# Year: 2018

The end of a year. Time to reflect. How did those beginning-of-year “must do better” plans work out? Not that I’m usually an enthusiast for them: but, for once, this last year I did make two fairly serious resolutions.

The first was relatively easy to keep — to lose quite a bit of weight. The secret? You just eat fewer carbs, exercise more. And keep going. Who could possibly have guessed? So, primed with this wonderful new insight, I’ll now have to write my sure-to-be-best-selling life-style-and-diet book. Move over Gwyneth. I expect to make a fortune.

The other resolution initially took significantly more will-power to put into practice. Keep off the internet from mid-evening. For a start, try to ignore Twitter, the newspapers, the political magazines, the political blogs, and the rest. Read novels and the like instead. In these days of Trump and Brexit and more, I really can’t recommend this enough for the sheer improvement to eudaimonia and well-being. Try it for a week (you can do it!), and then another week … You won’t regret it, trust me!

And I’ll add, don’t read the novels onscreen: there’s still something about sitting down with a real printed book that seems to engender a different level of engagement (and I don’t think that that’s just me).

I keep a list. I didn’t quite get to a book a week (unless I’m allowed to count the likes of Dombey and Son as more than one!). But I have read exactly twice as many novels and other books this year as the year before, and I feel I’m reconnecting to that much earlier self who seemed to have endless time for non-work-related reading. The novels’ explorations of our  human world, the delights of encountering wonderful writing, the sheer fun of getting caught up in a story, have all given great pleasure.

There’s no plan to the reading, other than a rough intention to mix up classics and recent books, and first readings with re-readings. And sheer chance plays a large part:  what turns up in a favourite haunt, the beautifully run Oxfam bookshop in Saffron Walden? Indeed, such serendipitous finds have been among the most enjoyable — Madeline Miller’s Circe, the collected poems of U.A. Fanthorpe, the novels of Helen Dunmore (I’m reading through an as-new set of the first ten, bought for a pound each …).

And what am I reading as the  year ends? Clive James’s long poem The River in the Sky (which isn’t entirely working for me, but has its magical moments); Sue Prideaux’s I am Dynamite (not that I am a Nietzsche fan, but it promises to be a rollicking read); and, not least, this winter’s Dickens, Our Mutual Friend. All more than good enough to keep me happily sticking to that resolution to avoid frittering time (and to avoid getting stressed!) on the internet.

## Three tweets wiser

I asked three different questions on twitter recently. Pity not to pass on what I learnt in a slightly more long-lasting form! In ascending order of likely  interest:

1. How do you pronounce “wff” in the classroom? Approximately woof seemed the majority view. Which is how I’ve always pronounced it. Some, oddly to my mind, prefer wiff. Some, apparently, spell it out w-f-f. Joel Hamkins wondered why we should use “wff” at all – why not just “formula”? Which is a very good question. The habit of a lifetime makes me a bit resistant to change, however!
2. What’s a neat example of a written sentence with different meanings in different languages? — (approximate homophones are familiar, but I wanted a nice example that worked on the page). Thomas Brouwer offered the lovely “David Hume was slim”. Falsely saying in English that the bon viveur was svelte of figure, truly saying in Dutch that he was smart!
3. “How many books has J.K. Rowling sold?”, “How many books has J.K. Rowling written?”. We need the distinction between tokens and types to properly construe the likely questions here. And we all know that Peirce was responsible for the nowstandard terminology for this distinction. But surely the distinction is an old one: who first made it (whatever the terminology)? Surely the stoics or other Greek writers talking about words, sentences, lekta, etc. would have somewhere made a type/token distinction? Or what about the medieval writers on logic? My learned twitter friends had no specific pointers to give. Which was a real surprise. What were we missing?

## A Christmas card

With every good wish for a happy Christmas and a peaceful New Year.

## LaTeX for Logicians updated

Setting aside the other logic-related things I really ought to be doing, I’ve just been going in for a bit of constructive procrastination, systematically checking through the LaTeX for Logicians pages for the first time in almost three years. There’s some very minor re-arrangement, some renewing of broken links, and just a few new links.

As I’ve said before, whatever one’s issues and reservations about LaTeX for more general use, it is still surely quite invaluable for logicians. The LaTeX for Logicians pages continue to be heavily visited; so do please let me know how these pages can be improved, what new LaTeX packages of use to logicians that I have missed, etc.

