Peter Smith's Blog, page 28

In the midst of life …

The post In the midst of life … appeared first on Logic Matters.

The Annotated Gödel

Some years ago, Charles Petzold published his The Annotated Turing which, as its subtitle tells us, provides a guided tour through Alan Turing’s epoch-making 1936 paper. I was prompted at the time to wonder about putting together a similar book, with an English version of Gödel’s 1931 paper interspersed with explanatory comments and asides. But I thought I foresaw too many problems. For a start, not having any German, I’d have had to use one of the existing translations, which would lead to copyright issues, and presumably extra problems if I wanted to depart from the adopted translation by e.g. rendering Bew by Prov, etc. And then I felt it wouldn’t be at all easy to find a happy level at which to pitch the commentary.

Plan (A) would be to follow Petzold, who is very expansive and wide ranging about Turing’s life and times, and is aiming for a fairly wide readership too (his book is over 370 pages long). I wasn’t much tempted to try to emulate that.

Plan (B) would be write for a much narrower audience, readers who are already familiar with some standard modern textbook treatment of Gödelian incompleteness and who want to find out how, by comparison, the original 1931 paper did things. You then wouldn’t need to spend time explaining e.g. the very ideas of primitive recursive functions or Gödel numberings, but could rapidly get down to note the quirks in the original paper, giving a helping hand to the logically ept so that they can navigate through. However, the Introductory Note to the paper in the Collected Works pretty much does that job. OK, you could say a bit more (25 pages, perhaps rather than 14). But actually the original paper is more than clear enough for that to be hardly necessary, if you have already tackled a good modern treatment and then read that Introductory Note for guidance in reading Gödel himself.

Plan (C) would take a middle course. Not ranging very widely, sticking close to Gödel’s text. But also not assuming much logical background or any prior acquaintance with the incompleteness theorems, so having to slow down to explain ideas of formal systems, primitive recursion and so on and so forth. But to be frank, I didn’t and don’t think Gödel’s original paper is the best peg on which to hang a first introduction to the incompleteness theorems. Better to write a book like GWT! So eventually I did just that, and dropped any thought of doing for Gödel something like Petzold’s job on Turing.

But now, someone has bravely taken on that project. Hal Prince, a retired software engineer, has written The Annotated Gödel, a sensibly-sized book of some a hundred and eighty pages, self-published on Amazon. Prince has retranslated the incompleteness paper in a somewhat more relaxed style than the version in the Collected Works, interleaving commentary intended for those with relatively little prior exposure to logic. So he has adopted plan (C). And the thing to say immediately — before your heart sinks, thinking of the dire quality of some amateur writings on Gödel! — is that the book does look entirely respectable!

Actually, I shouldn’t have put it quite like that, because I do have my reservations about the typographical look of the book! Portions of different lengths of a translation from the 1931 paper are set in pale grey panels, separated by episodes of commentary. And Prince has taken the decidedly odd decision not to allow the grey textboxes containing the translation to split themselves over pages. This means that an episode of commentary can often finish halfway down the page, leaving blank inches before the translation continues in a box at the top of the next page. And there are other typographical choices while also quite unfortunately make for a somewhat  unprofessional look. That’s a real pity, and does give a quite misleading impression of the quality of the book.

Now, I haven’t read the book with a beady eye from cover to cover; but the translation of the prose seems quite acceptable to me. Sometimes Prince seems to stick a bit closer to the Gödel’s original German than the version in the Works, sometimes it is the other way about. For example, in the first paragraph of Gödel’s §2, we have

G, undefinierten Grundbegriff: W, primitive notion: P, undefined basic notion,

while in the very next sentence we have

G, Die Grundzeichen: W, The primitive signs: P, The symbols.

But these differences are relatively minor.

