# FOM: g.i.i.; f.o.m.; genetic defects; ignorance; the boxing match

Stephen G Simpson simpson at math.psu.edu
Sun Jan 11 20:15:28 EST 1998

```General intellectual interest:

Colin McLarty writes:
> Those on fom who insist "fom is of more general interest than other
> math" have still not even distinguished the many things this could
> mean, let alone picked one it does mean. Does it mean:
>
> 1) More people actually want to hear about fom than other math.
>
> 2) More people would want to hear about fom than other math, if
>         they knew more about it.
>
> 3) People would benefit more from learning fom than other parts of math

All of statements (1)-(3) are true.  And it is also true that

(4) F.o.m. (foundations of mathematics) is of much greater general
intellectual interest than pure mathematics.

But (1)-(3) are not the essence of (4).  They are only consequences of
(4).  And to understand (4), we need to first understand very clearly
what f.o.m. is.  After that, the reasons for (4) will be obvious.

Definition of "foundations of mathematics":

What is f.o.m.?  So far as I am aware, nobody on the FOM list,
except me, has proposed a definition of f.o.m.  I proposed such a
definition way back in September and October, when the FOM list was
just getting started.  I have said repeatedly that I would welcome
discussion of alternative concepts of what f.o.m. is, but nobody
here has offered one.

To repeat my definition:

F.o.m. is the systematic study of the most basic mathematical
concepts and the logical structure of mathematics, with an eye to
the unity of human knowledge.

A couple of explanatory notes:

1. The word "basic" refers to the hierarchy of concepts.  All of
human knowledge is organized in a hierarchy of concepts, which is a
partial ordering.  This is elaborated in my essay

http://www.math.psu.edu/simpson/Hierarchy.html

The "most basic mathematical concepts" are those mathematical
concepts that are closest to the base of the conceptual hierarchy.
As such, the most basic mathematical concepts are necessarily
involved in all connections between mathematics and the rest of
human knowledge.  This is why they are essential for the unity of
human knowledge.

2. As a tentative list of the most basic mathematical concepts, I
offered the following:

list 1:

number
shape
set
function
algorithm
mathematical axiom
mathematical proof
mathematical definition

Naturally this list is open to revision, but I think the intent is
clear.

With this much explanation, I think it's clear why f.o.m. is of such
enormous general intellectual interest.  The reason is that some
knowledge of concepts such as those on list 1, qua basic
mathematical concepts, is essential to an understanding of the unity
of human knowledge.  Nobody can claim to be educated without some of
this insight.

"Genetic defects":

Some FOMers (Pillay, Marker, McLarty, ...) have raised a radical
objection to list 1.  They have asserted that list 1 is radically
incomplete and needs to be supplemented as follows:

list 2:

cohomology
projective analytic variety
Riemannian manifold
...

(List 2 seems to be open ended.)

It's clear to me that this proposal has no merit.  I think it's
pretty obvious why the concepts on list 1 are much more basic than
the concepts on list 2.  One way to appreciate this is to note that
the usual textbook definitions of the concepts on list 2 are
formulated in terms of chains of concepts that begin with concepts
on list 1.  It's also clear that there is a need for a field of
study devoted to basic mathematical concepts a la list 1.  That
field of study is f.o.m.  List 2 is of a very different character.

I'm at a loss to understand what might be called the "list 2
mind-set".  I don't know how anyone can claim that, for example, the
concept "Riemannian manifold" is just as basic as the concept "real
number".  My best analysis of the situation is that these people
"just don't get it", i.e. they don't understand the concept "basic
concept".  Perhaps they don't appreciate the idea of a hierarchy of
concepts.  Are they under the spell of Tennant's "Quinean holism"?
I don't know.

In making this point about the list 2 mind-set, I once engaged in
some hyperbole or figurative language.  I said that some people are
"congenitally incapable" of grasping the distinction between basic
and non-basic concepts.  Here of course I wasn't referring to any
genetic defect, but rather to a certain mind-set, the list 2
mind-set, which appears to prevent them from understanding a crucial
distinction.  This mental condition may even be correctible.
Perhaps all that is needed is for these people to admit (to
themselves at least) that f.o.m. is a legitimate field of study,
distinct from pure mathematics.

Ignorance:

I'm not the only one who has used rough language.  Both Dave Marker
and Lou van den Dries have accused their opponents of being
"ignorant" of certain mathematical developments.  Dave later
apologized.  Lou's latest accusation is as follows:

> My *experience* in the course of 30 years has indeed made me
> suspicious of certain wide spread instincts that Harvey and Steve
> may have in mind here, and which I share: these instincts,
> covered by a thin veneer of questionable philosophy, are often
> used to justify ignorance of major developments in mathematical
> thought of the last 200 years that are outside of the *relatively
> minor* and *exceedingly familiar* FOM-line: Cantor, Frege,
> Goedel, ...

I don't know how one could "justify ignorance" of major mathematical
developments.  Obviously students of f.o.m. need to be aware of
mathematical developments insofar as they have an impact on
foundational issues.  On the other hand, if the developments in
question are of interest only for particular branches of pure
mathematics, then non-mathematicians need not be concerned.

I also don't know what "wide spread instincts" Lou is referring to,
or in what sense Lou "suspects" and/or "shares" those instincts.
Lou, I think you need to explain yourself.

The boxing match:

I'm completely on Harvey's side in the boxing match with Lou.  I
can't understand how anyone can deny that f.o.m. questions such as
P=NP and the role of large cardinal axioms in finite combinatorics
are of incomparably greater general intellectual interest than
Fermat's theorem, etc.

Sincerely,
-- Steve

Name: Stephen G. Simpson
Position: Professor of Mathematics
Institution: Penn State University
Research interest: foundations of mathematics