[FOM] The irrelevance of Friedman's polemics and results

[Neil Tennant]

Arnon,

Might you be misinterpreting the dialectical background to Harvey's
"polemic"?

As I understand it, the background is as follows. Core mathematicians
(CMs) tended to downplay the significance of the G"odel-phenomena because
(they claimed) the independent sentences were horrifically cumbersome,
non-natural statements. So, CMs thought the Goedel-phenomena somehow
"irrelevant" to their core mathematical concerns. Within core mathematics
(that body of natural-seeming/elegant/deep/beautiful conjectures and
theorems) CMs proceeded on the working Hilbertian assumption: "Wir m"ussen
wissen! Wir werden wissen!" The sorts of problems that they, the CMs,
wanted to work on were surely soluble by usual methods---or, at least,
some very modest and plausible extension of those methods, as might be
proposed should the need be felt to arise (in order to obtain the
consensually conjectured solution). That, anyway, was how the CM-line of
thought went.

So, that was "thumbing of the nose" #1 on the part of CMs to
metamathematicians and foundationalists (MFs).

"Thumbing of the nose" #2 concerned set theory. This pet discipline of MFs
was disparaged by CMs as guilty of excessive, extreme postulation.
Surely, the CM thought went, those gargantuan cardinals won't ever be
relevant to core mathematics? How could stratospheric facts within the
cumulative hierarchy ever "feed down" to our concrete concerns within core
mathematics (*especially* at the Pi-0-1 level)?

It seems to me that Harvey is confronting these two CM-dismissals head-on.
His method is as follows. He finds Pi-0-1 statements of some area of core
mathematics that are simple/beautiful/elegant/natural-looking *to CMs
themselves*. He gets leadings CMs to confirm this status of the statements
in question, so that the "thumbing-of-the-nose #1"-type of objection to
his extended G"odel program cannot be raised. Then he shows that the
statements in question, in order to be "decided" the way one would expect
(on the basis of extrapolation, say, from finite to infinite cases; or
because of syntactic homologies with other statements in the neighborhood
that *can* be decided by simple means) cannot actually be decided without
assuming the (1-)consistency of some very large cardinal in set theory.
That meets the "thumbing-of-the-nose #2"-type of objection to his extended
G"odel-program.

Furthermore, his own proof of the latter result is carried out "very low
down" in the hierarchy of theories (ordered by consistency strengths). The
proofs are carried out in exponential function arithmetic. So, he is not
"begging the question" methodologically against CMs who might complain
that he, Harvey, is using outlandish and implausible methods to obtain
these results---or "using large cardinal to show that one must use large
cardinals". No, *that* kind of circularity is not involved at all.

Does this shed any light on the shape of the polemics? Harvey's challenge
seems to be (to CMs): "You want to do business as usual? So you want in
due course to solve a problem like this?: XXX (where XXX is
natural/elegant/beautiful and Pi-0-1). Buddy, do I have a surprise for
you!---You're going to have to use very large cardinals, just to force the
facts the way you want them down here at the Pi-0-1 level!...and on this
ntural/elegant/beautiful fact to boot, which you would just love to have
in your very own core mathematics."

Whether you, as a predicativist, choose to be impressed by the Pi-0-1
statements and/or salivating to get it, is not really relevant to the
assessment of such a profound breakthrought. As you well know,
predicativists are a minority within the core mathematical community.
Harvey is speaking to this wider community, *refuting* their dismissals
both of G"odel-phenomena as irrelevant to CM, and of the resources of
higher set theory as not needed for CM. His polemic is not directed
against predicativists uninterested in any mathematical theorems except
those that can be proved by predicative means. After all, one might also
decide to be interested only in those theorems whose shortest proof in a
given system have a prime number of primitive symbols; or, less
facetiously, only in those statements that have a constructive proof, or a
feasible proof ... or whatever.

___________________________________________________________________
Neil W. Tennant

http://people.cohums.ohio-state.edu/tennant9/

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