This and that

If you have downloaded a PDF copy of one of the Big Red Logic Books, found it useful, and have been thinking you might buy a printed paperback (or even a hardback), then now might be the moment to do so!

Amazon have announced that the printing costs of their KDP print-on-demand service will go up next month (for the first time in six years, apparently: whatever we think of Amazon, they haven’t been price-gouging on this service). Since I price the paperbacks more or less at cost to me, rounding up just a bit, it could well be that the price I have to charge will need to increase. So hurry, hurry, while the lower prices remain!

IngramSpark who print the hardbacks of IFL and GWT have already raised their printing prices (for the UK and EU in particular) to the point I have been losing money on each copy of IFL. So those prices have already had to go up by a little. But I note that — today at any rate — Amazon UK are still selling the hardback of IFL at the old price of £20 (as against £22.50). And they have an offer on the hardback of GWT, selling that for £9.75 (instead of £15), which is a bargain, though I say so myself!

I’ve reached the point where I can seriously think about another Big Red Logic Book, a paperback version of Category Theory I (it come in a bit over 200 pages, but I hope under £5). I’ll try to put it out in the world by the end of next month as a minimal-cost sort-of-beta-version, alongside the freely downloadable PDF, since a substantial number of people do find it much nicer to work from a printed copy. And indeed, if you have a properly designed PDF, it is almost no work to set up another KDP print-on-demand book. Hopefully making the draft available in hard copy will help increase the quantity of feedback/comments/corrections I get. When that paperback is sorted, I’ll then move on to having a serious bash at Category Theory II. Oh what fun!

But logical matters have been going much more slowly than I’d hoped since we got back from Perugia, with the after-effects of Covid taking its toll. Things have thankfully improved quite a bit in recent days, however, and — fingers crossed — life is returning much more to ‘normal’. So, in particular, I hope I’ll be able to hit that self-imposed end-of-June deadline. We’ll see …

There can only be one choice of musician this week, Menahem Pressler who died on 6th May at the age of 99. He played with the wonderful Beaux Arts Trio for 53 years (leaving a stunning series of recordings of the core repertoire). And when they eventually disbanded, and Pressler was 84, he launched another career as a soloist.

We heard him a few years later at a very memorable concert in the intimate Peterhouse Theatre in Cambridge, when he played Beethoven’s A-flat major sonata, Op. 110, Debussy’s Estampes, and then Schubert’s last piano sonata D. 960. As I wrote at the time “He talked touchingly at the beginning of the evening, and this was evidently music that meant a great deal to him. Pressler’s … desire to communicate with his audience is undimmed. The Schubert in particular was very affecting: in the second movement, the poignancy of an old man now 87 playing the searing music of a young man facing early death was almost too much to take.”

The photo here is of him with Elisabeth Brauß, characteristic of them both. And Pressler was by all accounts a wonderful teacher and encourager of young players, and held in much affection.

What to choose of his to watch and listen to again? There is a wonderful too brief excerpt here of the Beaux Arts playing the slow movement of the Schubert B flat Trio — their CD recording is so fine. But perhaps for sheer joy, I have to pick this performance he gave with Quatuor Ébène at his 90th birthday celebration, of Schubert’s Trout Quintet. What a delight.

Another birthday gone. It is, as the t-shirt has it, weird being the same age as old people.

As a present for myself, I sent off weeks ago for the catalogue for the Rijksmuseum Vermeer exhibition, but only opened it on the day. And I’m very glad that I hadn’t instead dropped heavy hints to Mrs Logic Matters that I would really like a copy, because it would have been difficult to hide my initial disappointment.

Many catalogues of major exhibitions are wondrous art-objects in their own right. But this one so strangely isn’t.

Some complaining Amazon reviewers have put this down to the use of a matt paper, but actually I think that’s a mistake. Putting pages side by side with the reproductions in other books on Vermeer we have then — when not placed so as to catch reflections — you can in many cases hardly tell the matt and the slightly glossy pages apart. True, independently of the matt/gloss finish, in a number of cases the colours in this volume seem slightly dulled. But maybe these reproductions are in some sense truer to the paint surface; we have just become so used to seeing paintings rather artificially glowing in modern gallery lighting.

No, I don’t think it is principally the exact level of colour reproduction which makes this catalogue unsatisfactory as a visual object. Rather, there is a very odd choice of a dark, modern, sans serif font. So instead of the extensive blocks of text receding with quiet grace, as in the lovely catalogue for the Fitzwilliam’s Vermeer’s Women a few years ago, we get page after page visually shouting (and with the print overwhelming the inset — and oddly small — supplementary illustrations). If this was a catalogue for an exhibition by Kandinksy, say, it might have worked excellently. But for Vermeer of all artists? It simply looks off-puttingly uncomfortable.

