# RE: this very moment

From: Chris Simmons <cps102.domain.name.hidden>
Date: Thu, 11 May 2000 20:50:10 +0100 (BST)

Hello. In reply, here's some new (?) stuff that I've been thinking about,
which hopefully you find appropriate to this discussion - especially given
that there has been a previous descussion about the ultimate theory of
everthing possibly consisting of nothing more than all mathematical
constructs.

So the problem is, what would 'all mathematical constructs' consist of? I
think it would be a good idea to firsty figure out what the essential
ingredient of all mathematical constructs to date are, to whit are some of
my thoughts.

Firstly we're going to make a few simplifying assumptions, which hopefully
you wont object to too much. They seem pretty sensible to me. These are
that, at any given time, our mathematics in its entirety can be considered
as a language (in the formal sense) L over some finite alphabet A, and in
particular, *this language is finite*. To clarify, an alphabet A is just
a non-empty set, and a language L over an alphabet A is simply any subset
of A^+, where A^+ denotes all formal (non-empty) sequences of elements of
A.

I'll explain why I think these assumptions are okay. Basically, we could
let A = {0,1}, and if we assume that any mathematical text could be
translated into, say, LaTex or some other appropriate system for
representing maths, every mathematical text can be reduced to a sequence
of bits. The language L we are refering to, then, would consist of the
languange of all text files which LaTex compiles into a piece of correct
mathematics (which has already been written, note!).

Now the big question is, what's so special about this language L?? Is
*anything* special about it? In particular, given some such finite
language L, how would we decided whether it is mathematics or not? Do
these questions make sense??? :)

I suspect that the language is to some extent unimportant. After all, any
finite language can be recognised by a finite state automata, and who
would like to claim that mathematics is nothing but structures recognised
by FSA's? It seems to me that the only way of reconsiling this is by
accepting that, when someone reads a mathematical text, we are performing
*computations*, and it is this additional structure, along with the
language, which is important. This would mean that we might consider a
finite language L to be 'mathematical' if it is sufficiently
well-structured as to allow us to use it to describe abstract constructs
(ie we can 'parse' the language, and imagine that in doing so, the words
in the language each describe some structure).

But, to be honest, I really don't know. What do you lot think?

Chris Simmons.
Don't bother, the web page is crap ;)
http://www.york.ac.uk/~cps102
Received on Thu May 11 2000 - 12:54:56 PDT

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