FOM: quasi-empiricism and anti-foundationalism
Stephen G Simpson
simpson at math.psu.edu
Thu Sep 17 16:22:38 EDT 1998
Reuben Hersh 15 Sep 1998 10:32:37 writes:
> I am unable to look up your reference to a message of mine in
> times past.
Then you need better software. All FOM postings are archived and
searchable at http://www.math.psu.edu/simpson/fom/. Without the
ablity to access previous postings, it's impossible to carry on a
serious extended discussion. The FOM list can be a powerful medium
for exchange of ideas, but only with adequate software.
> If you think I failed to answer your earlier questions, I hope you
> agree that this message is respomsive.
I agree that this message is more responsive than your earlier one.
But I still think that your views on what you call "foundationalism"
> There is a practically insignificant but philosophically crucial
> difference between indubitability and high standards of rigor.
What is the philosophically crucial difference?
Your term "indubitability" is slippery, or rather, as Martin Davis
said in 16 Sep 1998 11:30:27, a straw man. Indubitable means
impossible to doubt, but any assertion whatsoever can be doubted at
any time by any yokel. Indubitability in this sense is a non-issue.
Instead of indubitability, let's talk about certainty. Do you accept
the fact that, because of high standards of rigor in 20th century
mathematics, mathematical truths are typically much more certain or
solidly established than truths in other sciences? If you accept this
fact, then what do you make of it? Do you think that this fact is of
philosophically crucial significance? Do you think that the issue of
certainty in general is of philosophical significance?
You have taken a stand against foundationalism, citing Frege, Russell,
Hilbert, Brouwer, .... Let's get down to cases. I will list some
specific high points of f.o.m. Could you please tell me which ones
you think you are opposed to, and why?
Frege's invention of the predicate calculus
G"odel's completeness theorem
Heyting's intuitionistic arithmetic and analysis
Hilbert's program of finitistic reductionism
G"odel's incompleteness theorems
Zermelo's study of the axiom of choice
Russell's type theory
axiomatic set theory
consistency of the continuum hypothesis
independence of the continuum hypothesis
the large cardinal hierarchy
Turing's work on computability
the MRDP theorem (a.k.a. Matiyasevic's theorem)
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