FOM: resp.to Tait
Robert Tragesser
RTragesser at compuserve.com
Thu Mar 19 09:01:43 EST 1998
Bill(Tait),
Your reprimands (to put it mildly)
are undoubtedly just. I am undoubtedly in
over my head (indeed,
as you suggest, weighed down by a thick skull).
[1] Self-feeding/self-containedness of philosophy
of mathematics and f.o.m.. Here is why
I thought that: I learned some bits
of mathematics and have been curious about how
f.o.m. and phil.of math. are/can be illuminating
about, e.g., mathematical proof as it occurs. IN
FACT, I've experienced everything from indifference
to hostility to my questions. ["This
in not the [sic.] philosophy" someone wrote to me. On
the one hand I do not understand P.Maddy's circa
'90 naturalism, yet Field's nominalism/physicalism.
I don't understand Dreben's violent denunciation of
Goedel as a philosopher... The hostility to Hao
Wang. On the other, I do not understand H.Friedman's
insistance
that there is no issue about all mathematical
proof necessarily going over into ZF, and without
loss. Where there is so much I don't get. . .]
[2] I've not understood g.i.i. wrt f.o.m.
It has seemed that it is used whenever anyone
raises a tough
philosophical question or a question of the
significance for mathematics. It seems to be
used to preserve f.o.m. as an autonomous science
in the Aristotelian spirit, an autonomous domain
of inquiry.
I do not doubt [but on the contrary
am very struck by]
the Friedman connections between finite combinatorics
and large cardinals. I just
was not happy about what seemed the g.i.i. attitude
toward these results, that neither their mathematical
nor philosophical significance are worth [deeply]
troubling onesself with. Philosophically, shouldn't
it make sense to wonder if these results don't
show us that the metaphors of "largeness" or "size",
as well as those of "finite" and "infinite" might
after all not be giving us the right picture of
what the foundational problems are? As to mathematical
significance, surely (or so I think) Feferman's
careful discussion of Friedman's earlier results in
this vein -- in "Infinity in mathematics: Is
Cantor Necessary?" against the background of the problem
of understanding the mathematical significance of
Goedel's PM-I deserved a more thoughtful response?]
[3] Finally. Plato had a quite rich field of "political
appearances" to work with. I see in his writings
considerable back-and-forth between these appearances
and his reasoning toward the political good (this
back and forth is to me much in evidence in The
Stateman.) I imagine that if Plato had our "mathematical
appearances" [that is to say, modern mathematics in toto]
to think about], one would see equally much back
and forth -- aiming at producing a good mathematician in
the way that Plato aimed at producing a good leader
in his works on political philosophy; viewing him
in this way, I just couldn't see him working an
autonmous philosophy of mathematics, rather than
a philosophy of mathematics that attempts to address
the mathematical appearances in their every detail,
carefully evaluating each.
robert tragesser
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