As for the texts themselves — mostly essays on different groups of paintings, written by many hands — some are a bit banal, but mostly they are engaging and illuminating. Which makes it all the more a pity that the mode of typographical presentation is, to my eyes, so very misjudged. I’ve got a bit more used to the look of the book: but it’s not fontastic!

I intended to return from Perugia full of renewed energy, imagination, and zest. It hasn’t quite worked out like that. We have Covid. So far, it’s largely a matter of massive fatigue. Let’s hope that’s the worst it gets. So perhaps not much here on logical matters for a week or two.

It’s the blog’s birthday! Let’s party!!

The very first post was back on March 9th, 2006 (and I have just restored the link there to a short talk on Gödel’s theorem for mathematicians which I’d quite forgotten about ). Since then there have been — rather astonishingly — 1630 posts.

Of course, many of these were of merely passing interest (if that). But over the years, there have been some blog posts, or series of posts, of perhaps less ephemeral interest. So I’ve decided to rework the webpage that used to be unhelpfully labelled ‘Varia’ to become more of an ‘Archive’. I plan over the coming weeks not only link to a few papers, book reviews, and other pieces, but also to ‘best bits’ from the blog. It will take a while to organize this; but I’ll hopefully enjoy finding out what earlier time-slices of me thought!

Heavens: I’ve just realized that if I start the clock when I got my first job, the blog has been going for about a third of my academic life. Doesn’t time fly when you are having fun.

Long ago, I posted a piece here under the mournful title “Logic disappearing over the horizon ….”.

I’ve just had an invitation to give a talk at the University of X, a distinguished place, with a philosophy graduate community of about fifty (according to their website). So I checked out how much logic/phil maths is going on, what I could reasonably take as given. Zilch. Apart from a first year course perhaps approaching the level of my intro logic book, nothing at all, as far as I can tell. Which leaves me a bit bereft of anything to go to talk about. But more to the point, it means that for students at X a central swathe of the work of lasting value from the last hundred years has disappeared over the horizon. Which is, shall we say, a pity.

My sense is that this is happening more and more in UK universities. I’d be delighted to learn that I’m wrong.

There was some discussion on the blog at the time, not very cheering. And my current sense is that the situation is getting worse and worse. How many logic-orientated posts in philosophy departments have been advertised and filled in the UK in the last dozen years? Very few, as far as I know.

And things are just as grim, if not more so, with the philosophy of mathematics. Here is Jeremy Avigad, in a recent essay which I’ve just noticed:

A recent analysis of tenure-track positions advertised in Jobs for Philosophers in the 2021–2022 academic year doesn’t even mention philosophy of mathematics in its categorization. Digging into the data shows that the phrase “philosophy of mathematics” occurs in only three of the 201 advertisements, in each case listed among multiple areas of potential interest. Surely this is an indication that the field is no longer viewed as important. It is sad that a discipline that was so central to the philosophical tradition from ancient times to the middle of the twentieth century now barely registers a pulse.

An Avigad now would, it seems, have a pretty hard time getting an academic post in philosophy. Sad indeed.

Is it always the fate of those in their declining years to think important bits of their world are falling apart? Probably so. But we’re not always wrong to thing so ….

I thought that this piece on chatbots from New York magazine was worth reading and thought-provoking.

I was amused/alarmed by one Christopher Manning who seems to think we are stochastic parrots, and “the meaning of a word is simply a description of the contexts in which it appears.” Really? Apparently, the idea that meaning has anything to do with how words hook up to the world and to our interactions with the world and each other is antiquated, the “sort of standard 20th-century philosophy-of-language position.” Well, that’s us told.

I was pointed to this piece by a post on Mastodon. I still occasionally look at the old bird site because that’s where posts about music and other cultural stuff still are mostly to be found. But these days, quite apart from not wanting to have too much to do with Musk Enterprises Inc., the more genial atmosphere of Mastodon, on my instance anyway, suits me fine.

I asked, “What is the mistake in Gödel’s faulty proof of the incompleteness theorem?”

Gödel’s incompleteness theorem is a fundamental result in mathematical logic that states that no consistent formal system that includes arithmetic can prove all true statements about arithmetic. Gödel’s proof of this theorem is considered one of the most important results in the history of mathematics.

However, Gödel did make a mistake in an earlier attempt to prove the incompleteness theorem, which is often called Gödel’s faulty proof. In this earlier attempt, Gödel attempted to show that a consistent formal system that includes arithmetic must be incomplete by constructing a statement that is both true and unprovable in the system.

