FOM: Platonism v. social constructivism

[Neil Tennant]

So Wiles devoted seven years to proving FLT because of the prestige
conferred on the problem by the mathematical community? All that this
shows is that Wiles is human after all, not that mathematical reality
cannot transcend (and is completely conditioned by) human social
reality. It does *not* count even as a tiny bit of evidence supporting
social constructivism as a philosophy of mathematics. The historicity
of our collective mathematical experience is irrelevant to questions
about the ultimate nature of mathematical objects and structures.

Suppose for the sake of argument that Platonism, or constructive
logicism, or some other opposing philosophy of mathematics were
correct.  Then it would be completely unsurprising if (human)
mathematicians, in apprehending different aspects of the abstract
realm, were to develop a strong consensus as to which problems were
deep and difficult, and which solutions elegant and profound. Indeed,
one could even imagine that the consensus itself could depend not only
on the Platonic facts of the matter, but also on the neuro-cognitive
architecture of the human mind/brain. (The dependence need not be
causal.)

Our perceptual system gives rise, in ways at present beyond our
scientific and philosophical understanding, to a brand of
phenomenological experience that is analog-from-digital. Our
conceptual system, with its sortal and classificatory concepts
enabling us to individuate and distinguish and re-identify objects,
gives rise to a mode of thought that is both digital-from-analog and
digital-from-digital. Geometry, topology, and the theory of the
continuum probably feed off the analog phenomenology; number theory
and set theory probably feed off the digital/classificatory. But that
does not make the mathematical realities themselves in any way the
exclusive province of human thinking.

To repeat: pointing to any social-institutional factors shaping the
history of mathematics as thus far developed by human beings is
IRRELEVANT to the truth of Platonism (or any other philosophy of
mathematics opposed to social constructivism). The same goes for
whatever social-institutional factors might have shaped the history of
mathematics in any other extra-terrestrial civilization. Rational
beings, of whatever embodiment, can be expected to develop a consensus
over what is worthy of intellectual exploration. The consensus will
depend on at least

(a) the structure of Platonic mathematical reality, or objective
and truly universal patterns *in intellection*, patterns that are
reflexively accessible to intellection;

(b) the neuro-cognitive structures of the thinking beings in question
(or phi-structures, for whatever phi is appropriate, given their
embodiment); and 

(c) various social-institutional factors contingent upon the past
history of the mathematical community in question.

Even with the strictest methodology of interpretation or translation
being respected, one could find that (b) and (c) vary across
extra-terrestrial mathematical communities. But such variation could
not count in any way against the universal presense of factor (a).

Just because it's we who do the thinking doesn't mean that what we are
thinking about depends entirely on us.

Neil Tennant