From: rickert@mp.cs.niu.edu (Neil Rickert)
Subject: Re: Sets in mathematics
Organization: Northern Illinois University
Date: Sun, 12 Apr 1992 05:36:48 GMT

In article <s85diINNnet@agate.berkeley.edu> epfaith@twinkies.berkeley.edu (Edward Paul Faith) writes:
>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.

 I don't like to put words into David's mouth.  But I do think you are
misstating his position.  I didn't find anything in his messages which
implied that mathematicians do not understand what they are doing.  He
does perhaps question whether they understand the philosophical
underpinnings of what they are doing, but that is very different.

>Mathematicians make mistakes, but these are eventually recognized as
>such without need for deep reflection as to the nature of mathematics.

 Russell's paradox pointed out some mistakes which did require deep
reflection on 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.

 This is a rather naive statement.  It is true that a theorem, once
proved, stays proved.  But there have been many things that were thought
to have been proved, yet it later turned out that they were not.

 Quite apart from that, a considerable amount of old mathematics is of
little use.  In Euclid's era geometry was used to prove the equivalent
of
	(a + b)^2  = a^2 + 2ab + b^2

but it would be difficult to make a case that this approach to algebra
is useful today.

>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?

 Is it so self-correcting?

 You only hear of the cases where the errors were discovered and corrected.
You don't hear of the errors which have never been discovered, because
the fact that they have never been discovered means that this information
is not readily available.

>                                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?

 I don't believe this is unique to mathematics.  It happens in other
disciplines too.  It is probably related to the way the mind learns
by building new knowledge on a foundation built from prior knowledge.
Thus, with some sharing of knowledge, the same preconditions for a
new discovery may occur at about the same time in different people.

-- 
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  Neil W. Rickert, Computer Science               <rickert@cs.niu.edu>
  Northern Illinois Univ.
  DeKalb, IL 60115                                   +1-815-753-6940