When I was a lad, a long while since, the usual English alternative to tea seemed to be a beverage known as ‘milky coffee’. Black coffee was suspiciously foreign or downright louche — and indeed had to be drunk as a bohemian pose rather than for enjoyment since the coffee itself seemed almostly universally awful and (the cost of beans being as it then was) very weak.

Now of course, we have a sophisticated coffee culture here. Sort of.

Well, you can get a fairly decent espresso macchiato in Caffè Nero (if you gently nudge the barista to doing it properly — the clue is in the name, guys: mark with a just bit of steamed milk/foam, but don’t try to fill the espresso cup to overflowing!). But I couldn’t help but notice that today what is usually the majority choice was universal: everyone else in the busy place had either a huge cappuccino (surely three times the size of anything you’d get in Italy) or a giant mug of latte. Fifty-something years on, milky coffee still rules, OK?

I’ve raved here about the Pavel Haas Quartet before. But we had another chance to see them live at the Wigmore Hall last night. And they were as terrific as ever.

The Quartet were on fire from the start, playing Schnittke’s String Quartet No.3 with passion, commitment, authority and utterly convincing musicality. The only flaw in the performance was that just before the end, Veronika Jaruskova broke a string: they had to stop and rewind a page or so before finishing. I would have been very happy to have heard a lot more again, and I can only hope they record this.

Then they played the Shostakovich String Quartet No.8 (a piece I know a great deal better). I have never heard a more emotionally overwhelming performance. To borrow a phrase from another reviewer on another occasion, “they seamlessly drifted between fervor and introspection”. The Pavel Haas seem to sit on stage in a unusually tight semi-circle, and the interaction between them is a wonder to watch and gives such emotional intensity to their playing.

The Shostakovich was perhaps the high point of the evening, even compared with what was to come after the interval, when we heard Beethoven’s Quartet in B flat Op.130 with the Grosse Fuge played as the final movement. Primed as we were by the Schnittke and Shostakovich, this late quartet sounded even stranger, more ‘modern’, than usual, and it was played with the same level of emotional drive. I have perhaps heard more wrenching performances of the Cavatina (from the Lindsays in particular), but then the Pavel Haas launched themselves into Grosse Fuge with more fire and extreme attack than even the Lindsays used to give it. Quite astonishing. The audience was besides itself at the end.

The best string quartet now performing? Many think so. Their discography is astonishing. So it was a simply wonderful chance to hear the Takács Quartet live in Cambridge last night in the very intimate surroundings of the Peterhouse Theatre which seats just 180. They played the two Quartets Op 51 by Brahms, and Haydn’s Quartet in D, Op 76 no. 5. Haydn apart that was not, to be honest, a programme I was particularly looking forward to, Brahms not being a favourite of mine. But the performances were wonderful, with such rhythmic verve and expressive range, such seamless ensemble, such emotional attack, that I was much more convinced by the Brahms than I’d ever been before. While the Haydn was simply the most joyful experience. The audience was obviously bowled over, and the Takács seemed to respond warmly to the considerable intimacy of the setting, which must be unusual for them these days. Three cheers, and more besides, to the very generous donors who make the Camerata Musica series possible. Next up, András Schiff playing the last three Beethoven sonatas …

[The Guardian has reviewed the Takács’s two Brahmsian programmes at the Wigmore Hall.]

So here’s how to spot IGT2 out in the wild, lurking shyly on the bookseller’s shelves, trying to hide its gaudy plumage … Distinctive yellow flashes, and quite a bright blue-green spine with white markings. Multi-coloured on the front. Thought you’d like to see.

OK folks: you just know you want one for yourself. Cambridge locals will find it already in the CUP bookshop (and can get 20% off the official price of £19.99 for the pbk, with their uni card). Amazon UK has it for 12% off and free delivery. Amazon USA has a rather more measly discount, and you’ll have to wait a few weeks for the book to arrive across the Atlantic. But it is still a bargain for a book that weighs in at xvi + 388 larger format pages, and even contains some timeless truths.

Now I’ve just got to finish getting all the exercises up on the website …

Philosophia Mathematica seeks to publish half a dozen survey articles on current and emerging areas of interest in philosophy of mathematics written by early-career philosophers.

If you are interested, please submit a 500-word sketch of the area that you propose to survey, along with a CV containing details of your publications, to the editor, Robert Thomas by one of two deadlines, March 31 and September 30, 2013. (At March 31, submissions will be considered and some topics assigned. Until Sept. 30, further submissions will be considered as they arrive and may be assigned, subject to not duplicating a previous assignment.)

