FOM: Probes rather than Foundations

[Robert Tragesser]

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