Year: 2020

Logic: A Study Guide — Naive set theory

I’ve decided to divide the coverage of set theory in the Guide into three different chapters. There will now be two chapters in Part I. A short initial chapter on naive set theory, meaning the bits and pieces of notation, concepts and constructions that are often taken for granted in even very elementary logic books. Mathematicians shouldn’t need the chapter, but it could well be useful for philosophers without much mathematical background. This chapter therefore now comes before the chapters on FOL, model theory, and arithmetic. Then, after those chapters, there will be the main chapter on elementary set theory (a first real encounter at the level of e.g. Enderton’s book or a little more). A later chapter on hard-core set theory (large cardinals, forcing, and the like) belongs in Part III.

So I’ve now inserted the draft chapter on naive set theory (and made a few changes too to other chapters, responding to a few comments and suggestions). Here then is the current version of Part I of Logic: A Study Guide, still lacking its main chapter on set theory, which I hope will follow fairly shortly.

A Christmas card, from a small corner of Cambridge

From the Fitzwilliam Museum

So here we are. I don’t need to tell you that it’s been a troubling year in so many ways. Though as I have said before, compared with too many people, we personally are very fortunately placed — staying healthy (thanks for asking), no small children or very aged relatives to be deeply anxious about, in funds, well-housed, delightful walking on our doorstep (if you have to be locked down in a city, central Cambridge is one of the better options), and indeed with some gently lovely countryside still accessible not many minutes away. Friends and relations are there frequently on Zoom and FaceTime. The thing we miss most is being able to travel, and in particular to meet up in person with The Daughter who lives abroad. The weeks do drag, and the sameness can be enervating. But we mustn’t, and (mostly) don’t, complain. There is, in the circumstances, still much to be grateful for. But there is no question but that it is going to be a long,  long, winter.


The last months have certainly concentrated the mind on what really matters. Family and being in closer contact with nature seem to be very high on most people’s list: they certainly have been on ours.

Have any philosophers recently been writing particularly well on lockdown themes? I don’t know. But I wouldn’t entirely bet on it, given philosophers’ propensities for daftness of one sort or another. I was struck the other day by David Papineau’s report of Bernard Suits’s pretentious  The Grasshopper (a book I gave up on very quickly): “The overall argument of the book is that in utopia, where humans have all their material needs satisfied at the push of a button, what we would do would be play games, and therefore playing games is the ideal of human activity. Freed from all the necessities of having to do things we don’t want to do in order to get the material means of life, we’d do nothing but play games.” How profoundly silly is that? Not to say philistine. To be sure, some people sometimes enjoy games. But many of us, me for one (and actually most of the people of near my generation that I know well), have more or less zero interest in sports or games. And the idea of doing nothing but play games would fill us with horror — apart from spending time with family and friends, there are so many books to read, so much great music to listen to again, art to see, theatre to go to, wonderful countryside to be explored, new cities, new countries, to visit,  … Given the alternatives, spending time on games has very little appeal.


As I said, so many books to read. And re-read. Indeed, I mostly seem to have been re-reading since lockdown. But this is the season when all those lists of Books of the Year are published, depressingly emphasizing how few recent books have come our way. I certainly won’t be adding to those lists of obscure titles you mostly have never head of. Of books which were published this year, I’ve in fact most enjoyed two that very many others have equally enjoyed and recommended: Maggie O’Farrell’s Hamnet and (quite in a league of its own) Hilary Mantel’s The Mirror and the Light. I had just previously re-read with huge enjoyment the first two books in Mantel’s trilogy: but this final part is stunningly good.

On my desk too, dipped into at random times, have been some of Alice Oswald’s books of poetry. I do not find her at all easy or comfortable to read. But her work is deep and challenging and rewarding.


No concerts to go to. Wigmore Hall’s series of streamed concerts has included some wonderful occasions, most recently Mitsuko Uchida’s playing of two Schubert sonatas. Initially, being able to see so many concerts online in lockdown seemed terrific: but lately, I’ve been feeling that they somehow emphasized what we were missing by not being able to go to a live performance shared with an audience. Others have said the same.

