FOM: g.i.i.; Tragesser; philosophical exercises
Stephen G Simpson
simpson at math.psu.edu
Thu Mar 19 09:00:18 EST 1998
This is a response to part of Robert Tragesser's posting of 17 Mar
1998 04:28:32.
The bulk of Tragesser's posting consists of a sincere but unjustified
attack on Harvey Friedman's philosophical ability, summarized as
> it is easy to imagine that he has little appreciation
> of the nature of philosophical problems (or at least no
> patience with them).
I of course disagree with Tragesser, but I don't want to get into a
full discussion of this here and now. Let me say only that Harvey's
towering achievements in f.o.m. are driven largely by philosophical
considerations concerning the nature of mathematics and its place in
the broad structure of human knowledge. In this respect Harvey
follows in the footsteps of G"odel.
What I'd like to comment on now is Tragesser's bewildering attack on
the concept of general intellectual interest (g.i.i.). Tragesser
speaks of
> the philosophical puerility [professional judgment, not rhetoric]
> of the ideology of "general intellectual interest"--a cunningly
> designed slogan for levering all manners of evading the most
> difficult philosophical problems we face.
and
> the philosophically subversive ideology of "general intellectual
> interest".
When I read this, I was taken aback. Isn't Tragesser's thinking
imbued with the ideals of classical philosophy? Perhaps I'm naive,
but I would have expected classical philosophers more than anyone else
to appreciate g.i.i. and seek it out whenever possible. Thus, I'm
somewhat shocked by Tragesser's attack on g.i.i.
Let me try to make this point using Tragesser's own stated ideals.
Tragesser speaks of
> "philosophy" in Plato's sense, as a thinking about mathematics that
> aims to make us wiser about and at mathematics -- that aims to make
> mathematicians fully GOOD mathematicians.
and I agree; I read this as a high-level mission statement for
philosophy of mathematics and f.o.m. (In my Tymoczko posting of 1 Feb
1998 22:17:36 I spoke of "professional mathematicians"; the term "GOOD
mathematicians" would also have served.) My question for Tragesser
is: doesn't g.i.i. have a role here? To my way of thinking, g.i.i. is
a touchstone of excellence in foundational studies generally.
Foundations of subject X deals with the most basic concepts of X and
the place of X in the human knowledge as a whole. The purpose of
studying foundations of X is to make us better X-ists. New
foundational insights will necessarily have a degree of general
intellectual interest, because they will have implications for the
entire human intellectual enterprise.
Am I missing something? I'd like Tragesser to comment on this aspect
of g.i.i., while avoiding anti-Harvey rhetoric to the extent possible.
To bring this issue back to f.o.m. specifically, let me ask Tragesser
to comment the "philosophical exercises" at the end of my "judging
g.i.i." posting of 17 Mar 1998 13:13:02? There I presented a concise,
high-level formulation of the philosophical aspect or g.i.i. of
Friedman's recent results, and I said:
> As a useful series of philosophical exercises, let's try to give
> comparable g.i.i. formulations of other well known high points of
> f.o.m. research.
Here I had in mind f.o.m. advances such as:
1. Frege's invention of the predicate calculus
2. G"odel's completeness theorem
3. G"odel's incompleteness theorems
4. Turing's work on computability
5. consistency and independence of the continuum hypothesis
6. the large cardinal hierarchy
7. the MRDP theorem
8. determinacy and large cardinals
etc.
My question for Tragesser is: Don't you think these exercises can help
us FOMers to raise our philosophical awareness and make us better
mathematicians, in the Platonic sense?
-- Steve
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