## Postcard from the Bahamas

It’s a long flight, and not the most comfortable (thanks, BA). But lovely to be in the Bahamas again, on a family visit. Brexit news inescapable but all seeming particularly mad from this distance. Time for a lot of reading. I’ve particularly enjoyed Jonathan Raban’s Coasting (again!), Jill Paton Walsh’s A Desert in Bohemia  and am devouring Sally Rooney’s rightly much praised Normal People. Time too for re-thinking the dratted intro logic book: another month, another conception emerges of how best to organize things — but I hope there is at last convergence on a satisfying solution rather than just a continuing random walk though the options!

## Tim Button, Loving the Universe

I was having coffee this morning with Thomas Forster, and we were talking — as you do — about theories with a universal set, and the claim that (a version of) Church’s Theory with a universal set is in fact synonymous with ZF. Doing a bit of googling around after our chat, I find that Tim Button has recently given a characteristically clear and lively talk about very closely related ideas, including developing some of Thomas’s. Here it is.

## Six concerts, and a few CDs

For a small city, there is an extraordinary amount of music-making going on here in Cambridge  — there are many days during term when you can choose from half-a-dozen or more concerts, often of quite stellar quality.  And if you have the great good fortune to live near the centre, as we do, everything is all within easy walking distance too.

In the last few weeks, then, I have been able to hear Mitsuko Uchida play Schubert (though I find her playing has now become more than a little mannered and occasionally excessive in dramatic emphasis), and hear Angela Hewitt play Bach (the second book of the 48, which made for a very demanding evening! — though the wonderful control, the way Hewitt is able to bring out the fugal structures, combined with the emotional range she found from joyful to sombre contemplation, were all terrific). Then was a delightful concert from the Academy of Ancient Music and the BBC Singers of excerpts from Rameau and Lully operas — sheer enjoyment. Which isn’t ever quite the word to apply to a performance of Winterreise, but the prize-winning young baritone Samuel Hasselhorn was impressive and moving (I wasn’t so taken though with the playing of his accompanist on this occasion, the well-established Malcolm Martineau).

Then there were two quartet recitals I had been much looking forward to. The Jerusalem Quartet, however, I found distinctly disappointing. To be sure, their performances (Mozart ‘Hunt’, Beethoven ‘Harp’, Schubert Quintet with Gary Hoffman) were polished — well-engineered as it were. But even though it was the best possible setting, the very intimate space of the Peterhouse Theatre, I just couldn’t emotionally engage. The Takács started too in a rather disappointing way, with a routine-seeming Mozart K387 — though to be fair, this time I was at the back of a largish church, and the setting was far from ideal. But for Shostakovich’s  Quartet No. 4 they were on fire (enlivened by their younger new second violinist?), playing now with drive and intensity and emotional depth. And after the interval, an equally driven Mendelssohn  Quartet No. 6.

As it happens, I’d just been listening to a recommendable new CD of Mendelssohn quartets, from the Doric Quartet (whom I much admire) which also includes a performance of the sixth quartet — though the Elias Quartet or Quatuor Ébène perhaps  bring out the drama more. And talking of Quatuor Ébène, I’ve very much enjoyed this disc which I’ve only just disovered on iTunes, of the quartet playing with Manahem Pressler, as a 90th birthday celebration.

But the stand-out recent CD really has to be the wonderful Chiaroscuro Quartet playing Schubert. As a visceral experience, their Death and the Maiden — though transformed by the different sound world of their gut strings — is up there with the Pavel Haas Quartet’s award-winning recording. Extraordinary.

## Getting Things Right

I used to much enjoy contributing to the admirable Ask Philosophers site (you can check out my efforts here!) Rather sadly I had to give that up. But it seems that I can’t resist the pedagogic imperative. So my energies got  diverted instead to another  admirable question-and-answer site,  math.stackexchange; I occasionally do my bit there, when the spirit moves, to answer some elementary logic questions, trying to  Get Things Right.

A lot of my answers are to questions of no lasting interest. But  I’ve linked to an assorted 140 answers which are of perhaps more than ephemeral interest. There should be something here to amuse, even instruct, students at various levels.

Is contributing to a question-and-answer site like that worth doing? Well, there are far worse ways of procrastinating on the internet! But anyway, I’ve just noticed the estimate for the number of readers for my answers. And while we all know that idle browsing doesn’t in general mean that we are paying much attention, someone is unlikely to be visiting math.se and clicking on the link to an answer without some level of interest. I hope. Anyway, the stats are that approximately one million people have now viewed my answers there. Heavens!