Where P’s translation of G departs most is not in rendering the German prose but in handling symbolism. W just repeats on the Englished pages exactly the symbolism that is in the reprint of G on the opposite page. But where G and W both e.g. have “Bew(x)”, P has “isProv(x)”. There’s a double change then. First, P has rendered the original “Bew”, which abbreviated “beweisbare Formel”, to match his translation for the latter, i.e. “provable formula”. Perhaps a good move. But Prince has also included an “is” (to indicate that what we have here is an expression attributing a property, not a function expression). To my mind, this makes for a bit of unnecessary clutter here and elsewhere: you don’t need to be explicitly reminded on every use that e.g. “Prov” expresses a property, not a function.

Elsewhere the renditions of symbolism depart further. For example, G has “” for the wff which says that is an instance of Axiom Schema II.1. P has “isAxIIPt1(x)”. And there’s a lot more of this sort of thing which makes for some very unwieldy symbolic expressions that I don’t find particularly readable.

There are other debatable symbolic choices too. P has “” for the object language conditional, which is an unfortunate and unnecessary change. And P writes “” for the result of substituting for where is free in . This may be a compsci notation, but to my untutored eyes makes for mess (and I’d say bad policy too to have arrows in different directions meaning such different things).

Other choices for rendering symbolism involve more significant departures from G’s original but are also arguably happier (let’s not pause to wonder what counts as faithful enough translation!). For example, there is a moment when G has the mysterious “p = 17 Gen q”: P writes instead “p = forall(, q)”. In G, 17 is the Gödel number for the variable : P uses a convention of bolding a variable to give its Gödel number, which is tolerably neat.

There’s more to be said, but I think your overall verdict on the translation element of Prince’s book might go either way. The prose is as far as I can judge handled well. The symbolism is tinkered with in a way which makes it potentially clearer on the small scale, but makes for some off-putting longwindedness when rendering long formulas. If you are going to depart from Gödel’s symbolism, you could be snappier. But as they say, you pays your money and makes your choice.

But what about the bulk of the book, the commentary and explanations interspersed with the translation of Gödel’s original? My first impression is definitely positive (as I said, I haven’t yet done a close reading of the whole). We do get a lot of helpful framing of the kind e.g. “Gödel is next going to define … It is easier to understand these definitions if we think about what he needs and where he is going.” And Prince’s discussions as we go along do strike me as consistently sensible, accurate, and will indeed be helpful to those who bring the right amount to the party.

I put it like that because, although I think the book is intended for those with little background in logic, I really do wonder whether e.g. the twenty pages on the proof of Gödel’s key Theorem VI will gel with those who haven’t previously encountered an exposition of the main ideas in one of the standard textbooks. This is the very difficulty I foresaw in pursuing plan (C). Most readers without much background will be better off reading a modern textbook. But, on the other hand, for those who have already read GWT (to pick an example at random!), i.e. those who already know something of Gödelian incompleteness, they should find this a useful companion if they want to delve into the original 1931 paper — though some of the exposition will now probably be unnecessarily laboured for them, while they would have welcomed some more  “compare and contrast” explanations bringing out more explicitly how Gödel’s original relates to standard modern presentations.

In short, then: if someone with a bit of background does want to study Gödel’s original paper, whereas previously I’d just say ‘read the paper together with its Introductory Note in the Collected Works’, I’d now add ‘and, while still doing that, and depending quite where you are coming from and where you stumble, you might very well find some or even all of the commentary in Hal Prince’s The Annotated Gödel pretty helpful’.

The post The Annotated Gödel appeared first on Logic Matters.

In praise of Truss …

Praise not for our new Prime Minister, about whom the less said the better, but for her admirable logician father John Truss.

By chance, I had occasion recently to dip into his 1997 book Foundations of Mathematical Analysis (OUP) which I didn’t know before and which is excellent. It is my sort of book, in that there is a lot of focus on conceptual motivation and Big Ideas, and a relatively light hand with detailed proofs. E.g. if you want just a very few pages on Gödelian incompleteness, his treatment in the first chapter is exemplary. Or jumping to the end of the book, there is e.g. a really helpful broad-brush section on the ingredients of Gödel’s and Cohen’s independence proofs in set theory, and a very good chapter on constructive analysis and choice principles. In between, we get a story that takes us from the naturals to the integers to the reals to metric spaces and more (e.g. a nice chapter on measure and Baire category). OK, this is a tale which is often told, but I think John Truss’s version is particularly insightful and good at bringing out the conceptual shape of the developing story. So, recommended!

