[FOM] Re: Tim Gowers work

[Dean Buckner]

Bill:
> I'm sorry to see it [Gowers work] called Wittgensteinian though, as what I
know of him I dislike intensely!

Try the bits in between Tractatus and Investigations.  It's not so bad.  For
example:

Wittgenstein:
§ 181  " I'm tempted to say, the individual digits are always only the
results, the bark of the fully grown tree.  What counts, or what something
new can still grow from, is the inside of the trunk, where the tree's vital
energy is.  Altering the surface doesn't change the tree at all.  To change
it, you have to penetrate the trunk which is still living.

"Thus it's as though the digits were dead excretions of the living essence
of the root.  Just as when in the course of its vital processes a snail
discharges chalk, so building onto its shell.

"§ 188   There is no number outside a system.  The expansion of pi is
simultaneously an expression of the nature of pi and of the decimal system.

Compare with Gowers: " the real numbers, defined as decimals, can be added,
multiplied and so on, in a natural way, and that adopting this natural way
as a definition, one finds that there is indeed a number x that squares to
2. In other words, I didn't just define the square root of two. Rather, I
defined an entire number system and showed, by which I mean actually proved,
that the square root of 2 exists in that system. "

Bill:
> While noting that this must always be only what we *mean*, we can never
say it directly,

You mean, we can as it were *mean* it, but not *say* it?  That sounds v.
Wittgensteinian!  Actually, Gowers goes further:  "... it is still tempting
to think along the following lines: yes, the paradoxes go away if one is
precise about what is meant by `definable', but, given any precise notion T
of definability, we can always stand outside it and genuinely understand the
definition of a number, even if it is not T-definable. For example, if I
grasp the definition of T-definable, then I will grasp the definition of the
non-T-definable number y produced by a diagonalization process".  He then
considers " the set of all real numbers that can be defined, not necessarily
according to some formal system, but just in some unambiguous way that any
mathematically trained person could in principle understand. " - and gets a
diagonalisation out of that!!

> "definable" objects
> are somewhat clear-ish to the intuition, but can never be totally
> encompassed therein.    Briefly, "definability" is not definable!

Except we have the same Gowerish objection, regarding "set of objects
somewhat clear-ish to the intuition, but which can never be totally
encompassed therein".

(Although I'm not quite sure what you mean by the quoted description)

Gowers:
>  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 ...

Bill:
> Exactly so, again!

Me:  What does "definable bijection" mean?  Is there any other sort?

Is it like this:  I construct a program that goes through all possible
finite strings of symbols, starting with strings of length 1, then length 2,
then 3 and so on forever (I have a special infinitely fast processor and
infinite RAM).  For each generated string, I then have a subroutine that
checks whether the string is a sensible algorithm for generating digits.
Note that any string including a sequence like "the nth digit of the nth
sensible algorithm " will be thrown out, because this string, if sensible,
will have a number m, cannot therefore generate an mth digit, and therefore
cannot be sensible.  However we can then write a second program that takes
the (infinite) array of algorithms generated byt the first program, and
which uses some diagonalisation to generate a new number that was not
generable by any algorithm in the array.  And then I could write a third
program that ...

So we could not write a program (i.e. a finite sequence of symbols) to
generate all finite programs.  But could there be an infinite program?
Elsewhere G. writes "a set X of positive integers is a rule which tells you,
for each positive integer n, whether or not n belongs to X, except that the
rule can be totally arbitrary and impossible to specify in a finite time . "
What does that mean?  And is such an arbitrary rule the opposite of
"definable bijection"?

I looked up "definable bijection" on Googol and got a small number of
things.  Looking up "non-definable/indefinable/arbitary bijection" I go
nothing.


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.


Dean



Dean Buckner
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