Next week’s post on Beginning Mathematical Logic will be more exciting (promise!) — an all-new chapter on intuitionist logic. But before we get there, here is the revised chapter on set theory. Again, I have done some minor tidying since the last edition, but there is no real novelty, as I’m currently reasonably content with the chapter. But I’m sure that doesn’t mean it couldn’t be improved — so, as always, all comments gratefully received.
Another revised chapter for the Study Guide. And again, there is little substantial change from the previous version, except to significantly cut down the length of the “overviews”, which were getting a bit out of hand! Anyway, here is the latest version of the chapter on computable functions, formal arithmetic, and Gödelian incompleteness.
These revised chapters I am posting are being downloaded a significant number of times, but comments (either here or by email) are few and far between. I‘m rather hoping that that’s because people aren’t finding my overviews on topic areas gruesomely misleading or my recommendations for reading too outlandish! But, as I’ve said before, if you do think I’m leading your students horribly astray (if you are a logic teacher) or think the Guide could be more helpful (if you are a student), now really is the time to say!
I have just read Stephen Budiansky’s Journey to the Edge of Reason: The Life of Kurt Gödel (OUP 2021). And no, I’m not immediately breaking my self-denying resolution to concentrate on finishing the Study Guide — I wanted to know if this biography should get a recommendation in the historical notes of the relevant chapter!
I’ll be brisk. Budiansky’s book comes much praised. But, to be honest, I’m not really sure why. You’ll certainly learn much more about Gödel’s ideas and intellectual circles from John Dawson’s reliable and thoughtful Logical Dilemmas: The Life and Work of Kurt Gödel (A. K. Peters, 1997). When Budiansky occasionally ventures to explain something in Gödel’s work, he too often just gets it wrong. And a lot of the scene-setting background in his book about Brno in 1910s, Vienna in the 1920s and 1930s, Princeton during and after the war, is routine stuff (and a bit too much of it is just padding). For example, the anti-semitism in the Austrian universities in the thirties, the speedy rehabilitation of Nazi collaborators after the war, is still shocking: but there is no depth to the familiar story as recounted here. Yes, the book rattles along and is easily readable. But does it add much to our understanding of things that matter? I certainly don’t feel any better or wiser for knowing a few more of the sad details about Gödel’s later health and fragile mental state. When does biographical curiosity slide into shabby prurience? If you want to read about Gödel’s life, stick to Dawson.
There used to be a little subscription form in the sidepanel of the old-look Logic Matters where you could subscribe to get email notifications whenever there was an update here. I’ve decided against replicating this. Instead of auto-generated mailings, I’ll just send out occasional short Newsletters to alert people to the more interesting new postings or series of postings. Here’s the first Newsletter, as sent out to previous subscribers. If you want to get future such mailings — and I promise your email box will not be cluttered! — then you can subscribe by following this link. (There’s a permanent link in the footer of those pages which have them.)
It’s arrived! Ever since it was announced, I’ve been very much looking forward to seeing this new book by Paolo Mancosu, Sergio Galvan and Richard Zach. As they note in their preface, most proof theory books are written at a fairly demanding level. So there is certainly a gap in the market for a book that presents some basic proof theory taking up themes from Gentzen in a more widely accessible way, covering e.g. proof normalization, cut-elimination, and a proof of the consistency of arithmetic using ordinal induction. An Introduction to Proof Theory (OUP, newly published) aims to be that book.
Back in the day, when I’d finished my Gödel book, I had it in mind for a while to write a Gentzen book a bit like IPT (as I’ll refer to it) though in parts a couple of notches more technical. But when I got down to work, I quickly realized that my grip on the area was really quite embarrassingly shallow in places, and I lost all confidence. What I should have done was downsize my ambitions and tried instead to write a book more like this present one. So I have a particular personal interest in seeing how Mancosu, Galvan and Zach write up their project. I’m cheering them on!
Some brisk notes, then, as I read through …
Chapter 1: Introduction has three brisk sections, on ‘Hilbert’s consistency program’, ‘Gentzen’s proof theory’ and ‘Proof theory after Gentzen’.
The scene-setting here is done very cogently and reliably as far as it goes (just as you’d expect). However, on balance I do think that — given the intended readership — the first section in particular could have gone rather more slowly. Hilbert’s program really was a great idea, and a bit more could have been said to explore and illuminate its attractions. On the other hand, an expanded version of the third section would probably have sat more naturally as a short valedictory chapter at the end of the book.
But then, one thing I’ve learnt from writing my own introductory books is that you aren’t going to satisfy everyone — indeed, probably not even a majority of your hoped-for readers will be happy. Wherever you set the dial, many will complain that you take things at far too slow a pace, while others will complain that you lost them only a few chapters in. So, in particular, these questions of how much initial scene-setting to provide are very much a judgement call.
