From: israel@math.ubc.ca (Robert Israel)
Newsgroups: sci.math
Subject: Re: What is mathematics, really?
Date: 12 Dec 1997 02:28:29 GMT

In article <881794611.805371906@dejanews.com>, Ted R Shoemaker <shoematr@uwec.edu> writes:
 
|> Do you mind some questions?  I have read of different views among
|> mathematicians, regarding this question.  I do not claim to understand
|> these positions correctly. 

You are correct in not claiming to understand them.

|> One view is "formalism", which, if I
|> understand, means that mathematics is the manipulation of symbols,
|> regardless of meaning.

No.  Formalism was the attempt to put mathematics on a secure footing by
(1) producing a formal system which is capable of expressing all of mathematics.
Within the formal system, proofs consist of manipulations of symbols according
to fixed rules, which do not take into account any notion of meaning.  This
is not to say, however, that the mathematical objects themselves lack meaning,
or that this meaning is not important. 
(2) proving that the formal system is consistent, i.e. will never prove both
"A" and "not A".     

Formalism died in the 1930's as a result of the work of G\"odel.

|> "Constructionism" holds that the *idea* of a
|> number is irrelevant; nothing exists until someone actually makes it. 
|> The example given was: Take N arbitrarily large.  Pi does not exist,
|> because we do not know pi to N decimal places.	Don't try to prove the
|> existence of pi by using circles; they don't exist either.

If you mean "constructivism", that basically says that you can't claim to have
proved that a mathematical object exists unless you have an algorithm
to construct it.  This "construct" is not to be taken too literally: we may 
never know the 10^(10^10)'th digit of pi, but we have an algorithm that
would in principle produce it if we had enough time and memory available.
In the case of a real number, "construct" means essentially that you have
a procedure that will produce arbitrarily close approximations to it.  
So pi is a perfectly acceptable number for a constructivist.  On the other
hand, almost all of the classical analyst's real numbers are not acceptable
to the constructivist, because they don't come with such a procedure; in 
fact there is no way to put any kind of an identifying "label" on them, because
there are too many numbers and too few labels.

|>  The
|> constructionist says that math is an arbitrary, artificial product of the
|> human mind.

No.  If it was arbitrary, there would be no point in arguing about
which methods are valid and which are not.

Robert Israel                            israel@math.ubc.ca
Department of Mathematics             (604) 822-3629
University of British Columbia            fax 822-6074
Vancouver, BC, Canada V6T 1Z2