[FOM] Work of Timothy Gowers

Dean Buckner Dean.Buckner at btopenworld.com
Sat Jul 5 11:19:16 EDT 2003

Alasdair writes that I have misrepresented Gowers' point of view as

> I haven't had time to study all [Gowers'] material, but I don't believe
there is
> the slightest reason to ascribe anything resembling a Wittgensteinian
> to Gowers.

Perhaps he should have taken a little time.  Gowers writes

> I should confess that I am a fan of the later Wittgenstein, and I broadly
agree with > his statement that "the meaning of a word is its use in the
language". [Philosophical > Investigations Part I section 43  ("Does
mathematics need a philosophy")

It is true that what I quoted was from a dialogue, but I did say (29 June)

> this is my slant on his [Gowers'] slant, see for yourselves.

and I did say (3 July)

>I should add that I took the Gower quotes from a dialogue, where different
>characters represent different views.  The quoted views may not be his

Surely that's sufficient health warning?  Anyway, a careful reading of the
dialogue against Gowers' other work, show the three characters represent
sides to his own thinking (as with all good dialogues, which should resemble
an internal debate).
Nobody "wins",  Shortly after the passage I quoted, he writes (L = logician,
U = undergraduate)

>L. Well, the most obvious thing to say is that U. is on the way to
rediscovering so->called constructible set theory. The details are more
complicated than U. is >making them, but the basic thought is sound: when we
talk about the set of all >subsets of an infinite set, it is not clear what
we mean. Most mathematicians just >don't worry about that, and accept a
notion of "arbitrary sequence, whatever that >might turn out to be". But a
perfectly consistent (at least if normal set theory is
> consistent) set theory results if one interprets the power set operation
as giving >you all definable subsets of a given set.

So, the logician says that "the basic thought [the undergraduate's] is
sound".  If the logician's view represents Gowers' own, then it's quite
wrong of Alasdair to say "The quotation given above is attributed to the
character U (the undergraduate) and hence U's remarks in no way represent
Gowers's own view."

Also, I believe Gowers dislikes mathematicians' habit of turning relations,
which are grammatically much more like verbs, into nouns.  Which is
Buckner/Slater, rather than Wittgenstein, but that's even better!

Dean Buckner

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