Inquiries are welcome. Publication will occur individually as articles are ready. Submissions of articles and shorter discussion notes on the subjects of special issues and any other topic in philosophy of mathematics are always welcome.

Suppose, just suppose, that one of the very greatest and most loved actors of his (or her) generation was tackling one of the high peaks of the repertoire, returning perhaps to a previous near-triumph with a quarter of a century’s more experience. Every broadsheet would have extensive reviews, though telling most readers about a performance they will never see (even if in reach of London and in funds to make the journey), because the tickets have long since sold out. There will be interviews with the actor, even colour-supplement spreads. You know how it goes.

Now here is one of the very greatest and most admired pianists of her generation, still at the very height of her powers, returning to Schubert’s last piano sonata, a quarter of a century after a very fine earlier recording. You would have thought that the broadsheets with arts pages would at least notice this major event. After all, nearly all their readers can afford the CD, which is so accessible that it will arrive at the click of a button. So here surely is something worth reviewing. Here too perhaps is even an occasion for a retrospective pieces on a remarkable artist.

But no. As far as I’ve seen, nothing.

Which is not at all unusual these days. It is difficult not to feel (as you do when look round your fellow concert-goers, noting all the grey heads) that slowly but inexorably a deep engagement with classical music is becoming less and less central to our cultural life.

Well here, let it be said, is the most extraordinary music-making, indeed inviting the most personal engagement. Maria João Pires offers us a performance of the A Minor Sonata D845 played with intense intimacy. There is nothing declamatory here: she is sitting across the room, playing for us listening, close around her. There’s a care to every phrase which isn’t mannered, wonderful lightness of touch when called for, and moments when time is slowed to a pause (her magical way with the Trio of the third movement). For this sonata alone you will want the disc.

But then there is D960. What is to be said? For many, this is very high on the list of the music that matters the most, that has to be returned to time and again. We have a most wonderful inheritance of recordings from Schnabel onwards of this “music which … is better than it can be performed”. Brendel, Richter and Imogen Cooper, all more than once, Lupu, Kovecevich and Uchida — all are stunning in their different ways, all their recordings are to be listened to repeatedly. And Pires herself recorded the sonata 25 years ago.

It would be absurd (or at least absurd for me) to try to make comparisons. Let me just say that this new recording is surely a more than a worthy addition to that great inheritance. This is not one of those more brooding performances where we are made conscious from the beginning that this is the work of a dying man (performances which give the first movement such ominous weight as to unbalance the whole sonata). There is an intimate directness to her undeclamatory opening: again, we are sitting with Pires — she is not addressing us across a concert hall. And it is only slowly that the intensity is ratcheted up in the first movement (especially about 11 minutes in), and then we are gripped by a new tension as the opening theme and development return once more. The following Andante sostenuto is unsentimental, played with a rhythmic delicacy that becomes magical. The Scherzo is played vivace con delicatezza as Schubert asks: but Pires deconstructs the Trio with bass emphases which I can’t recall being made quite so aware of before, harking back to previous movements. The final Allegro has moments of lightness again but also a certain weight and drive giving a more-than-satisfying balance to the whole sonata, the tumbling final chords finishing in a sudden silence.

This won’t replace your other recordings of D960, how could it? But you will want to listen and listen again, and you will hear more from Pires each time. Wonderful.

Some 300 people have downloaded the February version of the Teach Yourself Logic Guide, so I guess I’m not entirely wasting my energy. Sure, the Guide won’t be a particular exciting project for many: so I must try to get back to blogging about more interesting stuff. But for those who are following, here — far out of chronological sequence — is a draft entry for the Big Books chapter on Goldstern and Judah’s The Incompleteness Phenomenon.

A number of people have recommended this book to me, so I thought I should take a closer look. Well, I have, and I wasn’t bowled over. What am I missing?

Half of The Incompleteness Phenomenon: A New Course in Mathematical Logic by Martin Goldstern and Haim Judah (A.K. Peters, 1995: pp. 247) is a treatment of first-order logic. The rest of the book is two long chapters of just the same length, one on model theory, one on incompleteness and a little on recursive functions. So the emphasis on incompleteness in the title is somewhat misleading: it is at least equally an introduction of some model theory. I have had this book recommended to me more than once, but I find myself immune to its supposed charms (I too often don’t particularly like the way that it handles the technicalities): your mileage may vary.

