Peter Smith's Blog, page 76

Button & Walsh: Philosophy and Model Theory

It is always pleasing to be able to warmly recommend a new book. So let me enthusiastically draw your attention to this newly published book by Tim Button and Sean Walsh, Philosophy and Model Theory, just published by OUP. (The pbk is £30 — but this is xvi + 517 larger-format and action-packed pages, so we certainly can’t complain!)

This is a unique book, both explaining technical results in model theory (eventually at a pretty non-trivial level), and exploring the appeals to model theory in various branches of philosophy, particularly philosophy of mathematics, but in metaphysics more generally (recall ‘Putnam’s model-theoretic argument’), the philosophy of science, philosophical logic and more. So that’s a very scattered literature that is being expounded, brought together, examined, inter-related, criticised and discussed. Button and  Walsh don’t pretend to be giving the last word on the many and varied topics they discuss; but they are offering us a very generous helping of first words and second thoughts. It’s a large book because it is to a significant extent self-contained: model-theoretic notions get defined as needed, and many of the most philosophically significant results are proved.

The book only arrived yesterday, so at this point I have to report just having read the opening four chapters (two carefully, two more quickly) and dipped very speedily in and out later in the book. But it seems to me that — in fact, as you’d expect from these authors — the expositions of the techie stuff is quite exemplary (they have a good policy of shuffling some extended proofs into chapter appendices), and the philosophical discussion is done with vigour and a rather engaging style. The breadth and depth of knowledge brought to the enterprise seems to be remarkable.

So first impressions: this book looks as if it is an outstanding achievement. Logic-minded philosophers should find it fascinating; and — with judicious skimming/skimming (the signposting in the book is excellent) — so should mathematicians with an interest in some foundational questions. Make sure it is in your university library (and at that price, in your library!).

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On Frege seeing what is in front of his nose

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

(1)

(2) If , then or ,

— and fancier things, of course!

Whatever unclear thoughts about ‘variables’ people may or may not have had once upon a time, they have surely been dispelled well before the 1870s, if not by Balzano’s 1817 Rein analytischer Beweis (though perhaps that was not widely enough read?), at least by Cauchy’s 1821 Cours d’analyse which everyone serious will have read. Both Bolzano and Cauchy take claims like (1) and (2) to be true just if each instance is true, plain and simple, and clearly gloss various claims written with variables as claims holding ‘for any value of x’. Maybe it is worth noting, then, that the mathematicians of the day, at least when on their best behaviour, could be very decently clear about this!

But then it seems to be only the tiniest of steps to say outright that an ideal notation for such claims might have the form ‘for any value of , ’, and that such an explicit form is true when ‘’ is true whatever ‘’ names. 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) and (2), 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 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 wisdom. It is as if his now standard treatment of generality is a very happy result of Frege’s knowing rather little of previous logic and philosophy.

I have often wondered, then, if Frege’s philosophical commentators have seen his basic discovery of a quantifier(-for-scoping)/variable notation as more exotic than it is. 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 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 we are already more or less there!

Now, in Begriffsschrift and pieces written around that time, Frege’s concern seems to be very much with regimenting mathematical language (formalizing the logical bits of it to go along with the 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 structure of ordinary language generalizations is arguably considerable over-interpretation: Frege seems at least in Begriffsschrift to be more in the business of giving us a tidied up replacement for informal ways of talking, useful in regimenting science — one modelled, to borrow his phrase, upon 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 I would very much like to know if there are any discussions in the history of maths or history of logic literatures on the how variables were regarded in 19th century mathematics.

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Names and quantifiers

‘Socrates is a philosopher’ gets rendered into an appropriate formal language of predicate logic by the likes of ; ‘Someone is a philosopher’ gets rendered by . The syntactic difference between a formal name and a quantifier-tied-to-a-variable vividly marks a semantic difference between the rules for interpreting the two resulting formal wffs. And there isn’t the same sort of immediately striking syntactic difference between the vernacular name ‘Socrates’ and the quantifier ‘someone’.

That’s agreed on all sides, I guess. But many have wanted to say more — namely that, in English, there is no syntactic difference between the name and the quantifier. Thus,

Quine writes that ‘one of the misleading things about ordinary language is that the word “something” masquerades as a proper name’. Well, presumably there is no masquerade if you can easily tell them apart by their surface look. So, more carefully, Quine’s idea is presumably that the English quantifier is just like a proper name as far as surface sentence structure is concerned. Or as Michael Dummett explicitly puts it: ‘As far as the sentence-structure of natural language is concerned, signs of generality such as “someone” and “anyone” behave exactly like proper names — they occupy the same positions in sentences and are governed by the same grammatical rules.’

But this is just not true (as Alex Oliver, for one, has had fun pointing out). Thus, contrast ‘Something wicked this way comes’ with the ungrammatical ‘Jack wicked this way comes’, or ‘Someone brave rescued the dog’ with ‘Jill brave rescued the dog’. Or contrast ‘Foolish Donald tweeted’ with the ungrammatical ‘Foolish someone tweeted’. Compare too ‘Senator, you’re no Jack Kennedy’ with ‘Senator, you’re no someone’. Or what about ‘Hey, Siri!’ compared with ‘Hey, someone!’?

