Peter Smith's Blog, page 10

Back, and in reasonable working order …

With huge thanks to the terrific (and uniformly kind) medical staff at the much admired Royal Papworth Hospital, here I am back home, after needed but non-urgent open-heart surgery, complete with a patched ascending aorta and a new heart valve. Some residual aching and annoying post-operative insomnia, but I was already able to walk normally half-a-mile each way to sit by the river, nine days after surgery. Which surely counts, in fact, as more than reasonable. So fingers crossed for a continuing rapid recovery. (But I don’t expect there will be logical posts here for a week or two!)

The NHS might have its real problems, especially in poorer regions of the UK: but for me here in Cambridge it has worked quite brilliantly. I am duly deeply grateful.

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A musical intermission

I’m going to be out of commission for a while,  as the date for planned surgery has rolled around. Hopefully, I’ll be back in reasonable working order next month. Meanwhile, let’s have a musical intermission, from two of my very favourite pianists from different generations.

First, the transcendentally great Maria Joāo Pires playing Beethoven’s Sonata in D minor, Op 32 ‘The Tempest’, a few years ago:

And then another real poet of the keyboard, the young Elisabeth Brauss, equally honest and utterly true to the music, playing Beethoven’s Sonata in E Major Op.109. Again wonderful:

“Playing the piano is not a mechanical act, but a spiritual journey,” wrote Schnabel, and it is surely so here in these two performances.

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Frege on seeing what is in front of his nose, revisited

There’s a new piece just published by Jamie Tappenden with the promising title ‘Following Bobzien: Some Notes on Frege’s Development and Engagement with his Environment’ (History and Philosophy of Logic https://doi.org/10.1080/01445340.2024...). But, for me a bit disappointingly, this turns out to be — yet once more — mostly about Frege’s engagement with, in a broad sense at any rate, philosophers. Yet didn’t the early Frege think of himself as a mathematician? So I’d certainly like to know more about his mathematical environment: for example, what texts on analysis was  Frege initially most directly acquainted with and how did these feed into and shape his work in Begriffsschrift?

A few years ago, I posted here a note relevant to this, and here it is again, slightly tidied up.

Take a mathematician of Frege’s generation, accustomed to writing the likes of

(1)

(2) If , then or ,

and much fancier things, of course!

Whatever confused thoughts about ‘variables’ people may or may not have had once upon a time, they had surely been more or less dispelled before the 1870s, if not by Balzano’s 1817 Rein analytischer Beweis (though probably that was not widely enough read?), at least by Cauchy’s great 1821 Cours d’analyse which everyone serious will have read.

Both Bolzano and Cauchy will take a claim like (1) to be true just when the equation holds for each value that x can take (each real number, as it might be, or each integer, say) — i.e. they clearly gloss such claims written with variables as claims holding for any value of x. Likewise, (2) will be glossed as saying that there are just two values of x, namely the stated ones, where the equation holds. The mathematicians of the day, at least when on their best behaviour, could be pretty decently clear about this. (Yes, reading Cauchy’s Cours, we will be very struck that after only a few pages we encounter “infinitely small quantities”: indeed the very first numbered theorem is about such quantities. So we are definitely in a pre-Weierstrassian world! But on the other hand, I don’t think we encounter anything that we should find too uncomfortable about variables and their values.)

But then it seems to be only the tiniest of steps to say outright that an ideal notation for such claims as (1) might explicitly have the form ‘for any value of , ’, and that such an explicit formulation is true when ‘’ is true whatever ‘’ might name. So — looked at from this angle — the wonder is not that Frege came up with his basic account of the logical form of expressions of mathematical generality like (1) but that no one had quite said as much before.

If you go back to Bolzano’s 1810 Beyträge, what happens there (with hindsight) seems very odd indeed. After all, here is someone who within a few years — in his 1816 Der binomische Lehrsatz and (even more) the 1817 Rein analytischer Beweis — is very clear indeed about variables and their use, when making essential practical use of quantification in talking about continuity etc. Yet in the earlier Beyträge, his Contributions to a Better-Grounded Presentation of Mathematics, when Bolzano turns to talking about logic and the principles of deduction, he looks quite antediluvian — and as far as I know he never revisited the logical basics in order to try to do better. OK, he says the mathematician will need more principles than we’ll find in the traditional syllogistic, and your hopes rise just for a moment: but the additional principles he comes up with are such a very limited and disappointing lot — the likes of A is an M, A is an N, so A is an M-and-N. It seems that the Bolzano of the Beyträge is — despite his disagreements and amendments — so thoroughly soaked in Kant and in broadly Aristotlean logic that he just can’t see what is right in front of his nose in the mathematician’s use of variables.

