FOM: quasi-empiricism and anti-foundationalism
Stephen G Simpson
simpson at math.psu.edu
Sat Sep 12 10:57:25 EDT 1998
Reuben Hersh has rejoined the FOM discussion, this time with an olive
branch to "Harvey Friedman and his circle".
Let me also put forth an olive branch: Hersh writes in a provocative
way, and he draws attention to some interesting issues.
However, I still find Hersh's views incoherent and peculiar.
> In describing foundationalism in the sense of Lakatos, (Frege,
> Russell, Hilbert, Brouwer) I did not consider or have in mind the
> research in logic and set theory of Harvey Friedman and his circle.
Is Hersh aware of the extent to which current research in logic and
set theory is a continuation of the pioneering work of Frege, Russell,
Hilbert, and Brouwer? Before attacking the line of research begun by
Frege et al, maybe Hersh ought to learn something about it.
> I do not regard this as part of foundationalism, because the
> restoration of indubitability is not its goal.
Speaking for myself at least, I would be proud and happy if my own
f.o.m. research could contribute to a heightened sense of certainty
and rigor in mathematics. In Hersh's demonology, perhaps I myself am
one of the dreaded "foundationalists".
> ... --logicism, formalism, and intuitionism. But they all
> encountered grave difficulties, and a consensus emerged that the
> enterprise of restoring the foundations (restoring indubitability)
> was hopeless.
On the contrary, there is now a massive consensus in the opposite
direction, in favor of indubitability. Virtually all mathematicians
accept and take pride in current high standards of mathematical rigor.
The definition-theorem-proof template is almost universally accepted,
including its formal explication as provability in ZFC. There is
little contention about when mathematical conclusions have been
rigorously established. Such contention is much less in mathematics
than in other sciences. Dissenters such as Bishop exist, but they
have been virtually marginalized. The consensus of mathematical rigor
has been dominant since the early 20th century.
Why do quasi-empiricists deny that this consensus is dominant? It's
fine for Hersh to view mathematical rigor as an oppressive right-wing
conspiracy if he wants to, but why can't he at least acknowledge that
the definition-theorem-proof model is widely accepted? If the
quasi-empiricists can't acknowledge evident sociological facts such as
this one, then how can we expect anything of value to emerge from
their other sociological investigations?
> the fundamental cleavage in philosophy of mathematics is between
> those who think mathematics is a human activity and a human
> creation, and those (the dominant view) who think mathematics is in
> some way or other superhuman or inhuman. This disagreement goes
> back long before Lakatos or Mill; at least as far as Aristotle
> vs. Plato.
I reject this dichotomy. It's true that Plato's forms and the
quantitative universe are non-human systems. But that doesn't
preclude rigorous mathematics from being a human activity motivated by
human concerns. Rigorous mathematics is just as human as other
sciences which study non-human phenomena: chemistry, biology,
Hersh's insistence on demonizing "foundationalism" as anti-humanistic
is not only irritating and counterproductive, but also false. And
when he drags in left-wing politics (as in his book and, presumably,
his lectures), it gets truly bizarre.
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