The mistake in Gödel’s faulty proof was that he used a self-referential statement that was not well-formed in the system. Specifically, the statement asserted its own unprovability in the system, but the system could not handle statements that referred to their own provability or unprovability.

In his later and correct proof of the incompleteness theorem, Gödel used a different technique to construct a statement that is true but unprovable in the system. Instead of using a self-referential statement, he used a statement that essentially said “this statement cannot be proved.” This statement is well-formed in the system and is true if and only if it is unprovable in the system. By constructing such a statement, Gödel showed that the system is incomplete.

I was intrigued! So I followed up “Where did Gödel give his earlier faulty attempt at a proof?”

Gödel’s earlier attempt at a proof of the incompleteness theorem, which is often referred to as “Gödel’s first incompleteness theorem,” was presented in a series of lectures at the Institute for Advanced Study in Princeton, New Jersey in 1931. In these lectures, Gödel attempted to prove that any consistent formal system that includes arithmetic must be incomplete by using a self-referential statement that was not well-formed in the system. However, after recognizing the mistake in his proof, he corrected it and presented a different proof of the incompleteness theorem in his landmark paper “On Formally Undecidable Propositions of Principia Mathematica and Related Systems” published in 1931. This corrected proof is now referred to as “Gödel’s second incompleteness theorem.”

Really? News to me! “Who first showed that Godel’s earlier attempted proof was wrong?”

The mistake in Gödel’s earlier attempted proof of the incompleteness theorem was first pointed out by the American mathematician and philosopher Ernest Nagel. In 1934, Nagel and his collaborator James R. Newman published the book “Gödel’s Proof,” which presented a detailed and accessible explanation of Gödel’s second incompleteness theorem, as well as a critique of Gödel’s earlier attempted proof. They argued that the proof was flawed because it relied on a self-referential statement that was not well-formed in the system, and they provided a detailed analysis of the mistake. Their book helped to popularize Gödel’s incompleteness theorems and made them accessible to a wider audience outside the specialized field of mathematical logic.

Hold on! “But if, as you said, Gödel’s mistake was first found in 1934, how did Gödel give a correct proof in 1931?”

I apologize for the confusion in my previous response. To clarify, Gödel did not give a correct proof of the incompleteness theorem in 1931. In his original 1931 paper “On Formally Undecidable Propositions of Principia Mathematica and Related Systems,” Gödel presented a correct proof of the first incompleteness theorem, but this was after he had recognized and corrected the mistake in his earlier attempted proof.

Gödel’s earlier attempt at a proof of the incompleteness theorem, which is often referred to as “Gödel’s first incompleteness theorem,” was presented in a series of lectures at the Institute for Advanced Study in Princeton, New Jersey in 1930-31. This proof was flawed, as it relied on a self-referential statement that was not well-formed in the system. After recognizing the mistake in his proof, Gödel corrected it and presented a different proof of the incompleteness theorem in his 1931 paper.

So, to summarize, Gödel’s incorrect proof was presented in his lectures in 1930-31, and his corrected proof was presented in his 1931 paper. Ernest Nagel and James R. Newman’s critique of Gödel’s earlier attempted proof was published in their book “Gödel’s Proof” in 1958, long after Gödel had presented his corrected proof.

You will recognise the dunderhead bullshit artist that is ChatGPT. But I hate to think how very many minds are going to get addled by this sort of farrago. Or how very many hours are going to get wasted by teachers wading through course work sprinkled with excerpts. Reason number 42 for being quite glad to be retired and out of the game.

It is an odd business writing books (other than research monographs which get reviewed in the journals and perhaps with luck discussed by a few colleagues).

You send your best efforts out into the wider world to take their chances and have precious little idea of how they are really received. For example, the second edition of IFL is downloaded over 10K times a year. I don’t think one reader in a thousand lets me know how they found the book.

So I take comfort and encouragement where I can! And I’ve just spotted a new review on Amazon for Godel Without Too Many Tears (the first for its second edition, though the review is linked to the original edition, which I’ll try to get changed):

Clearly the best concise introduction to Gödel’s theorems ever written. I bought the first edition and found in a very few places errors and margin for a more felicitous presentation. Nothing, however, that could stump an astute reader. In the second edition … this is all corrected. The book really does accomplish the miracle of being self-contained, though, of course, a reader may come up with questions not covered in the book. In that case it may be helpful to consult the author’s more complete treatment in “An Introd. to Gödel’s Theorems” (also cheaply available from Amazon) or a book like Boolos, Burgess & Jeffrey.

I recommend the book not only to students but also to academic teachers. It is a model of how logic ought to be taught.

I can very happily live with that!