This and that

Another year, another Christmas card. Like last year, this is a picture to be found down the road at the Fitzwilliam Museum, a Nativity long attributed to Bastiano Mainardi, a member of Domenico Ghirlandaio’s workshop and his brother-in-law. But latterly, the Fitz has claimed to see in it the hand of the master himself. Though I do wonder!

Ghirlandaio has been in my mind recently, in part prompted by reading Paola Tinagli’s Women in Renaissance Art (which, among much else, has interesting things to say about the iconography of the great frescos in Santa Maria Novella). And I was prompted to read that book by the references that Maggie O’Farrell gives at the end of her wonderful novel The Marriage Portrait.

The portrait is of the young Lucrezia de’ Medici. Here she is:

Lucrezia is taking her seat at the long dining table, which is polished to a watery gleam and spread with dishes, inverted cups, a woven circlet of fir. Her husband is sitting down, not in his customary place at the opposite end but next to her, close enough that she could rest her head on his shoulder, should she wish; he is unfolding his napkin and straightening a knife and moving the candle towards them both when it comes to her with a peculiar clarity, as if some coloured glass has been put in front of her eyes, or perhaps removed from them, that he intends to kill her. She is sixteen years old, not quite a year into her marriage.

How can you not read on? By some way, my favourite novel of this last year. So that’s my warm recommendation of a book to treat yourself to this Christmastide!

Distracting fictions of one kind or another have been necessary more than ever, no? As Eliot says, “Humankind cannot bear very much reality.” To mention just one reality befalling the world, who would have predicted, twelve months ago, that we would be three hundred days into a simmering land war in Europe? Grim times.

But just for a few days let’s try not to dwell too much on that! May you and yours indeed have a very happy Christmas, and I earnestly hope your New Year is a peaceful one.

One episode in the Beginning Mathematical Logic Study Guide which I must radically revise in the next edition is the final section on type theories, which was tacked on almost as an afterthought. But it will, indeed, take quite a bit of work to organize a better overview of what is a messy area, and devotees of varieties of type theory are not always the clearest of advocates to help us along.

Egbert Rijke has an Introduction to Homotopy Type Theory coming out soon with CUP. This is a textbook aimed at quite a wide readership; “it introduces the reader to Martin-Löf’s dependent type theory, to the central concepts of univalent mathematics, and shows the reader how to do mathematics from a univalent point of view. … The book is entirely self-contained, and in particular no prior familiarity with type theory or homotopy theory is assumed.” Which sounds promising. And a late pre-print can now be downloaded: here’s the link. This may very well be very useful to some readers of this blog, depending on your background. And all credit to Rijke for making his text freely available on the arXiv.

The first part of the book is, as announced, on Martin-Löf’s version of type theory. I’ve dived into this 98 page introduction hopefully. But I can’t, to be honest, say that I have wildly enjoyed the experience — Rijke makes it no easier than others for a conservatively-minded logician to happily find their way in. He acknowledges “Type theory [or at least, type theory of this stripe] can be confusing for people who are new to the subject,” but this means that many of us could do with rather more explanatory chat than we get here. For a small but not insignificant example, right at the outset we are told without further ado that “The expression 𝑎 : 𝐴 is … not considered to be a proposition, i.e., something which one can assert about an arbitrary element and an arbitrary type, but it is considered to be a judgment, i.e., an assessment that is part of the construction of the element 𝑎 : 𝐴.” Is the distinction between a proposition and a judgement transparently clear to you? No. Me neither. (Amusingly, I asked ChatGPT to give a simple explanation of the difference between propositions and judgements in Martin-Löf, and it was a lot clearer. On its first attempt. Though it lost the plot a bit when the question was re-asked.)

Well, meaning is use and all that, and eventually the mists clear somewhat as the notions of proposition and judgement get used later in Rijke. More generally, if you have in fact already encountered a bit of type theory, his explanations will probably serve well for revision and consolidation. But we still await (OK, I still await) a really introductory text on dependent-type-theory-for-old-fashioned-logicians.

A small package of hardback copies of Gödel Without (Too Many) Tears has just arrived. One copy I’ll give to the Moore Library here (the university’s main library for the mathematical sciences); another will go to my long-ago college (once a Trinity mathmo, always a Trinity mathmo). A third will be sent off to Boston Spa, because you are legally required to deliver a copy of any book you publish to the British Library.

