FOM: Re: Simpson on Mattes

[Michael Thayer]

Steve writes:


>My perspective on what has been going on the last few days:
>
>Josef Mattes gave a list of 9 diverse mathematical subjects and asked
>me whether I would consider them foundational, i.e. within the scope
>of f.o.m[...].  In that case, my answer is that I wouldn't *automatically*
>consider these subjects foundational.  In each case, someone would
>first need to explain clearly and honestly, starting with concepts
>that are of obvious general intellectual interest, exactly how subject
>X is related to the most basic mathematical concepts, and exactly what
>is the general intellectual interest of subject X

I take this to mean that something could be foundational , but not
foundational for mathematics, and thus that both the fundamental nature AND
the relevance to mathematics must be shown.  Thus in most cases on Josef's
list, one could argue that we are seeing a mathematical result which
suggests some fundamental insight about the world (e.g. Goedel's
relativity model) but that this is fairly typical of applied mathematics,
and while these results MIGHT be fundamental for some other science (which I
find debatable for many on that list) they would still not be f.o.m.

This seems quite reasonable, but I do have a question about Steve's own list
:

  1. number
  2. shape
  3. function
  4. set
  5. algorithm
  6. mathematical proof
  7. mathematical axiom
  8. mathematical definition

It would seem that we could argue that the following are more fundamental
than any of these:

1. Difference
2. Structure
3. Congruence

My argument being that without the ability to distinguish different objects,
there is no need for number, that structure is necessary for thought, as
well as being sometimes thought to be the subject matter of mathematics, and
that congruence (in the sense of a congruence relation - "I will consider
these object to be identical for the purpose of studying these structural
relations) is what gives us our mathematical objects.


No you may wish to dispute these philosophical points (and you certainly
could find ammunition to do so), but GRANTING that I am correct, are these
concepts then part of your list, or would you say they were foundational for
some other area , say epistemology or psychology?

I would think there are limits to f.o.m. in BOTH directions.  Do you agree?