Some details Ch. 1 starts by talking about inductive proofs in general, then gives a semantic account of sentential and then first-order logic, then offers a Hilbert-style axiomatic proof system.

Early on, the authors introduce the notion of $latex \mathcal{M}$-terms and $latex \mathcal{M}$-formulae. An $latex \mathcal{M}$-term (where $latex \mathcal{M}$ is model for a given first-order language $latex \mathcal{L}$) is built up from $latex \mathcal{L}$-constants, $latex \mathcal{L}$-variables and/or elements of the domain of $latex \mathcal{M}$, using $latex \mathcal{L}$-function-expressions; an $latex \mathcal{M}$-formula is built up from $latex \mathcal{M}$-terms in the predictable way. Any half-awake student is going to balk at this. Re-reading the set-theoretic definitions of expressions as tuples, she will realize that the apparently unholy mix of bits of language and bits of some mathematical domain in an $latex \mathcal{M}$-term is not actually incoherent. But she will right wonder what on earth is going on and why: our authors don’t pause to explain. (A good student who knows other presentations of the basics of first-order semantics should be able to work out after the event what is going on in the apparent trickery of Goldstern and Judah’s sort of story: but this isn’t the way to start, without adequate explication of the point of the procedure.)

Ch. 2 gives a Henkin completeness proof for the first-order deductive system given in Ch. 1. This has nothing special to recommend it, as far as I can see: there a lot of more helpful expositions available. The final section of the chapter is on non-standard models of arithmetic: Boolos and Jeffrey (Ch. 17 in their third edition) do this more approachably.

Ch.3 is on model theory. There are three main sections, ‘Elementary substructures and chains’, ‘ultra products and compactness’, and ‘Types and countable models’. So this chapter — less than sixty pages — aims quite high to be talking about ultraproducts and about types. You could read it after working through e.g. Manzano’s book: but I certainly don’t think this chapter makes for an illuminatingly accessible first introduction to serious model theory.

Ch. 4 is on incompleteness, and the approach here seems significantly more gentle than the previous chapter. The authors make things easier for themselves by adopting a version of Peano Arithmetic which has exponentiation built in (so they don’t need to tangle with Gödel’s beta function). And they only prove a semantic version of Gödel’s first incompleteness theorem (the authors don’t say anything about why we might want to prove the syntactic version of the first theorem, and don’t even mention the second theorem). The proof goes as by showing directly that — via Gödel coding — various syntactic properties and relations concerning PA are expressible in the language of arithmetic with exponentiation (in other words, they don’t argue that those properties and relations are primitive recursive and then show that PA can express all such properties/relations). But this isn’t done particularly well: I think this sort of more direct assault on incompleteness is better handled in Leary’s book (recommended in the Guide).

The book ends by over-briskly introducing the ideas of primitive recursive and recursive functions.

Summary verdict The first two chapters of this book can’t really be recommended either for making a serious start on first-order logic or for consolidatory reading. The third chapter could perhaps be used for a brisk revision of some model theory if you have already done some reading in the area. The final chapter about incompleteness (the title of the book might lead you to think that this will be a high point) isn’t a helpful introduction: it could be skimmed through to see how the authors approach things, but it doesn’t really go far enough for more serious purposes.

Right. I really must get on with other work (in particular the task of writing exercises for the Gödel book awaits). But, as an exercise in constructive procrastination I’ve just uploaded the February 2013 version of the Teach Yourself Logic Guide.

Chapter 1 on The Basics is in a reasonably stable state, and I’ve only tinkered with that in small ways since the last version as uploaded back in November. I’ve added another section to Chapter 3, ‘Exploring Further’. But the big change is that I’ve started work on the new Chapter looking at some of the Big Survey Books on mathematical logic. There’s quite a long list to work through — I must be mad to have taken this task on myself! — so don’t hold your breath waiting for the entry on your favourite book. Still, between you and me, it has been enjoyable to dive in, blow the dust off some old acquaintances, and remind myself what they get up to. So I do plan to continue adding entries sporadically.

