FOM: Philosophy and platonism

[Michael Zeleny]

Martin Davis has responded to my argument against his foundational
pragmatism by citing Frege's private racial animadversions as
reflecting the practical upshot of his ethical principles, observing
that "the process of developing formal and symbolic methods for
dealing with genuine mathematical issues produces a momentum of its
own, a kind of "analytic continuation" that impels one beyond the
original subject matter", and wishing that I had been persuaded to
withdraw my flawed analogy between mathematics and politics.  I would
carefully consider withdrawing my analogy upon notification of any
specific flaw inhering therein.  Until then, I shall extend it with
the following considerations.  It seems to me that all foundational
concerns, pertaining as they do to the nature and warrant of claims
to knowledge, cannot but share a single basis irrespectively of their
purview within any specific domain of human action.  Similarly, the
perpetual likelihood of error, which Frege adduced as his knock-down
refutation of reducing norm to praxis, arises equally in each of our
endeavors.  In this connection, it is quite pointless to invoke his
private political peccadilloes as if they countermanded the rule of
principle that underlies his foundational program.  More to Davis'
point, mistakes have been made throughout the history of the modern
foundational research, even at its inception in the study of Euclid's
Parallels Postulate.  Recall that Gauss had deliberately abstained
from publishing his original contribution to the demonstration of its
independence, in the name of protecting the Boeotian sensibilities.
So it came to pass that two outliers, Bolyai and Lobachevsky, earned
priority in demonstrating the possibility of non-Euclidean geometries.
Similarly, and more relevantly to the working mathematical mainstream
uncontaminated by omphaloscopic philosophical concerns, Galois might
have emulated his eminent predecessors in wasting time on the search
for the general closed-form solution to quintic equations, instead of
approaching the problem in a more general and abstract fashion, indeed
exemplifying Davis' favored process of "developing formal and symbolic
methods for dealing with genuine mathematical issues".  But I should
not have to belabor the merits of essential deviation from traditional
practices and deliverances of one's predecessors in an assembly that
contains prominent contributors to negative resolution of at least two
of Hilbert's problems.  In this connection, I can only conjecture that
not even Cantor's paradise is likely to yield the final answers to the
metaphysical predicaments that gave rise to its original discovery.
The generic origin of foundational studies in the three paradoxes of
Eubulides of Miletus, antedates their resolution within set theory by
2.4 millennia, and the existence of several genuine alternatives to
the predominant approaches therein, as exemplified by non-Archimedean
analysis and set theory with a universal set, suffices to demonstrate
that "pragmatic practice by working mathematicians" can in no way be
streamlined in the service of agenda for identifying "the ONLY guide
we have."  The history of every human practice is replete with error
and strife.  Our real guide is disinterested concern for the truth.

cordially
Mikhail Zeleny at math.ucla.edu
All of old.  Nothing else ever.  Ever tried.  Ever failed.  No matter.
Try again.  Fail again.  Fail better.               -- Samuel Beckett