Of CDs released in recent months, I’ve kept coming back to Supraphon’s boxed set of the Smetana Quartet playing the Beethoven quartets (recorded between 1976 and 1985), playing of the greatest humanity and insight. The tradition of Czech string quartets is indeed extraordinary.


A lot of reading and listening, then, in lockdown, while cautiously staying very close to home and trying to stay well. And on we will go, let’s hope, for more months like this. But still, the Tuscan wine selections for the holidays are looking very promising; we take our consolations where we can …!

With all good wishes for Christmas and for an eventually much better New Year. And stay well.

Logic: A Study Guide — Computability, arithmetic, Gödel

I mentioned a couple of days ago that, in the last four months, the newly available PDFs of my Intro to Formal Logic and Intro to Gödel’s Theorems have both been downloaded over 3.5K times (and that’s ignoring an initial flurry of downloads of the Gödel book by people who clicked on a probably misleading link posted elsewhere). In the same period — without any advertising at all — the Teach Yourself Logic Study Guide has been downloaded 7.5K times. I mention this to explain again why I feel I ought to give the TYL project some love and spend some quality time updating the Guide: if it is being downloaded that much, with a big surge at the beginning of semesters, it must be being recommended as useful. So I guess I really ought to make sure it is as useful as it can be, and indeed make sure it reflects what I now think about which texts to recommend.  The last full version was a pretty rough-and-ready layered accumulation of bits and pieces of various vintages: it is well past time for an end-to-end rewrite. But heavens, it’s necessarily a slow job, as I revisit texts old and new!

Anyway … here now is the latest version of the new-style Guide up to the rewritten Chapter 6. This reworked chapter covers three inter-connected topics: (a) the elementary informal theory of arithmetic computability, (b) an introduction to formal theories of arithmetic and how they represent computable functions, which leads up to (c) Gödel’s epoch-making incompleteness theorems

My reading recommendations for this chapter haven’t changed a lot. But a feature of the revised Guide is that (after the preliminary chapters), each chapter has a section (or two) giving an extended overview of its theme, from five to ten pages long. These overviews are supposed to be elementary indicators of some of the topics covered by the recommended reading.  They can certainly be skipped (that’s clearly signalled): the overviews are included just for those who might find this kind of preliminary  orientation helpful. It is difficult to know just how to pitch them, and I will no doubt later revisit the set of overviews to make them more uniform in style and level (so comments appreciated!).

I realize now that the Teach Yourself Logic Study Guide has been so-called for over eight years. Maybe I shouldn’t change the “brand” name after all ….

Gödel Without (Too Many) Tears published!

Brought to the front again — very short version: GWT is available on Amazon, print on demand.

Longer version: Gödel Without (Too Many) Tears is based on notes for the lectures I used to give to undergraduate philosophers taking the Mathematical Logic paper in Cambridge. Earlier versions were available here online, and have been much downloaded for a decade (and I know they have been used for seminars/lecture courses elsewhere). As occupational therapy in this time of pandemic, I have now very considerably tidied-up the notes into a book format — and many thanks to all those who have helped along with way with suggestions and corrections.

You can think of the result as a much cut-down version of my big Gödel book; it is just over a third of the length, but still aiming to explain some of the key technical facts about the incompleteness theorems. It should be rather more accessible too, as it cuts out some of the fancier digressions in the big book, and tries to make the rest of what it does cover as clear as possible.

The book is now available as a very inexpensive, at cost, print-on-demand book for less than $5/£4/€4.5. See e.g. US link, UK link; you can ‘look inside’ from the linked pages, which will give you a good impression of what the book covers. For other Amazons, use the ASIN identifier B08L5MQLRQ. 

(Small print: sorry about using Amazon; but they bought up the CreateSpace platform …! And they do a surprisingly decent production job at a very low price if you basically forgo royalties.)

Mitsuko Uchida plays Schubert at Wigmore Hall

Of recent concerts live streamed by Wigmore Hall, this stands out. Mitsuko Uchida plays Schubert’s Piano Sonata in C, D840 ‘Reliquie’, and the great Sonata in G, D894. This, surely, is Uchida at her very best, bringing out the depths, with miraculous moments. Truly impressive Schubert.

