FOM: Wigner; MacLane; Aristotle; incommensurables; topos challenges

[Stephen G Simpson]

I thank Sol Feferman for two excellent FOM postings, one on
foundations of naive category theory (25 Jan 1998 16:53:04) and the
other on Wigner's "unreasonable effectiveness" (25 Jan 1998 19:26:05).

Wigner's article is mentioned in my paper on Hilbert's program,
www.math.psu.edu/simpson/papers/hilbert/.  Basically, I regard
Wigner's position as disastrous, because it is so mystical and
unscientific.  It's terrible to say this about a physicist of Wigner's
stature, but there you are.  "The miracle of the appropriateness of
the language of mathematics for the formulation of the laws of physics
is a wonderful gift which we neither understand nor deserve."  This
descent into the world of MacLane (Shirley, not Saunders) makes me
heartsick.

What we need is a philosophy of mathematics that will account for not
only pure mathematics but also applications of mathematics to physics
and other applied areas.  This is why I see Aristotle's ideas as so
important and inspiring.  For Aristotle, mathematics is simply the
science of quantity, i.e. of quantitative aspects of the real world.
>From this point of view, there is no mystery or miracle about the fact
that quantitative knowledge is possible, no more than for any other
type of human knowledge.  The task of f.o.m. is to understand how
mathematical knowledge works and what are its relationships with the
rest of human knowledge.

I recognize that there are difficulties with this position, but I
believe they can be overcome.  This may be a rich source of technical
problems for f.o.m. research.  Bill Tait 21 Jan 98 00:21:03 has
mentioned one such problem: how to handle incommensurables in an
Aristotelean framework.  I think this can be done by developing a
rigorous theory of scale in measurement.  More on this later.

-- Steve


PS on McLarty topos challenges.

McLarty in his post of 26 Jan 1998 11:44:55 discusses my three topos
challenges and makes a tiny bit of progress toward answering them.
The three challenges are:

  1. fully formal statement of the topos axioms etc.
  2. topos + NNO vis a vis undergraduate real analysis
  3. motivating foundational conception for topos theory

Regarding challenge 1, McLarty promises a future posting on it.  (I
can't wait!)

Regarding challenge 2, McLarty calls on me to be more specific about
what I mean by undergraduate real analysis.  OK, fair enough, I'll do
that in terms of textbooks.  For calculus, Stewart 3rd edition (which
we are using for freshman calculus here at Penn State right now).  For
ODE, Coddington-Levinson 1955.  For PDE and Fourier series, Hans Sagan
1961.  There's nothing special about these books; they are typical
undergraduate textbooks.  Now I'd like McLarty to address the specific
point about the maximum of a real-valued continuous functions on
[0,1]; see my posting of 26 Jan 1998 12:15:34.

Regarding challenge 3, McLarty says that he has in mind a conception
that he finds coherent, but he doesn't explain what that conception
is.  I'd like him to explain it.