I suppose it quite likely that this third copy will disappear into the “Additional Storage Building” there, soon never to be seen again. In his strangely rambling but sporadically gripping Bibliophobia, Brian Cummings sets the scene:

From high in the roof, the book robot swings down in an arc in a vertical plane, 10 m or more in a single movement, between stacks ranking 20 m high. It pauses, chooses a stack, then hovers, humming all the time, hunting for the book that it is programmed to find. The scale of the building is difficult for a human to take in. The void is 24 m high by 24 m wide by 64 m long. In any case, the room is not designed for humans. Oxygen levels are kept at 14–15 per cent, which is similar to trying to breathe at the top of a Himalayan mountain. Visitors watch from a special cage, advised to leave after fifteen minutes for their own safety. This library is not a human environment, for it is designed for habitation by books and robots alone.

This is a library for the twenty-first century. … Since space at the main site [of the British Library] at St Pancras in London is at a premium, it is proposed to house books 262 km away and transport them between the Reading Rooms in both locations. Eventually, it is planned that seven million items will be held at the Additional Storage Building in Boston Spa. All of this makes logical sense …

The robotic crane adds a volume to a pile that it is assorting in a plastic bucket. In time, it delivers this to an airlock at the end of the room. There are 140,000 bar-coded containers. It is only at this moment that human intervention takes over, as staff retrieve items from the airlock to send to Reading Rooms, a maximum of 48 hours from ordering to arrival at a London desk. Yet if a human librarian wanted to enter the vault to retrieve a book, using either gigantic ladders or high-wire trapeze artists, it would not be feasible. The books are no longer on fixed shelves ear-marked for their location. Only the crane knows where the books are. It never puts a book back as it found it, absorbing the used item back into the system in the order in which it comes, then remembering where it was. The books are engaged in an eternal game of musical buckets, finding new neighbours, and slotting in accordingly. Only humans need shelf marks any longer: the shelves have gone. The retrieval system is fully automated …

Logical, but the stuff of Borgesian nightmare, squirrelling away everything in a limitless library, including the unread and the unreadable, published in their tens of thousands a year. Libraries, you feel (well, I feel), should be places of more homely comfort — whether in a private room or two with friendly familiars on the walls, or in a congenial public space for browsing along the shelves and reading and working with others similarly occupied.

I’m sure no one would notice if GWT2 never arrived at Boston Spa. But duty calls.

The non-fiction, non-logic, book which I have most enjoyed this year is the one I have just finished, Edmund de Waal’s remarkable The White Road (from 2105). I was enticed by the jacket description — the author “sets out on a quest – a journey that begins in the dusty city of Jingdezhen in China and travels on to Venice, Versailles, Dublin, Dresden, the Appalachian Mountains of South Carolina and the hills of Cornwall to tell the history of porcelain. Along the way, he meets the witnesses to its creation; those who were inspired, made rich or heartsick by it, and the many whose livelihoods, minds and bodies were broken by this obsession. It spans a thousand years and reaches into some of the most tragic moments of recent times.” It is indeed a quite remarkable read, intriguing and then — when it touches on some of the human disasters of the last century — distressing and moving. As you would expect from The Hare with Amber Eyes or Letter to Camondo, it is quite wonderfully well written — de Waal’s often short, beautifully crafted, paragraphs as carefully arranged as those white porcelain pots of his in their vitrines. Which makes the later pages all the more telling.

I hadn’t expected to bump into Leibniz on the road that takes us to seventeenth century Chinese porcelain and the attempts of the Jesuit missionaries to discern its secrets. But then I didn’t know that one of the few books Leibniz published in his lifetime was the Novissima Sinica — the very latest news from China! Unlike a Descartes who thinks that what is most fundamental is common to us all and is available through rational reflection, Leibniz is committed to the idea that the growth of knowledge will come from bringing together different perspectives, different ways of thinking, about our shared universe. So he has an extensive correspondence with the Jesuit mission, and (to quote de Waal) writes that

all this activity around China … is ‘un commerce de lumière’; enlightenment stretching both ways. This is a tremendous idea and a beautiful image, one of an equality of concerns, a correspondence of civilisations, of light.