OUP have disappointingly let the book go out of print. A Djvu file can, however, be found at your favourite file depository.

The post In praise of Truss … appeared first on Logic Matters.

GWT2 — a first full draft

I now have put together a first complete draft of the second edition of Gödel Without (Too Many) Tears. You can download today’s version of GWT2 here.

I need to do a careful read-through for typos/thinkos. I also need to update the index, make the typography more uniform between chapters, and e.g. decide on a more consistent policy about when I cross-reference to IGT2. That sort of fun to come over the next month or so. I’ll post updates from time to time.

It is too late to write a very different book, and after all this is supposed to be just a revised edition of the seemingly quite well-liked GWT1! This is not the moment, then, for radical revisions. But otherwise, all suggestions, comments and corrections, including quick notes of the most trivial typos, will be most welcome! Send to the e-mail address on the first page of PDF, or comment here. (Just note the date of the version you are commenting on.)

Actually, many readers of this blog will have better things to do than spend much time with this sort of intro-level enterprise (though massive thanks are due to a handful who have already been e-mailing comments). But if you aren’t a student yourself, you could well have students who would be interested to take a look and let me know what they find too obscure, and/or give other comments and corrections. (Back in the day, I lined up a whole team of volunteers to look at a couple of chapters each of IGT2: and there turned out to be precisely zero correlation between “status”, from undergraduate to full professor, and the usefulness of comments!) So do please spread the word to any students, undergraduate or graduate, who might be — or ought to be! — interested.

It’s not much of a bribe, I know, but those impoverished students who prompt the biggest corrections/improvements will get a free paperback in due course, as well as having their name in lights in the Preface!

The post GWT2 — a first full draft appeared first on Logic Matters.

PHQ, at the Edinburgh Festival

Photo by Petra Hajska

There was an extraordinary concert by the Pavel Haas Quartet at the Edinburgh Festival on Tuesday morning, with a BBC radio recording available for a month. They gave very fine performances of

Haydn: String Quartet in G major Op. 76/1
Martinů: String Quartet No 7 H314
Schubert: String Quartet in G D.887

The Schubert was particularly intensely felt. But what made the performances little less than miraculous was that they playing with (yet another) new violist. The gifted Luosha Fang was with them as recently as the East Neuk Festival in early July; and interviews when their Brahms Quintets disk came out a bit earlier gave every impression that after a year she was very much part of the Quartet. But now, it seems, more trials and tribulations for the Quartet and, for whatever reason, a sudden parting of the ways. The viola seat is now being occupied — at least for the rest of the year — by a Czech compatriot, Karel Untermüller. So there he was, just a few weeks into the role: yet the ensemble seemed (at least to my not very expert ears) to be as remarkable as ever. Which, as I say, was surely rather extraordinary.

There is a good piece on the concert here.

The post PHQ, at the Edinburgh Festival appeared first on Logic Matters.

NF is consistent

Randall Holmes has been claiming a proof for about a decade, and recently posted yet another improved update of his proof on arXiv.

A while back, there was some very interesting discussion about the possibility of formalising the proof using a proof assistant like Lean. There’s now some relevant local news, which I get from Thomas Forster.

Roughly speaking, the proof has three components. (1) Randall proved over twenty-five years ago that NF is consistent if what he called Tangled Type Theory, TTT, is. Arm waving, the latter is what you from the simple theory of types if you allow to be well-formed even when is more than one type higher (in other words, we relax the usual requirement that here is exactly one type higher than ), so there is an relation between any lower level and any higher level. This relative consistency claim is unproblematic.

(2) TTT is a seemingly rather wild theory. But Holmes now aims to present a Frankel-Mostowski-style construction that purports to be a model of TTT. The devil is in the contorted(?) detail: do we get a coherent description of a determinate structure?

