the film, who is also playing in the upcoming Wigmore Hall concert, read philosophy here at Trinity about ten years ago …)

I’ve been doing some proof-reading on the categories book, now with AI help from Claude as well as ChatGPT (I am temporarily subscribing to “pro” plans on both). In both cases I’m using a prompt like “Can you please proofread a LaTeX file for typos, ungrammatical English, and other mistakes. The file compiles correctly, so I don’t need comments on details of the LaTeX coding.” And then, after I’ve got a first suggested list of actual mistakes, I ask “Do you want to add any other, more stylistic, comments on that file?”.

I’m giving Claude and ChatGPT files that are late drafts which are already pretty polished. Claude is notably easier to work with and presents its feedback in a much more attractive and user-friendly way. While ChatGPT is astonishingly bad at actually accepting files to correct. Repeatedly I have uploaded a file, asked it to report back the chapter title in the first line, and it gets it wrong: it is obviously reading someone else’s file that is on category theory, but on different topics. ChatGPT is highly resistant to being told it is reading the wrong file and keeps telling me I must have uploaded an unintended file. Frustrating. So I have to chunk up a chapter, and simply paste portions into the chat window.

Those irritations apart, ChatGPT perhaps marginally shades Claude. But it is quite definitely worth using both because, except for gross typos like repeated or omitted words, there is almost no overlap in what they report as mistakes. Both over-report supposed errors; but both spot enough real infelicities to be pretty useful. (Their additional more “stylistic” suggestions in response to my second prompt are, however, in each case almost entirely unhelpful — too often, what I get are more like de-stylising suggestions of how to make the prose as banal or flat-footed as possible.)

As I read through again, checking a ridiculously long book from beginning to end, I find can live with the technical expositions and motivational chat, by far the bulk of the book. But I now realize that the author of some of the more conceptual/philosophical commentary in early chapters needs to have a good talk to the author of some of the later commentary. I’m going to need to do rather more work than I had hoped to make the text’s line consistent (overall more correct by my current lights). But first, a needed holiday to recharge the batteries.

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Schubert on Sunday 11: The Trout Quintet

A quite wonderful performance of the Trout Quintet by Patricia Kopatschinskaja, Anastasia Kobekina, and friends, made available by the Gstaad Digital Festival (to watch, you might have to register, but that’s free). Enjoy!

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Schubert on Sunday 11

A quite wonderful performance of the Trout Quintet by Patricia Kopatschinskaja, Anastasia Kobekina, and friends, made available by the Gstaad Digital Festival (to watch, you might have to register, but that’s free). Enjoy!

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