As Leibniz writes to his friend Sophie, the wife of the elector of Hanover:

I will thus have a sign placed at my door with these words: bureau of address for China, because everyone knows that one only has to address me in order to learn some news. And if you wish to know about the great philosopher Confucius … or the drink of immortality which is the philosopher’s stone of that country, or some things which are a little more certain, you only have to order it.

He is one of the gatekeepers. If you want to know about Chinese mathematics, the I Ching as a coding of chance events, Chinese characters and their relationship to hieroglyphs, you go to him. Leibniz has been to visit Father Francesco Grimaldi in Rome, just back from the emperor Kangxi’s court, and written up copious notes on fireworks, glass and metal. I realise to my surprise that my hero, the father of rationalism, is anxious to keep ahead in this new, congested field of China Studies.

So Leibniz knows about the problem of porcelain. Later, he corresponds with the mathematician Ehrenfried Walter von Tschirnhaus, who has a sideline experimenting on how to create porcelain here in Europe, and almost gets there. And the extraordinary story continues, that was all new to me, as I followed The White Road. 

Twitter could be pretty informative and entertaining, and I’d occasionally tweet posts myself. But the Elon Musk takeover seems to be Very Bad News on a number of fronts. So, like many, I’m backing off for the time being, though I’ve not actually deleted my account yet — the Musk era might or might not last. Hence I’ve removed the Twitter feed from the footer that shows on Logic Matters pages (when using a computer or tablet browser).

Instead, I’ll post occasional tweet-sized music-related snippets in the same place — about new CDs, old CDs I’ve been re-listening to, YouTube video links, etc.

Again like many, I’ve joined a Mastodon server, and there’s now a link in the footer to @PeterSmith@mathstodon.xyz. I’m seeing more mathsy stuff and fewer cats and otters. Still exploring who to follow in the wider fediverse. But so far, I’m enjoying the atmosphere and the occasional distractions.

“To thee, thou glorious son of worth”, from the new CD of music by Handel with the quite stellar Lea Desandre, Iestyn Davies, and Thomas Dunford’s Jupiter ensemble. Just wonderful.

A break from logical matters, away for half-a-dozen busy days in Athens, followed by visiting family on the island of Rhodes for a week. Both most enjoyable in very different ways. Then we needed a holiday to recover …

But I’m back down to business. The first item on the agenda has been to deal with some very useful last comments on the draft second edition of Gödel Without (Too Many) Tears, and to make a start on a final proof-reading for residual typos, bad hyphenations, and the like. I hope the paperback will be available in about three weeks. If you happen to be reading this while thinking of buying the first edition, then save up your pennies. The second edition is worth the short wait (and will be again as cheap as I can make it).

I’m still finding the occasional slightly clumsy or potentially unclear sentence in GWT. I can’t claim to be a stylish writer, but I can usually in the end hit a decently serviceable level of straightforward and lucid prose. But it does take a lot of work. Still, it surely is the very least any author of logic books or the like owes their reader. I certainly find it irksome — and more so with the passing of the years — when authors don’t seem to put in the same level of effort and serve up laborious and uninviting texts. As with, for example, Gila Sher’s recent contribution to the Cambridge Elements series, on Logical Consequence. 

Mind you, the more technical bits have to fight against CUP’s quite shamefully bad typesetting. But waiving that point, I really have to doubt that any student who needs to have the Tarskian formal stuff about truth and consequence explained is going to smoothly get a good grasp from the presentation here. And I found the ensuing philosophical discussion quite unnecessarily hard going. And if I did, I’m sure that will apply to the the intended student reader. So I’m pretty unimpressed, and suggest you can give this Element a miss unless you have a special reason for tackling it.

Another book which readers of this blog will probably want to give a miss to is Eugenia Cheng’s latest, The Joy of Abstraction:An Exploration of Math, Category Theory and Life (also CUP). This comes garlanded with a lot of praise. I suppose it might work for some readers.

But the remarks supposedly showing that abstract thought of a vaguely categorial kind is relevant to ‘Life’ are embarrassingly jejune. The general musings about mathematics will seem very thin gruel (and too often misleading to boot) to anyone who knows enough mathematics and a bit of philosophy of mathematics. Which leaves the second half of the book where Cheng is on much safer home ground “Doing Category Theory”.