(3) Assuming that stage (2) is successful in at least describing a kosher structure that a ZF-iste can happily accept as such, there then is the task of verifying that it really is a model of TTT.

Now, over the last few weeks, a bunch of maths students here have been working on a summer project arranged by Thomas (with a lot of Zoomed input from Randall) to formally verify the consistency proof in Lean, by first checking (2) that the model is coherently constructed, and then going on to check (3) it really is a model of TTT. And I understand the state of play to be this: that first of these stages is successfully more or less completed. And it has in the process become intuitively clear — said Thomas — that the defined structure is indeed a model of TTT. Dotting the i’s and crossing the t’s and implementing a Lean check that the model satisfies a certain finite axiomatization of TTT will take more time than there is left in this summer’s project (the students have lives to lead!). But with (2) secure, it looks as if Holmes indeed has his claimed proof, though its final best-form shape remains to be settled.

If that’s right, Holmes has settled one of the oldest open problems in set theory. Though quite what the wider significance of this, I’m frankly not so sure. Will a consistency proof (of a decidedly tricksy-seeming kind) really make us look much more kindly on NF? Should it?

The post NF is consistent appeared first on Logic Matters.

GWT2 — up to the first incompleteness theorem

I have now revised Gödel Without (Too Many) Tears up to and including the pair of chapters on the first incompleteness theorem. You can download the current version up to Chapter 13 here.

For info: the chapter on quantifier complexity has been revised (adopting a more complex definition of Sigma_1 sentences, so that I don’t have to cheat later in saying that primitive recursive functions can be defined by Sigma_1 sentences). Then the chapter on primitive recursive functions has been slightly revised yet again. I have tried to make the chapter that proves that primitive recursive functions can indeed be defined by Sigma_1 sentences a bit more reader-friendly (the key ideas are elegantly simple: implementing them is unavoidably a bit messy). The chapter on the arithmetization of syntax is little altered. And finally in this instalment, the two chapters on the semantic and syntactic versions of the first incompleteness theorem are more or less untouched.

I’m still on track for getting a second edition out by around the end of October. It goes without saying that all comments and corrections will be gratefully received (and do please alert any students who might be interested in reading through and spotting typos or unclarities). Many thanks once again to David Furcy and Rowsety Moid for corrections and suggestions.

The post GWT2 — up to the first incompleteness theorem appeared first on Logic Matters.

IFL as a free download, two years on …

The second edition of my An Introduction to Formal Logic was originally published by CUP. It is now exactly two years ago today that I was able to make the book free to download as a PDF and also make it available as a very cheap paperback, thanks to the Amazon print-on-demand system. How have things gone?

As with the Gödel book I really didn’t know what to expect. But from almost the beginning, IFL2 has been downloaded about 850 times a month, and has sold very steadily over 75 paperbacks a month (with numbers if anything creeping up). Which, on the one hand, isn’t exactly falling stone-dead from the press. But, on the hand, given the very large number of philosophy students who must be taking Logic 101 out there in the Anglophone world, it isn’t an overwhelming endorsement either. However, you can’t please everyone: there isn’t much consensus about what we want from an intro logic book (which is why unwise lecturers like me keep spending an inordinate amount of time writing our own, despite the best advice of our friends …). Modified rapture, then.

So what now? IFL1 was truth-tree based. IFL2 uses a Fitch-style natural deduction system. The intro book I’d ideally write would cover both trees and natural deduction. That wasn’t possible within the CUP page budget. But those constraints are lifted. A PDF can be as long as I want; and in fact the marginal additional printing cost of expanding the paperback by fifty or sixty pages wouldn’t make very much difference to the price. So an expanded IFL3 is certainly a possibility. But do I actually want to write a third edition?

OK, I confess I’m tempted! Not at all because I think the world stands in desperate need of such a book, but because (very sad to relate) I’d actually rather enjoy the exercise of getting things into the best shape I can, before I hang up my expository boots. A plan for 2023? If the gods are willing.

The post IFL as a free download, two years on … appeared first on Logic Matters.