Other quantifiers too aren’t interchangeable with names. Consider ‘Nobody’. We get similar failures of substitution: we can’t replace ‘Nobody wise …’ with ‘Jill wise …’, or replace ‘Foolish Donald …’ with ‘Foolish nobody …’, and preserve grammaticality, etc. And ‘Nobody ever finishes War and Peace’ constrasts with the ungrammatical ‘Jack ever finishes War and Peace’, while ‘Jill never finished War and Peace’ contrasts with ungrammatical ‘Nobody never finished War and Peace’.

And so it goes. Quine’s and Dummett’s claims simply overshoot. English grammar doesn’t treat names and quantifiers exactly on a par. But even if it did, ordinary language would only be “misleading” (in Quine’s word) if there was some tendency for us ordinary speakers to get misled. Now, Mark Sainsbury indeed talks of ‘our tendency to regard quantifiers … as names’. But what is the evidence is that we have such a tendency? This is shown, says Sainsbury, ‘by the fact that Lewis Carroll’s jokes are funny’. But that really is hopeless! After all, Carroll’s wordplay (you know the kind of thing: ‘I see nobody on the road,’ said Alice. ‘I only wish I had such eyes … To be able to see Nobody!’ etc. etc.), apart from being wearyingly unfunny, has nothing specifically to do with confusing quantifiers with names (as Alex Oliver notes, it’s pretty much on a par with the likes of ‘What’s your name?’ ‘Watt.’ ‘I said, what’s your name?’ ‘Watt’s my name’ …)

OK: Quine, Dummett are wrong that English names and quantifiers “are governed by the same grammatical rules”, and even if they weren’t, that would give us no reason to suppose that we are misled (Quine, Sainsbury) by the grammatical similarity or that we tend to regard quantifiers as names.

And yet, and yet … Even if the claim that “signs of generality behave exactly like proper names” is false au pied de la lettre, we are left with the feeling that the sort of exceptions we’ve noted are somehow quirks of idiom rather than deeply significant. We are left with the sense that there is something important and true which ought to be  rescuable from the very familiar kind of remarks from Quine Dummett and Sainsbury. But what is it? We say arm-waving things in our intro logic lectures — but what will we happy to put in black and white?

It is tempting to say this:  the syntax of our formal first order language more perspicuously tracks the semantics of the formal language than the syntax of English names vs quantifiers tracks their semantics. But in a way, this is just too easy to say. Of course  the syntax tracks the semantics in our formal language in a way that is  perspicuous even to beginners — we purpose-designed the language to be exactly that way! And who knows how things are in English when it comes to syntactic or semantic theory — of course that’s not perspicuous at all, as half a century of modern linguistic theory has shown?!

Now, Quine, Dummett and Sainsbury aren’t  aiming to contrast the known with the unknown: they make a positive (even if false) claims about English, after all. But then, to repeat the question, just what that is instructive and importantly true can we extract from those incorrect claims about quantifiers “behaving exactly like” (?!) names in English but not in first order languages? I have (or rather had) my lecture patter. But, as I think about the relevant bit of my revised IFL book, I’d be very interested to know what others say to their students!

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More Dvorak

If you loved the Pavel Haas Quartet’s recording of the Dvorak Piano Quintet No. 2, Op. 81 and String Quintet op. 97, then you  should also love this new Dvorak recording by the terrific Jerusalem Quartet (with Veronika Hagen and Gary Hoffman). They here play the String Sextet Op 48 and then the String Quintet op. 97 again. The Sextet was (I confess!) new to me, and is quite delightful: the performance is as good as you would expect.

The Quintet is again beautifully played. The Jerusalem’s playing is slightly gentler, slightly more restrained (I suppose) than the Pavel Haas’s: but warmly recommended too.

(You can listen on Apple Music: other streaming services are available …)

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Dry January

For my Dry January, I tried to quit reading stuff about Brexit (after all, surely nothing much was going to happen for a month).

Well, I miserably failed to quit outright. But I did cut down a bit, and I did get more Dickens read. But hell’s bloody bells, what an appalling mess on so many levels.

(That’s what you come here for, no? Incisive political analysis like this!)

Category Theory: A Gentle Introduction

At long last, I have updated my  notes Category Theory: A Gentle Introduction (now some x + 291 pages).

A good while ago, I received lists of corrections from a number of people, and just recently I’ve had another tranche of corrections, making over a hundred in all. Mostly these corrections noted typos. But there were also enough mislabelled diagrams, fumbled notation mid-proof,  etc., to have no doubt caused some head-scratching. So I can only apologize for the delay in making the corrections.

I have also added a new early chapter and restored a couple of sections that were in a rather earlier version but got lost in the last one (thereby breaking some cross-references and no doubt producing more head-scratching).

These notes were originally written for my own satisfaction, trying to get some basics clear. But I know some people have found them useful (despite their very obvious shortcomings, unevenness,  and half-finished character). So I hope some others will find the update helpful: you can download it from the category theory page here.

