FOM: Are Harvey's postings "Foundational"?
martin at eipye.com
Wed Mar 27 14:23:53 EST 2002
At 06:40 AM 3/27/2002 -0600, charles silver wrote:
> I have a question: Whatever is "foundational" about Harvey Friedman's
>copious, exceedingly technical postings?
Charlie raises two questions about Harvey's postings: what about them is
foundational? And, incidentally, why do they have to be so "technical"? I'd
like to briefly address both of these.
First, hooray for the introduction of "technical" methods into foundational
discussions. It was the great merit of the work of Frege, Russell, Brouwer,
and Hilbert that their philosophical discourse led to technical programs
which made it possible to view their ideas through a scientific lens.
In my opinion, the most important issue today in the foundations of
mathematics, what I like to call G\"odel's legacy, is the relevance of his
incompleteness theorem to mathematical practice. In his philosophical
writings G\"odel sketched an expansive open-ended view of mathematics. He
suggested that problems like the Riemann hypothesis may have remained
unresolved because they may require set-theoretic methods. Meanwhile actual
mathematical practice has been making great progress totally ignoring the
incompleteness phenomenon. The actual undecidable statements obtainable
using the usual direct approach via diagonalization are of no independent
It is in this context that Harvey's work is so exciting. He has been
obtaining simpler and ever more elegant propositions that are demonstrably
unprovable from the ZFC axioms, but which become provable when these axioms
are augmented by a large cardinal axiom. This work creates a genuine
dilemma for working mathematicians: either ignore these propositions
despite their evident mathematical interest or face squarely the
epistemological status of the large cardinal hierarchy.
Visiting Scholar UC Berkeley
Professor Emeritus, NYU
martin at eipye.com
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