A few thoughts about self-publishing

A very enjoyable walk down to my favourite library, the Moore library, in the winter sun. But not, sadly, to then read and write, and think, and idly look out of the windows, and take a coffee break, and write again. It will be a good while yet before all that is possible. I was just donating, via their dropbox, copies of IFL2 and GWT.

Time for an update, perhaps. How have things gone since I got the copyright back from CUP, and have been able to give away IFL2 and IGT2 as freely downloadable PDFs? I’ve just checked: since late August, IFL2 has been downloaded over 3.6K times. And after a quite crazy initial flood (when someone posted a direct link at Hacker News, without saying that the link was to a full book!), IGT2  has been downloaded another 4K times. The two books have sold well over 200 each of the inexpensive print-on-demand versions. (It is very early days for GWT … I’ll report back on that in the New Year.)

I didn’t at all know what to expect. Or rather, I was expecting something like that ratio of freely downloaded PDFs to bought copies: but I had little idea how many would be tempted by the books overall. I guess I am pretty pleased.

And it certainly seems to have been worth the small effort of making the print-on-demand versions available. I did ask online, and got enough responses to suggest that there is a significant minority of readers who significantly prefer to work from “real” books as opposed to onscreen PDFs (which is one reason that libraries should have hard copies available); and some of that minority said that they are prepared to pay a modest amount to get the hard copy too. And so it has turned out.

By the way, as I’ve remarked before, I wasn’t thrilled to bits to be using the Amazon-provided service. But for this kind of enterprise, it does seem the best and easiest option on various counts. And since sales are small, and I’ve only rounded up the price from the minimum possible by pence (in order to cover costs of getting proof copies, sending copies to copyright libraries etc.), you are at least not adding much to Amazon’s grossly undertaxed profits by buying a copy.

In some respects, then, isn’t this an ideal way of publishing book-length projects? Provide freely downloadable PDFs; and make as-inexpensive-as-possible print-on-demand copies available.

Well yes, but only up to a point. It works if you e.g. already have a book or two to your name and you don’t particularly need the imprimatur of a respected academic press for people to think that your book might be worth taking seriously. And if you don’t need that imprimatur for promotion purposes either. And  if you can find enough friends and acquaintances to give honest critical feedback at key writing stages (eventually doing the work of a publisher’s readers). And if you know your way around a document processing package like LaTeX well enough, and have a good enough design eye, to produce pages which look professional.  And if you can find enough other friends and acquaintances who will happily check for typos and thinkos (doing the work of a publisher’s proof-reader). And if you have enough internet presence via a blog or whatever to get the word out there beyond the small circle of those friends and acquaintances!

That’s quite a few rather big “if”s.

So traditional publishers do still have a role to play. Or at least some of them. Mind you, we can all think of publishers like Spr*ng*r where the quality control is minimal, and unheralded books (published at ludicrous prices) fall stone dead from the press. However, you can these days publish with an academic publisher and negotiate to be allowed to keep a PDF freely downloadable (some even put e.g. chapter-by-chapter PDFs open access on their website). CUP, OUP and MIT seem to allow this sort of thing sporadically, though I’m not sure what the principles of choice are. And then there is e.g. the very promising new BSPS Open initiative: the plan is to publish open access monographs under the supervision of an editorial board to maintain quality. It will be interesting to see how initiatives like that develop over the coming few years: for surely, with the gross pressure of costs on libraries (let alone the impoverishment of young academics) the days of publication solely by the £80 monograph must be numbered …

Meanwhile, if it can work for you, I can recommend the self-publishing route!

Logic: A Study Guide — Basic Model Theory

I’m continuing work on the update for Teach Yourself Logic: A Study Guide. So there are now five chapters in the new Logic: A Study Guide.

There are three preliminary chapters, giving an introduction for philosophers, an introduction for mathematicians, and a guide-to-the-Guide. Then there is a long chapter on FOL. I’ve previously posted versions of these.

The fifth chapter is on entry-level model theory. There’s an overview introducing a few elementary results, intended to give a flavour of the enterprise. There follows the usual sort of reading guide.

Here then is the Guide including this new  chapter. Need I add? — all comments very gratefully received.