To be continued (and I’ll return to The Many and the One in due course).
In Chapter 3, recall, Florio and Linnebo are discussing various familiar arguments against singularism, aiming to show that “the prospects for regimentation singularism are not nearly as bleak as many philosophers make them out to be”.
Now, it has always struck me that the most pressing challenge to singularism is actually that the story seems to fall apart when it moves from programmatic generalities and gets down to particulars. If the plan is, for example, to substitute a plural term referring to some Xs by a singular term referring to the set of those Xs, then how does work out in practice? How do we substitute for the associated predicate to preserve truth-values (without burying a plural in the new predicate)? Is the same treatment to apply to a plural term when it takes a distributive and collective predicate? The anti-singularist’s contention is that trying to substitute for plural terms ends up with (at best) ad hoc, piecemeal, treatments, and the resulting mess smacks of a degenerating programme (as Oliver and Smiley remark, having noted that e.g. Gerald Massey ends up giving four different treatments for four kinds of collective predicate, “where will it end?”). Now, this line of anti-singularist criticism might be more or less compelling: but in the nature of the case, that can’t be settled by a single counter-jab at one example. The devil will be in all the details — which is why I found F&L’s very brief treatment of what they call substitution arguments quite unsatisfactory.
But now let’s move on to consider another familiar anti-singularist line of argument that goes back to Boolos in his justly famous paper ‘To Be is to Be a Value of a Variable’. Here’s an edited version:
There are certain sentences that cannot be analyzed as expressing statements about sets in the manner suggested [i.e. replacing plural forms by talk about sets], e.g., “There are some sets that are self-identical, and every set that is not a member of itself is one of them.” That sentence says something trivially true; but the sentence “There is a set of sets that are self-identical, and every set that is not a member of itself is a member of this set,” which is supposed to make its meaning explicit, says something false.
F&L consider this sort of challenge to singularism in their §3.4.
One point to make (as F&L note) is that the argument here generalizes. Suppose we replace plural talk about some Xs with singular talk (not about the set of those objects) but by singular reference to some other kind of proxy object; and we correspondingly replace talk about some object o being one of the Xs by talk of o standing in the relation R to that proxy. Then it is easy to see that R can’t be universally reflexive if it is to do the intended work. So there will be some proxy objects such that any of the proxies which are not R to themselves is one of them. But this truth supposedly goes over to the claim that there is a proxy which is R to just those proxies which are not R to themselves. And it is a simple logical theorem that there can be no such thing.
But a second point worth making (which F&L don’t note) is that the quantificational structure of the Boolos sentence isn’t essential to the argument. Revert for ease of exposition to taking a singular term which refers to a set as the preferred substitution for a plural term, with membership as the R relation. Then consider the simple truth ‘{Jack, Jill} is one of the sets which are not members of themselves’. Supposedly, this is to be singularized as ‘{Jack, Jill} is a member of the set of sets which are not members of themselves’. Trouble!
OK. So how do F&L propose to blunt the force of this line of argument? They have two shots. First,
The paradox of plurality relies on the assumption that talk of proxies is available in [the language we are trying to regiment]. The lesson is that, if [the language to be regimented] can talk not only about pluralities but also about their proxies, then the regimentation validates unintended interactions of the sort just seen. To block the paradox, we would therefore have to prevent such problematic interactions. One possibility … is to refrain from making a fixed choice of proxies to be used in the analysis of all object languages. Instead, the singularist can let her choice of proxies depend on the particular object language she is asked to regiment. All she needs to do is to choose new proxies, not talked about by the given object language. In this way, the problematic interactions are avoided.
But hold on. I thought the the singularist was trying to give a regimented story about our language, using some suitably disciplined fragment of our language with enough singular terms but without the contended plurals? The proposal now seems to be that we escape paradox by introducing proxy terms new to our language, which we don’t already understand. Really? Usually singularists talk of sets, or mereological wholes, or aggregates, or whatever — but now, to avoid paradox, the idea is that we mustn’t talk of them but some new proxies, as yet undreamt of. It is difficult to see this as rescuing singularism as opposed to mystifying it.
F&L’s second shot is more interesting, and suggests instead that we discern “a variation in the range of the quantifiers involved in the paradoxical reasoning.” Thus, in the Boolos sentence “There is a set of sets that are self-identical, and every set that is not a member of itself is a member of this set” the proposal is that we take the ‘there is’ quantifier to range wider than the embedded ‘every set’ quantifier, and this will get us off the hook. On the face of it, however, this seems entirely ad hoc. Still, this sort of domain expansion is often put on the table when considering puzzles about absolute generality, and F&L announce they are going to return to discuss such issues in their Chapter 11. Fine. But so far, we have no hint about how the story is going to go.