At least here, then, Frege’s originality — it might be said — is not depth of insight but his unabashed willingness to take the mathematician’s usage of the likes of (1) and (2) at bald face value, and to not try to shoehorn such generalizations into the canonical forms sanctioned by received logical wisdom. It is as if his now standard treatment of generality is a very happy result of Frege’s actually starting off knowing little of previous logic and philosophy.

I have often wondered, then, if Frege’s philosophical commentators have tended to see his basic discovery of a quantifier(-for-scoping)/variable notation as a more stunning discovery than it was. Take the pre-Frege usage of mathematician’s variables-of-generality at face value, without preconceptions (“don’t think, look” as Wittgenstein might say). Then note that we must e.g. distinguish generalizing a negation () and negating a generalization (it’s not true in general that ) — so if we are going to use a symbol for negation we are going to somehow have to mark relative scopes. And — implementation details apart — we are already more or less there!

Moreover, in Begriffsschrift and pieces written around that time, Frege’s concern seems explicitly to be very much with regimenting mathematical language (i.e. formalizing the logical bits of it to go along with the already common formal expressions we use for the non-logical bits, showing how adding the logical bits allows us to neatly cut down on the non-logical primitives by giving us the resources to define more complex concepts out of simpler ones, etc. etc.). He says remarkably little — except in using a few toy examples like the ‘Cato killed Cato’ one — about ordinary, non-mathematical, language more generally. So e.g. Dummett’s reading of Frege from the very beginning as aiming for a story about the real underlying logical structure of ordinary language generalizations is arguably considerable over-interpretation: Frege seems at least in Begriffsschrift to be much more in the business of giving us a somewhat tidied up replacement for informal ways of talking which is useful in regimenting science — one modelled closely upon, to borrow his phrase, the formula language of arithmetic.

Saying all this is of course not for a moment to underplay the depth of Frege’s reflections consequent on his discovery of the quantifier/variable notation! But let’s start slowly: I would very much like to know more about the history of maths, in particular on  how variables were regarded in early 19th century mathematics. I’d like to know more about Frege’s mathematical environment, and so understand better the nature of the clarificatory step Frege was taking in regimenting informal mathematical discourse in quantifier/variable style as he does.

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Summer reading?

The newspaper culture pages have been full of recommendations for summer reading. Let me add my two-pennyworth on books I’ve recently particularly enjoyed reading or re-reading.

I’ve just devoured Rory Stewart’s Politics on the Edge (in the US, How Not to Be a Politician: A Memoir). Though I’m rather glad I left it until after the election — it would have been rather too depressing a read had the Tories still been in power! But the book of his that I’d recommend to anyone who hasn’t encountered it before — and I only read it a few months ago — is his astonishing The Places In Between. “In January 2002, Rory walked across Afghanistan — surviving by his wits, his knowledge of Persian dialects and Muslim customs, and the kindness of strangers. By day he passed through mountains covered in nine feet of snow, hamlets burned and emptied by the Taliban, and communities thriving amid the remains of medieval civilizations. By night he slept on villagers’ floors, shared their meals, and listened to their stories of the recent and ancient past. Along the way Rory met heroes and rogues, tribal elders and teenage soldiers, Taliban commanders and foreign-aid workers. He was also adopted by an unexpected companion – a retired fighting mastiff he named Babur in honour of Afghanistan’s first Mughal emperor, in whose footsteps the pair were following. Through these encounters – by turns touching, confounding, surprising, and funny — Rory makes tangible the forces of tradition, ideology, and allegiance that shape life in the map’s countless places in between.” Quite beautifully written and absolutely absorbing. (But now I look, it seems to be out of print, temporarily I assume. So if you can’t stumble across a second-hand copy, can I also warmly recommend his The Marches: Border Walks with my Father?)

One of the art books I got the most out of recently is The Italian Renaissance Nude. You may well recognize the author’s name, as Jill Burke last year published the very readable, entertaining, and instructive How to be a Renaissance Woman: The Untold History of Beauty and Female Creativity. Her earlier book on the nude is terrific, enlightening, and — as a good art book should — really changing how you look at familiar works. But this is now also out of print, I guess permanently, and you won’t now find a copy second-hand shy of $300. Which is a great pity.

I also much enjoyed two books published in the last year, Martin Gayforth’s Venice: City of Pictures and Laura Freeman’s Ways of Life: Jim Ede and the Kettle’s Yard Artists (the second of course has a local connection, Kettle’s Yard now being the University of Cambridge’s gallery for modern and contemporary art).