Reasons to be cheerful, of a reading kind

Janus La Cour, Olive Trees Near Tivoli

To the Fitz, to see their exhibition (on for another twelve days) “True to Nature”. Which we very much enjoyed (rather to our surprise, much more so than the trumpeted Hockney exhibition which is still continuing). Mostly minor works, to be sure, but the cumulative affect a delight, with some gems you would be more than happy to live with. Like these gnarled olive trees.

A list of forthcoming autumn books in one of the weekend papers. At the end of this month, we get Maggie O’Farrell’s The Marriage Portrait. “Winter, 1561. Lucrezia, Duchess of Ferrara, is taken on an unexpected visit to a country villa by her husband, Alfonso. As they sit down to dinner it occurs to Lucrezia that Alfonso has a sinister purpose in bringing her here. He intends to kill her.” OK, you’ve got me! I thought Maggie O’Farrell’s Hamnet was wonderful: so can’t wait for this. Then Elisabeth Strout has another novel, fast on the heels of the marvellous Oh William!: in October we get a sequel,  Lucy By The Sea. Then, not least, there is a new Kate Atkinson coming, Shrines of Gaiety. Enough said! Some happy autumn evenings ahead. Three reasons to be cheerful.

I was struck that the ones that stood out for me in that list of autumn books were all by women. And looking back at the list I keep, I see that most of the two dozen novels I’ve read this year so far have been by women. At least of the recently published ones, only one was by a man — Julian Barnes’s Elisabeth Finch. I must be missing out on something: but what?

A book taken down from the shelves one recent evening, The Faber Book of Landscape Poetry (a serendipitous as-new Oxfam purchase a while back). A real pleasure to dip into — some very engaging poetry, some familiar, some not at all. Perhaps not very challenging. Indeed, to be honest, perhaps a rather conservative selection. But then it was edited by a conservative, indeed a Conservative politician, the one-time Education Secretary Kenneth Baker. The thought strikes: which of the last six or seven Conservative Education Secretaries might have even had any literary interests, let alone sensibly edited some such book? Anyone?

Another forthcoming book: A. E. Stallings This Afterlife: Selected Poems. Something else to really look forward to. Pleasures of the reading kind will be plentiful, then. Outside books, the world isn’t doing so well, is it?

The post Reasons to be cheerful, of a reading kind appeared first on Logic Matters.

IGT as a free download, two years on …

I’ve just noticed that it is almost exactly two years since I was able to make An Introduction to Gödel’s Theorems, originally published by CUP, freely available to download as a PDF.

After a ridiculously large initial flurry of downloads, the book is now steadily downloaded about 600 times a month. As I’ve said before about such stats, it is very difficult to know how to interpret the absolute numbers: but this looks respectable enough, and  the trend is still upwards.

When I originally announced, without much fanfare, that the PDF was available, I added

I may in due course also make this corrected version of the book available as an inexpensive print-on-demand book via Amazon, for those who want a physical copy. But I doubt that there would be a big demand for that, so one step at a time

Well, of course, I soon did set up the POD paperback (very easy if you already have a publication-quality PDF), and I was proved quite wrong about demand. I thought any sales would be a tiny trickle given the availability of a completely free download; but in fact the paperback of IGT very steadily sells over 50 copies a month, over three times as many it was doing under the auspices of CUP. So I think we can count the experiment as a success!

Back in the day, when I was writing the first edition of IGT, I tried to do the whole thing from memory, reconstructing proofs as I went, on the principle that if an idea or a proof-strategy had stuck in my mind, then it was probably worth including,  and if it hadn’t then maybe not. I did fill in some gaps once I had a complete good draft; but that explains the relative shortage of footnotes to sources. When I look at the book occasionally, it is just a tad depressing to realize that I would struggle to rewrite it from memory now. That struck me forcefully yesterday when, reworking a section in Gödel Without (Too Many) Tears for its second edition, I consulted IGT and came across an important point that, at least for the moment, I’d quite forgotten the intricacies of. Ah well …

The post IGT as a free download, two years on … appeared first on Logic Matters.