[FOM] Work of Timothy Gowers
Alasdair Urquhart
urquhart at cs.toronto.edu
Fri Jul 4 11:18:59 EDT 2003
In his posting of Saturday June 28, Dean Buckner
represents Timothy Gowers as having "a Wittgensteinian
slant" and as espousing the view that all of real analysis
can be done using only definable real numbers. He gives
a lengthy quote from a dialogue posted on Gowers's
web site, expressing a view that he identifies as Gowers's
own point of view:
"The "theorems of analysis" you refer to concern objects I don't like and
don't have an obvious use for, like "arbitrary" sequences. I'd still be able
to do all the useful stuff wouldn't I? For example, the intermediate value
theorem would be true for definable continuous functions, and I'm not too
worried about any others. In fact, I could end up speaking exactly the same
language as you, but mean something slightly different by the phrase "for
all x", which for me would mean "for all definable x", whatever sort of
object x might be. In fact, I could even say that the reals were
uncountable! What I'd mean by this in your terms is that there is no
definable bijection between N and the definable reals, which there isn't
because then I could apply a diagonal argument and define a real not in the
image. " (Dialogue, p. 8)
An examination of the material posted on Gowers's web site shows that
this is a complete misrepresentation. The dialogue in question, which
is available at:
http://www.dpmms.cam.ac.uk/~wtg10/reals.html
"... is between three imagined characters: M, a mathematician
who takes the real numbers for granted, S, a sceptic who is not prepared to
learn about anything without being absolutely convinced that it is
necessary, and U, an undergraduate who has recently learnt basic analysis.
Towards the end, a logician, L, tries to sort out
some of the mess. (My apologies if, in my ignorance,
I put words into the logician's mouth that no self-respecting logician
would utter.) " (This is Gowers's own description.) 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.
By the way, I strongly recommend Gowers's web site to FOM subscribers.
It has a lot of interesting material on it. I haven't had time
to study all the material, but I don't believe there is
the slightest reason to ascribe anything resembling a Wittgensteinian philosophy
to Gowers.
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