🚀 go-pugleaf

=>   For us to *know* a truth of arithmetic, we'll need to see some
=>   effectively verifiable supporting argument--a proof.  If the set
=>   of acceptable proofs is decidable (or even r.e.), then the set
=>   of provable statements is r.e.  Here we are considering not simply
=>   proofs according to some fixed rules, but arguments that could
=>   *ever* be accepted (in theory) by human mathematicians.  Even so,
=>   this still seems to lead to an r.e. set.
=>
=>   To "know" that Fermat's last theorem is true, it's not enough to
=>   believe it and state it.  You need a supporting reason I can check.
=>
=>   --H. Enderton, hbe@ucla.math.edu

Although this sort of foundationalist view of knowledge has been
pretty thoroughly discredited, I can see how it might remain
attractive when restricted to mathematics.  I can also see that the
"socially knowable" arithemetic truths are r.e., given some unstated
assumptions having to do with the possible future evolution of
mathematics and mathematicians (and, of course, Church's thesis).

Perhaps my memory's faulty, but I thought you had claimed that the set
of all a priori truths was r.e.  Do you belive that your argument
generalizes, or that all a priori truths are mathematical?

Finally, I found the use of `us' and `we' in the reply interesting.
Was this meant to exclude the possibility of private mathematical
knowledge?  E.g., suppose Godel walked into your office--it turns out
that he didn't die, he just wanted people to leave him alone--and
claimed that, on close inspection of the natural numbers, he perceived
that Fermat's last theorem is true.  On questioning him, it turns out
that he can supply no supporting reason that you can check--he says
that he knows it on the basis of observation.  (Mystical Platonists of
Godel's ilk--for some reason, a lot of the set theorists of my
acquaintance seem to fall into this category--draw a clear distinction
between "I know" and "I can prove".)  Do you really want to claim that
it cannot be the case that he knows it because he cannot prove it?

							-- rar