But, most of all, I’m still enjoying Laura Cumming’s quite wonderfully written A Face to the World about self-portraits. Brimming with insights about art, portraiture, and human nature. Extraordinary. But perhaps too weighty a book — both literally and metaphorically — to take to the beach. There, you’ll surely want to read or re-read her slimmer, but no less wonderful, Thunderclap: A Memoir of Art and Life and Sudden Death. “On the morning of 12 October 1654, in the Dutch city of Delft, a sudden explosion was followed by a thunderclap that could be heard more than seventy miles away. Carel Fabritius – now known across the world for his exquisite painting, The Goldfinch – had been at work in his studio. He, along with many others, would not survive the day…” As Simon Schama puts it, [Cumming’s] pages are themselves lovely exercises in poetic vision and stay with you long after you finish.

I must try again with Emily Wilson’s The Iliad. Her introductions framing the epic, and explaining how she has approached the task of rendering into English are almost worth the price of entry in themselves. But I initially found her translation hard to take, and indeed ended up re-reading more of Christopher Logue’s War Music. I’ll return to the Iliad though when the nights draw in. Or so I tell myself.

I’ve read twenty-something novels so far this year, and to be honest the ones I’ve most relished are two by Rose Tremain which I’ve read before and which more than stood up to re-reading, Restoration (in which we meet the young medical student Robert Merivel finding favour at the court of King Charles II, rising and precipitously falling) and Merivel (in which Sir Robert finds and loses another life). Such an engaging flawed hero. The books are a real delight.

But the chance discovery I’d like to pass on — a serendipitous find in the wonderfully well-run Saffron Walden Oxfam bookshop — is Pereira Maintains: A Testimony by Antonio Tabucchi. We should have known about the Italian writer  Tabucchi, I suppose, but we didn’t: and we were both bowled over. “In the sweltering summer of 1938, with Lisbon in the grip of Portugal’s dictatorship of António Salazar, a journalist is coming to terms with the rise of fascism around him and its insidious impact on his work. Consumed by the passing of his wife and the child he never had, Pereira lives a quiet and lonely existence. One day, the young and charismatic Monteiro Rossi enters his life, changing everything. …” Philip Pullman wrote “Subtle, skillful, and clear. It’s so clear, in fact, that you can see a very long way down, into the heart of a flawed but valiant human being, into the sickness of a nation, into the depths of political evil. It’s the most impressive novel I’ve read for years.” Warmly recommended.

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And then there were five …

A printed copy of ICT has now arrived. So the fifth Big Red Logic Book — the longest yet — really exists!

I’m manfully restraining myself, at least for now, from looking at it too closely, because when I do — a pound to a penny! — I’ll immediately spot some silly typos. Why such foul-ups should leap off the page of a printed book but not from the home print-out of its PDF is one of life’s minor mysteries.

The print-on-demand quality from Amazon surely isn’t too bad at all for the price, though print density (if that’s the word) can vary, and text in Computer Modern can come out somewhat light and thin. Maybe I should swap to a version of Times, which would also slightly reduce the number of pages (but that swap isn’t quite as straightforward as it sounds). But still, the layout does on the whole does quite look attractive, methinks, though I could perhaps tinker with the size of some diagrams.

Drawing breath then, that’s IFL2 done in 2020, GWT at the very beginning of 2021, BML in 2022, then (after Part I came out in 2023), ICT finished in 2024. It would surely be tempting fate to say what comes next, but I do have fairly firm plans and still lots to say for a few years yet,  Deo volente. We shall see …

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How the continuum hypothesis could have been a fundamental axiom

Like many, I greatly admire Joel Hamkins’s terrific combination of technical prowess and expository ability as a mathematician. I’ve learnt a great deal from him. And I hope to learn more: he is promising us a book on ten ways of proving Gödelian incompleteness. Wouldn’t it be great, too, if one day he wrote an introductory book on forcing with his customary verve and insight?

I’m not always as convinced by his more philosophical pieces, but that’s philosophy for you. But I do think that his new approachable piece on “How the continuum hypothesis could have been a fundamental axiom” is particularly interesting and thought-provoking. In a close possible world in which Newton and Leibniz had said more about infinitesimals, there is a natural way our mathematics might have developed in which the continuum hypothesis would have come to be seen as axiomatic. Or so goes his story. You’ll find slides for a talk here.  And a preprint of quite a short paper on the arXiv here. Intriguing.

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The epigraph not used …

I was tempted, just for a moment, to preface all that abstract nonsense, all those higher ramblings, with

καὶ παίζειν ὅτε καιρός, ἐπαίξαμεν• ἡνίκα καιρὸς
οὐκέτι, λωιτέρης φροντίδος ἁψόμεθα.

When it was time for play, we played. Now that is no longer
we will apply ourselves to higher thoughts.