In particular I’m sure I can do better at the end of the displayed box on p. 34. I say earlier in the chapter that — although the focus is of course on standard first-order model theory — it is worth at this stage knowing just a bit about second-order logic/theories (so you get e.g. a glimmer of why first-order arithmetic isn’t categorical which a second-order arithmetic can be). But what short and accessible reading on second-order logic would you recommend at this stage? Later in the Guide we’ll be taking a serious look at the topic: but what brisk (perhaps arm-waving but still helpful) intro could be offered at this point?

Logic: A Study Guide — First Order Logic

Updated with corrected link!

I have started working occasionally on an update for Teach Yourself Logic: A Study Guide. It now has a slightly different format — and a marginally snappier title, Logic:  A Study Guide.

After three preliminary chapters — an “Introduction for Philosophers”, a shorter “Introduction for Mathematicians”, and a chapter on “Using this Guide” — the first substantial chapter of the new Guide gives, as you would expect, basic reading recommendations on first order logic. Here then is a draft of those preliminary chapters together with the new Chapter 4. (The earlier chapters will only be of any interest to those not familiar with the general intention of the Guide: everyone else can start reading at p. 12.)

All comments and suggestions very gratefully received, as always.


“Ok, it looks prettier, but the principal recommendations haven’t changed!” I’m afraid not. I have been doing a lot of enjoyable and indeed instructive re-reading over the last couple of weeks, but I do seem to have ended up not changing my verdicts about very much. Fancy that!

“So after all that effort, it’s a bit like Ford Prefect updating his entry for the Earth in the Hitchhiker’s Guide to the Galaxy from ‘Harmless’ to ‘Mostly harmless’?” Harsh but embarrassingly close to the truth …

… Still, there are enough minor changes, perhaps, to make it all worth while!

Philosophy of mathematics — a reading list

A few people recently have quite independently asked me to recommend some introductory reading on the philosophy of mathematics. I have in fact previously posted here a short list in the ‘Five Books’ style. But here’s a more expansive draft list of suggestions.

Let’s begin with an entry-level book first published twenty years ago but not yet superseded or really improved on:

  1. Stewart Shapiro, Thinking About Mathematics (OUP, 2000). After introductory chapters setting out some key problems and sketching some history, there is a group of chapters on what Shapiro calls ‘The  Big Three’, meaning the three programmatic ideas that shaped so much philosophical thinking about mathematics for the first half of the twentieth century — i.e. varieties of logicism, formalism, and intuitionism. Then there follows a group of chapters on ‘The Contemporary Scene’, on varieties of realism, fictionalism, and structuralism. This might be said to be a rather conservative menu — but then I think this is just what is needed for a very first introduction to the area, and Shapiro writes with very admirable clarity.

By comparison, Mark Colyvan’s An Introduction to the Philosophy of Mathematics (CUP, 2012) is far too rushed to be useful. And I would say much the same of Øystein Linnebo’s Philosophy of Mathematics (Princeton UP, 2017). David Bostock’s Philosophy of Mathematics: An Introduction (Wiley-Blackwell, 2009) is more accessible, but — apart from a chapter on predicativism — covers similar ground to the earlier parts of Shapiro’s book, but has little about more recent debates.

A second entry-level book, narrower in focus, that can also be warmly recommended is

  1. Marcus Giaquinto, The Search for Certainty (OUP, 2002). Modern philosophy of mathematics is still in part shaped by debates starting well over a century ago, springing from the work of Frege and Russell, from Hilbert’s alternative response to the  “crisis in foundations”, and from the impact of Gödel’s work on the logicist and Hibertian programmes. Giaquinto explores this with enviable clarity: this is really exemplary exposition and critical assessment. A terrific book.

Then, before moving on, I have to mention that most accessible of modern classics:

  1. Imre Lakatos, Proofs and Refutations (originally published in 1963-64, and then in expanded book form by CUP, 1976). Textbooks tend to present developed chunks of mathematics in a take-it-or-leave-it spirit, the current polished surface hiding away the earlier rough versions, the conceptual developments, the false starts. Proofs and Refutations makes for a wonderful counterbalance. A classic exploration in dialogue form of the way that mathematical concepts are refined, and mathematical knowledge grows. We may wonder how far the morals that Lakatos draws can be generalized; but this remains a fascinating read (I’ve not known a good student who didn’t enjoy it).

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