FOM: potential and interest of FOM

[Colin McLarty]

        jshipman at bloomberg.net: FOM: How McLarty and Thayer should parse
"more general interest" wrote:
 
>    A relatively minor point I was making is being misunderstood, so I will
>clarify it and put it back in context.  I am not trying to make the silly
claim >that the field of "Foundations of Mathematics" is in some sense
"bigger" or >"more important" than Mathematics itself.  Therefore counting
bestsellers and
>estimating votes for Goldwater miss the point.  What I WAS saying is that
>results in FOM are potentially of "more general interest" than results in Math
>itself for the simple reason that they require less specifically mathematical
>background.  "More" is an adverb modifying "general", not an adjective
>modifying "interest".  

        Two ambiguities remain, which have complicated discussions on this list
for a while. By "more general" you could mean that more non-specialists will
be interested--that barbers and men on the street will be more eager to
learn FOM than mainstream mathematics (some people on this list have even
tried to DEFINE "foundational" this way). Bestseller figures refute this
completely. Men and women on the street are decisive in those figures, and
they decisively favor number theory alone over all of foundations. They also
give chaos theory alone a large edge over FOM, and several other fields.

        Or you could mean FOM has a wider impact than "ordinary" mathematics
on our genral intellectual culture. I won't argue with a claim like that--it
is more of an ambition than an assertion and I like ambition.

        There are two important senses of "having an interest". It can mean
"being fond of" or it can mean "standing to benefit from". You could claim
that even barbers and people in the street stand to benefit more from
learning about FOM than from other math because FOM is about the nature of
reasoning.

        And the whole discussion has to start over when you decide to talk
about "potential" interest.  What kind of "potential" do you have in mind,
and what do we need to realize it?

        Which of these claims would you agree with, with what modifications:

1) A greater number of people not specializing in mathematics would enjoy
studying FOM than other University level math, if they tried it.

2) A greater number of people not specializing in mathematics would benefit
from         studying FOM than from other University level math.

3) Ideas coming out of FOM will have more important impact on the overall       
        intellectual conversation around us than will ideas from other
        University level math.  

        
>But the context of this point was that this distinction
>between FOM and ordinary Math ought to be a small one because ordinary Math can
>and should be made more accessible to non-mathematicians.

        I think this is fine. I even think elite math should be made vastly
more accessible to professional mathematicians. There is a good trend in
this direction now--shown in graduate textbooks like Eisenbud's COMMUTATIVE
ALGEBRA, exposition to the profession as in the MATHEMATICAL INTELLIGENCER,
and mass market popularization as by Ian Stewart and Keith Devlin.
Mathematics has not seen such a general outreach since the 1930's--with the
great texts by van der Waerden, Seifert and Threlfall, and others; the old
MATHEMATICAL INTELLIGENCER, and mass market popularization by Hilbert.

        I am especially glad to see Joe urge a smaller distinction between
FOM and ordinary math. I see a lot more FOM already in ordinary math than he
and many others do.

Colin McLarty