I suggest changing the thread, since it's a new subject.
>L over some finite alphabet A, and in
>particular, *this language is finite*.
This is not consistent with what you say later.
>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!).
But there is an infinite number of such files.
>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??? :)
They all make sense, and have been extensively worked on (except that I
don't see why you insist on your language being finite)
The answer is yes : you can run a program telling you if a mathematical
demonstration written in a formal language and using formal logic is correct
or not. The problem is that, except for very very simple theorems, no one
can really write math proofs in a formal language. So automatic
demonstration (which is a well developped branch of computer science) can't
be applied to "real" maths.
>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?
In principle, it is...and it is not ! What these automata tell you is that
the derivation of a theorem is correct or not. But this demonstration relies
on the axioms of the theory. So the choice of axioms is crucial, and these
automata will never tell you if you have chosen a good system of axioms for
what you had in mind (they can sometimes tell you that the system is
inconsistent if they happen to meet such an inconsistence, but that's all).
>From the formal point of view the axioms have no meaning, they are just
sentences in your language, but for a mathematician they have an intuitive
meaning (which may or may not be right).
>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.
I don't know if this is what you have in mind, but when you read a math
text, you do a lot of things : accept to use notations which are obviously
ambiguous, use many times the same name to call different things, fill the
blanks in every proof with trivialities the author did not care to write
down, and finally, write the proof yourself when the author says "this is
obvious" (almost 50 percent of the time) !
---------------------------------------------------------------
Fabien Besnard
http://perso.wanadoo.fr/fabien.besnard
Received on Thu May 11 2000 - 15:36:36 PDT