2018

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?

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

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!

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 …!)

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.

Suppose you read this exposition:

Frege’s conception of a function is closely related to his discovery that quantifiers like (“for all”) and (“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) and  and .

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)

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)

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)

not as simply ‘’  nor as ‘’ (neither does any gap filling!) but rather as something we might represent as ‘’  which is applied to the gappy ‘’.

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 …