And, more immediately, how do considerations about domain expansion engage with the not-overtly-quantified version of the Boolosian challenge that involves only a plural definite description. F&L just don’t say. They are, indeed, so far remarkably silent about plural terms and plural reference which, you might have supposed, would need to be a central topic in any discussion of plural logic.
We’ll have to wait to see what, if anything, F&L have to say later about e.g. plural descriptions. But for the moment, I think most readers will judge that the singularist’s prospects of escaping Boolos’s type of Russell-style paradox still look pretty bleak!
To be continued.
There’s a new book in the Cambridge Elements series on Wittgenstein’s philosophy of mathematics by Juliet Floyd. And for a few days it is freely available to read (and indeed download) here. I really rather doubt that it will appeal, though. Unless you like this sort of writing (the sixth paragraph, not at all untypical):
Aspects are modal, attaching to possibilities and necessities: fields of significance, opportunities for projecting and instantiating our concepts. We see through the picture to our own seeing of it as realizing one way among others. What we see is seen, but also we see. We rearticulate what we see, sometimes seeing it thereby anew. There is an active and a passive aspect to this. Aspects show themselves (the middle voice). What we are seeing is not simply an actual drawing on a page. We can also “see” in these drawings possibilities of projecting our concepts. Here we take modality as primitive, though up for investigation.
This ex-editor of Analysis most certainly wouldn’t have let that pass as acceptable.
We have been staying in a cottage near Monmouth for a few days. The countryside here is indeed a particularly green and pleasant land; we can sit outside the cottage looking over many rolling miles towards the Black Mountains. The ruins of Tintern Abbey are close by, as is Raglan Castle (both so very well looked after by Cadw for the Welsh Government, and both sites surprising quiet). There is a lot of wonderful walking here, through local woodlands, and in the Usk valley and the Wye valley. The weather has been kind. So a delightful escape from Cambridge.
With brilliant timing, the day before coming away, when I should have been concentrating on things domestic, Logic Matters got hacked. Or rather — since “logicmatters.net” then delivered a Chinese language advert, which hardly looked like a hack aimed at the typical reader here — I suspect some WordPress plugin had been hacked. (I’d been experimenting with different plugins while giving this site a fresh coat of paint). It took a while for me to find the source of the trouble; but then someone kindly recommended the Wordfence security plugin which quickly pinpointed where the evil code been added. Fingers crossed, but I hope the site is now more secure from such exploits. A long afternoon quite, quite wasted though. Not good for the blood pressure.
Calm was restored driving down to Monmouth and stopping at Kiftsgsate Court Gardens and then on to High Glanau, both gardens real works of art. We will visit Coton Manor Garden on the way home, which is even more stunning in its way (though perhaps I prefer the slightly wilder, less perfectly kempt style of Kiftsgate). I have said before, that gardens can be art-works of a kind that the English both do particularly well and particularly love. Is there, I wonder, anything attractively and insightfully written by modern philosophers on the aesthetics of gardens? (A genuine question!)
Update on the Logic Matters website. You can get lost down the infinite rabbit hole of WordPress customizations. But I’ve managed to escape, fixed on a theme, suppressed most of its fancy options, aiming for simplicity verging on starkness, and have got to work … Lots still to be done (for a start, in making more tablet and phone friendly), but you’ll get the basic idea. Any helpful comments/suggestions will of course be welcome.
Update on the hardback of Gödel Without (Too Many) Tears. Hooray! — a copy (from UK Amazon) has arrived at last, and another copy (from Blackwell’s, one to send on to the British Library) arrives tomorrow. UK and US Amazon are both now promising very speedy delivery.
I must say that I am very pleased with the result, it is really decently produced. So that, together with the sales figures, encourages me to organize hardback library copies of IFL and IGT. More about that anon. But for the moment, do please remember to get your local friendly librarian to order the hardback GWT for the library! — details here.
This Logic Matters site currently lives on Bluehost. But for various reasons, I’m in the middle of moving to a different hosting provider, Siteground: significantly more expensive (after the initial year’s discount) but by very many accounts also significantly better. Certainly, an experimental test version of the site runs there a lot faster, both on my iMac and even more so on an iPhone. As I’ve said before, the whole site could do with a good deal of tidying under the bonnet. So the needed update will keep me from fretting about the state of the world for a week or two.
I’ll need then to chose a modern WordPress theme that maintains the uncluttered look I like. I’ll ignore the pricey paid options. That only leaves about eight thousand free themes to choose from. So this is going to be dead easy. Down the rabbit hole I go …