From: epfaith@twinkies.berkeley.edu (Edward Paul Faith)
Newsgroups: sci.philosophy.tech
Subject: Re: Sets in mathematics
Date: 12 Apr 92 01:57:06 GMT
Article-I.D.: agate.s85diINNnet
Organization: U.C. Berkeley Math. Department.
NNTP-Posting-Host: twinkies.berkeley.edu

In article<15204@optima.cs.arizona.edu>, gudeman@cs.arizona.edu 
(David Gudeman) writes:

>Proofs that various mathematics can be modeled in set
>theory is not foundations, since sets are no more fundamental than
>other objects of mathematics.  If that was your original point, then I
>can agree with it.

Among other things, I meant to express that view.  I expressed it
badly, I know.

>I'm
>inclined to be more interested in the actual _practice_ of mathematics
>than in the (rather naive) philosophy of its practitioners.

What is the astounding fact here?  The astounding fact seems to be
that even though mathematicians don't understand what they are doing,
as you put so diplomatically, nevertheless what they do works.
Mathematicians make mistakes, but these are eventually recognized as
such without need for deep reflection as to the nature of mathematics.
Mathematics all the way back to Euclid is perfectly usable today. This
is not true of alchemy, which is one percent gold, 99 percent lead.
Even other maths, even Babylonian and Chinese maths, are pure gold.
Why is this so?  Mathematicians do the right thing despite themselves.
One question is, why is math so self-correcting?  That is, why is it
so eventually clear what is a mistake and what's not? Another question
is, why does math always give the right practical results (unless it
is due for a self-correction)?  A third of course is, why does it
always appear, and why do the various maths converge so much in their
results, so that two separate societies will often discover the same
theorems?

>The point is that mathematics is rife with hidden
>assumptions --or it was before the early work in foundational studies.
>The current state of affairs is that mathematics is rife with
>ill-founded assumptions that are at least recognized.  The purpose of
>foundational theories is to find ways to support these assumptions
>that are already being used.  But as I said, I do not mean to suggest
>that I think the current trends are doing a good job of it.

I assume you are saying that foundational mathematics is there to fix
math, not to justify it.  Foundational studies have as their raison
d'etre the reduction of the number of self-corrections that math has
to go through.  Mathematicians who study foundations want to be able
to trust math more, and this means, not that they want to understand
its nature better, but that they want to be sure that the methods they
develop are less likely to at some future date be refuted.  This is
how I interpret your use of "support".

But I also want to understand math, to understand why it is so free of
flaws, with or without foundational support, and to understand why it
can be used to say things about the real world. I think that you share
these concerns to some extent.


These two questions may have the same answer.  Mathematics works
partly because mathematical discoveries about mathematics can be
trusted to apply to the real-world activity of math. But if I write