(Actually, I don’t believe that figure at all — given I’ve answered less than a thousand questions and the average readership per question asked looks to be more like a hundred than a thousand: but even if it overcounts by a factor of ten, that’s still a goodly  number of readers.)

Does this count as “impact”, whatever the exact number? Who cares! It is enough encouragement to continue. (But excuse me, must dash,  someone is wrong on the internet, confusing entailment and the material conditional yet again …!)

## The Consistency of Arithmetic

There’s a nice new piece on the Consistency of Arithmetic by Timothy Chow in the Mathematical Intelligencer, which the author has made freely available. As he says in a FOM posting, he has put extra effort  into trying to make Gentzen’s proof accessible to the “mathematician in the street”. Of course, this kind of expository effort will always strike some readers as requiring too much of them to follow, and strike other readers as not going far enough into the details they want. But it seems an admirable effort to me, that students in particular could find pretty helpful.

## What Frege didn’t say about functions and quantification

Frege’s conception of a function is closely related to his discovery that quantifiers like $$\forall$$ (“for all”) and $$\exists$$ (“for some”) operate on what are now called open expressions — expressions containing free variables.

Say we’re interested in a series of calculations like this:

(A) $$\quad 3^2 + 6\cdot 3 + 1$$ and $$4^2 + 6\cdot 4 + 1$$ and $$5^2 + 6\cdot 5 + 1$$.

We soon begin to realize a pattern here; we are taking the square of a number, adding that to the result of multiplying the number by 6, and then adding 1. Following mathematical practice, we depict the pattern by replacing ‘3’ in first example in (A) by ‘x’:

(B) $$\quad x^2 + 6 \cdot x + 1$$

This example pictures a function. Contemporary logicians think of such examples as having a variable reference; when the variable ‘x’ is assigned a number, this will refer to the result of applying the function to that number. Frege thought of (B) as having an indefinite reference. It corresponds to a function, which is something incomplete or unsaturated. Saturation is accomplished, and reference — say, reference to the number 28 — is achieved when a referring expression like ‘3’ is substituted for ‘x’  in (B).

You wouldn’t, I hope, be particularly happy about this as an account of Frege’s thought from a student. Leave aside the fact that dots aren’t yet joined up (to tell us how, for Frege, quantifiers do apply to expressions for functions mapping to truth-values). For a start, you’d want to point out that what express functions for Frege are expressions with gaps not expressions with free variables. So, for example, rather than (B) he would use

(C) $$\quad \xi^2 + 6 \cdot \xi + 1$$

where the Greek letter is very clearly explained as a gap-marker, indicating that the two gaps are to be filled in the same way; and the Greek letters do not strictly belong to the concept-script, but are a convenient device in our metalinguistic commentary. And of course, Frege didn’t think that the likes of the gappy (C) as having indefinite reference. They have a definite reference to a function!

Now, it is true that — as well as the Gothic letters he uses as bound variables in his concept script, and the informal Greek gap markers  — Frege also uses italic Roman variables in his concept script. But Frege wouldn’t use them in an expression for a function comparable to (C) — for they are only to appear in expressions for assertible contents that can follow a judgement stroke.

Moreover, Frege’s Roman letters never occur in the scope of a corresponding explicit quantifier (in fact, they approximately function like parametric letters in natural deduction). For Frege, what quantifier expressions are applied to is — of course — open expressions in the sense of expressions with gaps, not to sentences with free variables. And — in modernized notation — we should think of the Fregean quantifier expression in e.g.

(D) $$(\forall x)(Fx \to Gx)$$

not as simply ‘$$\forall$$’  nor as ‘$$(\forall x)$$’ (neither does any gap filling!) but rather as something we might represent as ‘$$(\forall x) \ldots x \ldots x \ldots$$’  which is applied to the gappy ‘$$(F\xi \to G\xi)$$’.

And so it goes. It is a bit depressing, then, to report that the quotation above is lifted with only minor (and irrelevant) changes and omissions from p. 23 of a newly published CUP book aimed at linguistics students, Philosophy of Language, by Zoltán Gendler Szabó and Richmond H. Thomason. OK, I if anyone should know how difficult it is to write introductory logical stuff without corrupting the youth! But there is surely a boundary to how rough and ready you are allowed to be, and by my lights the authors overstep it here, given it would have been pretty easy to have been significantly more accurate without confusing the reader. And this sort of thing must make you wonder how trustworthy the authors are as guides elsewhere …

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