The Language of Category Theory

I’m taking a week or so off from on working the d****d second edition of my logic text (it’s quite fun, if you like that sort of thing, most of the time: but it is good to take a break). I’m instead updating, just a little, my Gentle Intro to Category Theory, about which more when the revised version is ready for prime time (within the week, I hope). So I’ve now had an opportunity to take a quick look at Steven Roman’s An Introduction to the Language of Category Theory (Birkhäuser, 2017) which in fact has been out a whole year.

This book is advertised as one thing, but is actually something rather different. According to the blurb “This textbook provides an introduction to elementary category theory, with the aim of making what can be a confusing and sometimes overwhelming subject more accessible.” We might, then, expect something rather discursive, with a good amount of the kind of informal motivational classroom chat that is woven into a good lecture course and which can be missing from a conventionally structured textbook. But what we get is actually much closer to a brisk set of lecture notes. For the book travels a long way — through the usual introductory menu of categories, functors, natural transformations, universality, adjunctions (as far as Freyd’s Adjoint Functor Theorem) — and all in just 143 pages before we get to answers to exercises. Moreover, these pages are set rather spaciously, with relatively few lines to the typical page. So certainly there isn’t much room for discursive commentary.

And I would have thought that the sequencing of topics would leave floundering some of those who would appreciate a gentler introduction. So we get to the Yoneda Lemma long before we eventually meet e.g. products (and that as part of a general treatment of limit cones). Yet aren’t products a very nice topic to meet quite early on?  — in talking about them, we  explain why it is rather natural not to care about what product-objects are intrinsically (so to speak) but rather natural  to care instead about how the product gadgetry works in terms of maps to and from products. Here then is a rather nice example to meet early to motivate categorial ways of thinking. But not in this book.

Still, look at this for what it is rather than for what it purports to be. In other words,  look at this as a  set of detailed lecture notes which someone could use as back-up reading for perhaps the first half of a hard-core course, to keep things on track by checking/reinforcing definitions and key ideas, with added exercises  (notes which could then later be useful for revision purposes). Then Roman’s book does seem to be  pretty clearly done  and likely to be useful for some students. But if you were wondering what the categorial fuss is about and wanted an introductory book to draw you in, I doubt that this is it.

[Two grumbles. The book is pretty pricey for its length. And why, oh why, in an otherwise nicely produced paperback have the category theory diagrams been drawn in such an ugly way, given the available elegant standard LaTeX packages?]

Georg Kreisel — partial bibliography

Clearing out an old box, I’ve come across a crumpled xerox of a bibliography of some two hundred papers and other pieces by Georg Kreisel, covering up to the early 1990s.

I believe this biblio was passed on to me by Dan Isaacson, though I cannot recall where he got it from. A quick internet search suggests that it isn’t readily available online. But it might well still be of interest to some, so I have scanned it and made it searchable, and here it is.

(Do let me know if there is a more complete biblio anywhere. I’ve always wondered what Kreisel’s reputation would now be had he had the expository facility — or at any rate, the desire to be understood — of e.g. a Putnam or a Feferman.)

Pavel Haas Quartet play Schubert

Pavel Haas Quartet 2016    Photo: Marco Borggreve

For another couple of weeks you can listen via the BBC website to a characteristically intense performance of the Schubert G major Quartet D887 by the Pavel Haas Quartet, from the Schubertiade last June, recorded at the Angelika Kauffmann Saal, Schwarzenberg. Catch it while you can.

Symbol for assignment of a truth-value?

Here’s an odd thing. There seems, browsing along my shelves, to be no really standard symbolic metalinguistic shorthand used in elementary books for assigning a truth-value to a wff (say, in the propositional calculus). You would have expected there to be some.

In the first edition of my Introduction to Formal Logic, I borrowed the symbol ‘‘

to abbreviate ‘has the value … [on some given valuation]’ and wrote the likes of e.g.

If and  then .

But on reflection this was pretty silly, given that the symbol ‘‘ is already overloaded (not in my book, but elsewhere — like on math.stackexchange! — where, for a start, some use it for the conditional, some use it in place of a turnstile, and some get in a tangle by using it ambiguously for both!). It seems wiser not to add to possible confusion, especially when readers might well simultaneously get to see the double arrow being used in one of these different ways.

A bit of notation that is used, not at all consistently but often enough, is square double-brackets, so ‘‘ is used for ‘the value of …’, and we write the likes of ‘‘. But this seems to me a bit cluttered for elementary purposes — I’m after readability, rather than portability to more sophisticated contexts. And it misses the dynamism(??) of some type of arrow.

So for the upcoming second edition, I’m tentatively minded to use the \mapsto symbol for value-assignment, and write instead

If and  then .

(I suppose a colon could be another possibility, but I’d rather have something more distinctive. And the likes of ‘T()’ isn’t so pretty/easy to read in bulk and is conventionally part of an augmented object language.)

Any objection to the revised arrow? Am I missing some sufficiently  established (or even just nicer) alternative??