I’ve added to the Guide the entries on the classic texts by Kleene, Mendelson and Shoenfield of which I posted drafts here. As a bonus I’ve also just added an additional entry on Joel Robbin’s 1969 book Mathematical Logic: A First Course. Yes, yes, that doesn’t really come next in chronological order and it isn’t exactly a Big Book either (the main text is just 170 pages). But it does cover an interesting amount in a short space. And having been a bit grouchy about Mendelson and very grouchy about Shoenfield, I’m inclined to be rather warm about this. My summary verdict is

A different route through this material [first- and second-order logic, primitive recursive arithmetic, PA_{2}, Gödel’s theorem], Robbin’s book is accessibly written and still worth reading. Look especially at Ch. 3 for the unusually detailed story about how to build a language with a function expression for every p.r. function, and at the last chapter for how to work in PA_{2}.

I think I might be going to regret this. I’ve just realized that there are over 25 “Big Books” on math logic on my shelves that could reasonably warrant a mention in the planned chapter in the Teach Yourself Logic Study Guide. So no promises to get everything finished quickly! Still, the chronologically next entry writes itself pretty easily. So let’s get it out of the way.

Joseph R. Shoenfield’s Mathematical Logic (Addison-Wesley, 1967: pp. 334) has, over the years, also been much recommended and much used: it is officially intended as ‘a text for a first-year [maths] graduate course’. This is hard going, a significant jump up in level from Mendelson, though often the added difficulty in mode of presentation seems to me not to be necessary. The book can probably only be recommended to hard-core mathmos who already know a fair amount and can cherry-pick their way through the book. It does have heaps of hard exercises, and some interesting technical results are in fact buried there. But whatever the virtues of the book, they certainly don’t include approachability or elegance or student-friendliness in the early chapters.

In a bit more detail, Chs. 1–4 cover first order logic, including the completeness theorem. It has to be said that the logical system chosen is rebarbative. The primitives are $latex \neg$, $latex \lor$, $latex \exists$, and $latex =$. Leaving aside the identity axioms, the axioms are the instances of excluded middle and instances of $latex \varphi(\tau) \to \exists\xi\varphi(\xi)$, and then there are five rules of inference. So this neither has the cleanness of a Hilbert system not the naturalness of a natural deduction system. Nothing is said to motivate this horrible choice as against others.

Ch. 5 is an introduction to some model theory getting as far as the Ryll-Nardjewski theorem. But this will done far too rapidly for most readers (unless you are using it as a terse revision course).

Chs. 6–8 cover the theory of recursive functions and formal arithmetic. Schoenfield defines the recursive functions as those got from an initial class by composition and regular minimization. As elsewhere, ideas are presented in a take-it-or-leave-it spirit, and no real motivation for the choice of definition is given (and e.g. the definition of the primitive recursive functions is relegated to the exercises). Unusually for a textbook at this sort of level, the discussion of recursion theory in Ch. 8 goes far enough to cover a Gödelian ‘Dialectica’-style proof of the consistency of arithmetic, though the presentation is not particularly accessible.

Ch. 9 on set theory is perhaps the book’s real original raison d’être. It is a quarter of the whole text and was (if I recall right) the first extended textbook presentation of Cohen’s independence results via forcing, from four years previously. The treatment also touches on large cardinals. This was surely an admirable achievement in its time: but it is equally surely not now the place to start with set theory in general or forcing in particular, given the availability of later presentations. [Or at least, that was my first thought, based on memory: but I’m inclined to go back and revise judgement at least on the set theory chapter.]

Summary verdict Now only for very selective dipping into by already-well-informed enthusiasts.

One of the most wonderful CDs of Bach keyboard music (indeed surely one of greatest Bach recordings ever) is Martha Argerich’s 1979 disc of the Partita No. 2, the Toccata BWV 911, and the English Suite No. 2. She plays with a quite extraordinary combination of clarity and sensitivity, of life and intensity. A hard act to follow.

Yet here in a new release is the young French pianist David Fray, also playing the Partita No. 2 and the same Toccata, but this time completing the disc (more generously) with the Partita No. 6. Surely the overlapping choice of programme here isn’t accidental. He is inviting a comparison. How does Fray’s disc stand up?

I thought (and still think) that his recording of the Schubert Impromptus Op.90 and the Moments Musicaux is stunning (much better in this repertoire than the mannered Paul Lewis). I haven’t, though, heard his previous much admired Bach discs. But I’m now a new convert here too. This disc is surely remarkable. Fray has said “We shouldn’t be afraid of acknowledging the expressiveness of Bach’s music”. And this is indeed expressive playing (yet within reason, without exaggeration), and again there’s great clarity and intensity. There is, for example, appropriate introspection in the 2nd Partita’s Sarabande and zest in the 6th Partita’s Air. This is overall a truly thoughtful and impressive recording. Warmly recommended.