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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

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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