FOM: Are Harvey's postings "Foundational"?

charles silver silver_1 at
Wed Mar 27 07:40:58 EST 2002

    I have a question: Whatever is "foundational" about Harvey Friedman's
copious, exceedingly technical postings?   A long time ago--I can't pinpoint
exactly when right now--it occurred to me that there were only three stellar
figures left in what seemed to me to be "real foundations."  This assessment
may well have been wrong, but that is what popped into my head at that time.
The three figures were (by age): Kreisel, Feferman, and Friedman.   I now
admit that there were probably several others deserving to be on the list
(for example, I am impressed by Tait's writings, but at the time I didn't
consider him, maybe because Chicago was simply too far away geographically
from my limited vision), but again: that's just what I thought at that time.
Nowadays, Kreisel has gone to who-knows-where to do who-knows-what.
Feferman still seems to retain to some extent a philosophical bent, for
example, in asking the question whether mathematics needs more axioms.
But, I believe--please correct me if I'm wrong--that Feferman's work has
become increasingly more technical over the years.   Perhaps that's the
inevitable evolution of mathematicians with philosophical interests: towards
increasingly more technical work, I don't know.    At any rate, it seemed to
me, anyway, that work in so-called "foundations" deteriorated into the
purely technical, where "results" were arrived at that fell into one of the
areas of mathematical logic.   Or perhaps belonging to a new area.   For
example, Harvey's and Steve's and others' work in reverse mathematics seems
to me to belong to a somewhat new area (though perhaps some persons would
assimilate it under proof theory).   As far as I can see, besides a nod in
the direction of first-order logic and some brief motivational talk of
various kinds of induction, work in r.m. is almost exclusively technical.
I come now to Harvey's postings.   I am not able to evaluate the importance
of Harvey's postings, because I do not have the requisite technical savvy.
I further doubt that the majority of people in FOM can understand his
postings either.   They seem to me to epitomize technical results for the
sake of technical results.   Harvey has described his recent results as
"beautiful".   Perhaps they are, I'm in no position to judge.  All that
seems clear to me is that there is nothing or at least relatively little in
his posts to explain their "foundational" content.   Now, perhaps they are
truly foundational, and perhaps many of  Feferman's recent papers are
foundational as well (Kreisel seems to have dropped off the face of the
earth).  Perhaps the implicit purpose, unstated, but in the minds of the
authors, is to address certain important foundational issues.  But for me to
be able to understand exactly what "foundational" issues are being
addressed, whatever has remained so-far implicit needs to be made fully
explicit--for me, anyway.   For, on the face of it, Harvey's postings (I'm
not really sure about Feferman's) seem simply technical (regardless of
whether they are "beautiful" technical results or not).   Perhaps my lack of
appreciation of the foundational significance of Harvey's (and Feferman's?)
work indicates an inordinate obtuseness on my part.  I freely admit this
possibility.   I am obtuse enough to have to be told exactly what in the
world is foundational about Harvey's extraordinarily technical postings.
So, I ask Harvey and anyone else to please explain this for me and for
others who may be similarly puzzled.  It is important (due to my obtuseness)
that the exact foundational nature of these results be spelled out in goodly
detail, because omitting anything of significance would still leave me, and
I would guess a good number of others, remaining in the dark.

    Thank you,

Charlie Silver

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