[FOM] Tim Gowers work (was: One Real Number)

Dean Buckner Dean.Buckner at btopenworld.com
Sat Jun 28 10:47:01 EDT 2003

Alexander Zenkin wrote:
>Dean Buckner raises, as usually, an important epistemological question as
>to Foundations of Mathematics (FOM).

I wish I could claim the insight.  Actually, I was reading some papers by
Tim Gowers, of whom more in a moment, which reminded me of a story by the
Argentian writer Jorge Luis Borges (1899-1986) about a library in which
every possible book has been written.  Borges point is, first, that the
library mostly contains rubbish (a whole book containing the repeated
sequence "MVC"), second, that it is all rubbish, simply because the
apparently meaningful stuff is mixed up with the rubbish.  Everything is
there.  (I nearly didn't send the posting in case people thought I'd been
smoking something.)

Onto Gowers.  He recently was awarded the Fields medal, which presumably
means something to people on this group.  But also, unusually for a
mathematician, he writes very good philosophy.  I recommend his site, to be
found somewhere on


particularly his piece "A dialogue on the need for real number system", and
the stuff on Richard's paradox.

Gowers has a Wittgensteinian slant, and has a healthy suspicion of set
theory.  For example, "If R is an equivalence relation on a set A and x
belongs to A, then the equivalence class of x is the set E(x)={y in A: xRy},
but as the proof proceeded, every time I wrote down a statement such as "z
is an element of E(x)" I immediately translated it into the equivalent and
much simpler non-set-theoretic statement xRz."

His most interesting stuff is on the idea of definable sets.  He has the
Wittgensteinian insight that perhaps we can do all analysis using just
"definable" reals.  And since we obviously can't say what non-definable
reals are, why bother with them at all?

"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)

Anyway, this is my slant on his slant, see for yourselves.

Dean Buckner

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