So writes Philodemus, the final two lines of one of his epigrams (AP 5.112). But I wonder who dips into the Greek Anthology these days?

Once upon a very long time ago, early in my Aberystwyth days, I found in Ystwyth Books (which still exists) an almost new copy of the Gow and Page edition of the Garland of Philip. I was tempted, and fell (how appropriate — falling into temptation being a recurrent theme of the epigrams!). And I have occasionally picked up the first volume — the poems and their (prose) translations — with pleasure, ever since. The second volume, though, is a rather stunning monument to old-fashioned scholarship, but not exactly light reading.

The Garland is the ancient anthology compiled by Philip of Thessalonica of epigrams, short poems from a period of about a century and a third, roughly 90 BC to 40 AD, one of the precursors of what we think off as the Greek anthology. As our editors tartly remark, “A few of the authors are of high quality, most are mediocre, a few are worthless”. But yet … the voices still echo down on the perennial verities, the brevity of life, the randomness of fate, the pleasures and follies of the human world, the need to seize the day:

Dead, you will lie under a yard of earth,
Far from daylight and all delighting,
So drain the cup, …

as Argentarius put it (as translated by Fleur Adcock).

And Philodemus of Gadara? He’s not just the author of the twenty nine epigrams — mostly decidedly racy — in Gow and Page (and perhaps another handful, as in David Sider’s more recent edition of The Epigrams Of Philodemos), but he is also an Epicurean philosopher of some note, a pupil of Zeno’s. A number of damaged but half-readable rolls of his philosophical writings were discovered when his patron Piso’s villa was excavated at Herculaneum. Philodemus wrote, among much else, On Methods of Inference discussing for example when it is appropriate to argue, as we would say, by induction from observed cases to new cases. He even gets an entry in the Stanford Encyclopaedia of Philosophy. A man of parts, then, who applied himself to higher thoughts!

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Introducing Category Theory — the paperback!

The headline news: there is now an inexpensive (but quite acceptably produced) paperback of Introducing Category Theory. Amazon-only to minimize cost, ISBN 978-1916906396: US $14.99, UK £10.99, DE €14.82, IT €14.40, etc.

I’m very sure this could be improved in all kinds of ways. As I say at the end of the Preface, the current text is quite certainly not set in stone. Indeed, think of it as a ‘beta version’, functional though surely not bug-free. But I need to pause work on it for a while, so I thought I’d get a paperback version out for those who prefer to work from one. All corrections and suggestions for improvement will continue to be very gratefully received.

The text of course remains freely downloadable as a PDF from the categories page. (The earlier paperback of Part I is, for the moment at least, withdrawn.)

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Introducing Category Theory — a full draft, at last!

There is now, at last, a full draft of Introducing Category Theory. You can download the PDF here.

The second half still needs more proof-reading and needs indexing. But I don’t envisage adding significantly to the content. After all — rather crazily for a book I didn’t originally set out to write — it is already 450 large format, small print, pages. Which is surely enough for an elementary, limited-ambition, introduction.

I hope, as a certain author once put it, it will provide interested readers with a ladder they can throw away after they have  climbed up it, now primed to tackle some of the standard books by real category theorists. (Though, unlike that author, I certainly don’t intend that  “anyone who understands me eventually recognizes [what I wrote] as nonsensical.”! Any nonsensical bits are plain mistakes.)

What’s the plan from here on? I want to complete the indexing pretty speedily, and do another proof-reading for the second half (though I seem to be increasingly bad at that!). And then I’m minded to promptly paperback it, though as a frankly acknowledged “beta version” with the expectation that I’ll certainly need to update it to correct typos and thinkos. But many readers will much prefer to work at least in part from a printed copy.

(I would have quite liked to have modest colour printing for some of the diagrams, to set off panels for theorems, etc. but that would more than double the book cost. The zero-royalties price for black-and-white Amazon print-on-demand, as with the other Big Red Logic Books, should be a tolerable £10.75, $14.90, €13.75 — still just the price of a few coffees.)

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Postcard from Aldeburgh

A few days away, staying in a familiar cottage near Aldeburgh, less than two hours from home. And the best weather we have ever had here. Usually, at least in our memory, it is windswept and intermittent rain: this time, warm sun and welcome breezes. So, favourite walks, and favourite places to eat. Very relaxing. We are fond of this place.

Late evening, after an early meal, walking back to the car, the splendid sight of the Aldeburgh lifeboat returning after, we guess, a training exercise, pulled up the beach by a tractor in a remarkably primitive-seeming way.

Another day, and then, batteries recharged, it must be one last push on with the blasted category theory book.  Anything to distract from the dire state of the political world. …

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