FOM: quasi-empiricism and anti-foundationalism
rhersh at math.math.unm.edu
Tue Sep 15 12:32:37 EDT 1998
I am unable to look up your reference to a message of mine in
times past. So I don't know exactly what I said then. I expect
it probably was the same that I think now.
There is a practically insignificant but philosophically crucial
difference between indubitability and high standards of rigor.
Go back to Plato. Why does he claim that knowledge of Ideas is superior
to knowledge of the visible world? Because, says Plato, knowledge of
Ideas is certain, indubitable, whereas knowledge of the visible world is
always open to doubt. He thinks knowledge of the visible world may
approach a high degree of certitude, but it can never be *absolutely*
His Ideal world consisted notably of religion--knowledge
of the Good--and mathematics. The claimed indubitability of mathematics
supported the claimed indubitability of religion.
Plato was long ago, but Platonism is still big. It's my opinion
that the linkage between mathematical indubitability and religious
conviction was a major factor in the great importance people like
Russell and Hilbert placed on restoring the foundations. (The
meaning of "foundations" in this context is not the same as
its meaning in the "f" of "fom".)
Of course, as you say, the consensus today values rigor in mathematical
proof. But it does not ascribe to it (at least on the part of a
substantial number, I haven't taken a ballot) a superhuman, perfect,
unquestionable authority. We seek the highest
degree of certainty we can attain. We don't insist on
perfect, indubitable, eternally unchanging certainty. In brief, there's
no contradiction between your claim about the consensus and my claim that
indubitability is no longer considered an appropriate goal for the
philosophy of mathematics.
Now politics. I didn't look up the amendment numbers. I was relying
on a fallible memory for the numbers associated with granting
suffrage to former slaves (14th?) and to women (20th?)
If you think I failed to answer your earlier questions, I hope you
agree that this message is respomsive.
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