FOM: advances in f.o.m.

[Harvey Friedman]

Franzen writes 12:54PM 3/17/98:

>    Harvey Friedman suggests that the general intellectual interest issue
>is "handled rather nicely" by the paragraph
>
>   >"A coherent body of discrete and finite combinatorial results, regarded as
>   >interesting, natural, basic, and simple by relevant practitioners, has
>been
>   >discovered and shown to be provable only by going well beyond the usual
>   >axioms of mathematics via standard axioms of higher infinities." A
>   >non-specialist can understand this finding.
>
>  A smallish group of non-specialists could no doubt understand this
>finding, in general terms. But why should they, or anybody else, care
>about it? Again what is lacking is any argument linking f.o.m. to
>the actual intellectual concerns of non-specialists.

Because, e.g., there is plenty of documented evidence that many
non-specialists relate very strongly to the incompleteness theorems and the
whole idea of unprovability. The above is a very strong and modern form of
earlier results which have stood the test of time in this regard. Also, the
general intellectual interest in findings relating to the "special
objective nature" of mathematics - construed as postitive or negative
findings - is completely self evident and well tested. The subject
"foundations of general intellectual interest" is a very real one, and not
really well developed. However, I personally have other committments, and
have not sought to develop this. Nevertheless, I fully believe that you
understand the notion and possess the intellectual capacity to usefully and
productively think in these terms. I wish you would refrain from
deliberately creating false doubts as to your understanding, solely for the
purpose of offending people who you think have offended you.

>It's unclear if
>you [Simpson] are suggesting that whether Friedman's results are an
>epochal advance in
>foundations does not, as I claimed, strongly depend on the applicability
>of the combinatorial principles and the epistemological status of the
>large cardinal principles.

Large cardinal principles, following Godel, are so ingrained in the
development of f.o.m., and form such a compelling coherent picture
(independently of wide open questions of ultimate justifications, etcetera)
that various results concerning them can be obviously crucial steps in the
development of f.o.m. totally independently of any future understanding of
their "epistemological status." Also, the importance of the Godel/Cohen
advance does not depend on the applicability of the continuum hypothesis,
although applications of CH - which are systemically limited - add somewhat
to the importance of the Godel/Cohen advance. By the way, when you use the
word "epochal" here, what do you mean?

Specifically, would you say:

"whether Godel/Cohen is an epochal advance in foundations strongly depends
on the applicability of the continuum hypothesis and the epistemological
status of ZFC"?

By the way, what, in your terms, is the epistemological status of ZFC? And
also, would you say:

"whether Godel's 2nd incompleteness theorem is an epochal advance in
foundations strongly depends on the applicability of the consistency
statements and the epistemological status of first order predicate
calculus"?

and what, in your terms, is the epistemological status of first order
predicate calculus?