So I tried to approach this part of the book with fresh eyes and without prejudice, shelving what has gone before. But, to my surprise, I found the level of exposition to rather less good than I was expecting (knowing, e.g., Cheng’s Catsters videos). She is aiming to get some of the Big Ideas across in an amount of detail, and I was hoping for some illuminating “look at it like this” contributions — the sort of helpful classroom chat which tends to get edited out of the more conventional textbooks. But I’m not sure that what she does offer works particularly well. Try the chapter on products, for example, and ask: if you haven’t met the categorial treatment of products before, would this give you a good enough feel for what is going on and why it so compellingly natural? Or later, try the chapter on the Yoneda Lemma and ask: would this give someone a good understanding of why it might be of significance? I’m frankly a bit dubious.

So that’s a couple of recent CUP books that I did acquire, electronically or physically, and am sadly not enthused by. But in their bookshop there is another new publications which looks wonderful and extremely covetable, a large format volume on The Villa Farnesina. On the one hand, acquiring this would of course be quite disgracefully self-indulgent. On the other hand …

I confess to have given little thought in the past to questions of just when objects of problematic provenance in our museums should be repatriated. But, better late than never, I realize I can’t conjure any cogent reason why the “Elgin Marbles”, the Parthenon Frieze and the rest, shouldn’t now be returned by the British Museum and displayed in the beautiful Acropolis Museum. That museum, as we found last week, is already worth a trip to Athens in itself, and the huge gallery waiting for the originals of the rest of the frieze is just stunning. Time the marbles went home.

A last call for comments/corrections (please!!) for the draft second edition of Gödel Without Tears — I plan to finalize and publish a paperback version around the end of the month. You can download the draft here (though I imagine that anyone interested will have done so by now).

I’ve mentioned before that my three hundred pages of introductory notes on category theory are downloaded surprisingly often — frequently enough for it to be rather embarrassing, given their current ramshackle state. So, with GWT2 simmering on the back burner while I wait to see if there are any last minute suggestions to deal with, I’m getting back to thinking a bit about categories.

I was, for a while, stumbling over two things when thinking about how to revise/develop the notes. Firstly, I didn’t have a clear enough conception of where I wanted to get to.  Secondly, I’ve become increasingly unhappy with the way the very opening chapters are handled (with those distracting sermons about set theory!). But I think that things are now falling into place rather better.

On the question of scope, of where to finish, I’m lowering my sights a bit. I had occasion, the other day, to be looking at the classic book on topos theory by Mac Lane and Moerdijk. It starts with a scene-setting fourteen pages of “Categorical Preliminaries” — a glorified checklist of what you need to bring to the party if you are planning to dive into the book. And that checklist more or less exactly corresponds to the topics sort-of covered (in rushed way towards the end) in the existing notes. So that’s persuaded me that maybe, after all, the notes do get to a sensible enough stopping point (and perhaps only need be rounded out with some brief pointers to routes onwards).

And on the question of how to start, I’ve decided that fussing at the outset about such issues as whether we should identify functions with their graphs just doesn’t make for a happy beginning. That’s largely got to go! But this makes for quite a bit of fiddly re-writing over the initial chapters.

I’m having a family break for a couple of weeks, so the new version of the first seventeen chapters or so won’t be ready for a few weeks. But I’m feeling decidedly cheerier about the project of improving those notes, at least enough for me to rest fairly content with the unambitious result.

The wider world continues to go mad and/or bad in various depressing ways. The most distractingly enjoyable novel I have read just recently? Perhaps Elspeth Barker’s atmospherically gothic O Caledonia. I dived in because of an enthusiastic recommendation by Maggie O’Farrell. I enthusiastically pass on the recommendation!

I have also been much distracted by Edmund de Waal’s The White Road, swept along by his obsession with porcelain and its origins (with walk on parts for Spinoza and Leibniz by the way). Strangely gripping I find!

Such a wonderful writer. The Wolf Hall trilogy is the extraordinary work of our times, that only strikes you as all the greater on rereading. And the many touching tributes to Hilary Mantel’s human qualities make her untimely death seem all the sadder.