FOM: Probes rather than Foundations
Robert Tragesser
RTragesser at compuserve.com
Sat Feb 14 13:21:40 EST 1998
To: FOM, Proposal for a Change of Root Metaphor
from Foundations to Probes.
I propose a change in the basic metaphor
from foundation on which one builds to probes
through which one finds out.
What is probed are as it were "naturally"
occuring mathematical theories/results, concepts,
or methods.
Probes are typically logical constructs (in
some very general sense). We "probe" with such a
construct (or congeries of constructs) by "interpreting"
or "translating" the mathematical
phenomena into them (typically not without some un-
interpreted or untranslated residue). Optimally
the probes are better understood than what is
probed. One can then understand why first-order
logic is good probe material. Well understood;
ease and naturalness of translation/interpretation;
in one way or another it has a universal character
in that all mathematical phenomena can be probed
with it-- translated into it, albeit informativeness
will vary.
This switch of metaphors has several advantages.
(1) it extends the sense of Stu Shapiro's "foundations
without foundationalism"; (2) it makes one more reflective
about what one is doing, what end one seeks, and about
whether the probes one chooses is optimal; (3) it dampens
ideological fervor forcing a more scientific and professional
considerations of ends and means; (4) it removes the temptation
to (I think) quite fruitless disputes about what are the
more natural or simpler "foundations" of a piece of mathematics
(replacing them with the question: what do we learn from
this "probe" that we don't from that). . .
Give such and such (what Sol Feferman calls) global
or local foundational problems, we are give pause to ask:
what sort of probes (applied to what mathematical phenomeno)
would show us the way to their solution.
RTragesser
Professor in the History and
Philosophy of Science and Mathematics
Connecticut College
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