FOM: Are Harvey's postings "Foundational"?

[Martin Davis]

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 
mathematical interest.

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.

Martin




                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
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