FOM: Re: Are Harvey's postings "Foundational"?

[wiman lucas raymond]

Hi,

>    I have a question: Whatever is "foundational" about Harvey
Friedman's
>copious, exceedingly technical postings?

I share (to some degree) your annoyance with Friedman's frequent and
technical postings to FOM.  I, like you, cannot really understand them,
and they do have a couple of problems:
(1) He claims they are self contained, but they frequently refer to
previous postings of his.
(2) They are fairly pompous.  It seems to me wrong somehow to
continually describe one's own work as "beautiful."  Even if true, I
think that this is a determination which should be left to others.

That said, I don't think there is anything inherently wrong with posting
technical results--mathematics is often technical.  The question of the
"foundational" nature of Friedman's work is somewhat irrelevant.  If
this were a list devoted only to discussions of deep foundational
results, then this list would be mostly dead.  The axiomatic system used
by the majority of mathematicians has not changed *at all* since the
proof of the undecidability of the continuum hypothesis and axiom of
choice.  In that time ZFC has sufficed for most tasks, and occasionally
ZFC+generalized continuum hypothesis.  Hence most results in set theory
and logic are of a technical nature.  

Yet this is fine.  Set theory and logic have in that time contributed a
great deal to the rest ofmath.  They have done this in the form of
solving problems
of interest to mathematicians other than set theorists and logicians.
Model theory has been used in many applications to many algebraic
geometry: the Mordell-Lang conjecture for function fields, etc.  Set
theorists proved the undecidability of Whitehead's problem in the
1970's, among other results.  These results were certainly significant
and fairly technical, but they were not of a strictly foundational
nature.  

I think that this is a positive step for "foundations".  The job of
creating a strong foundation on which mathematics can sit
was certainly a formidable and difficult challenge, but it was largely
met decades before I was even born.  

I'm not saying that there won't be significant results in foundations in
the future that might help shape mathematical thought.  I'm saying that
such results would almost certainly be of a very technical nature--and
there's nothing wrong with that.  Friedman's work may not (right now)
resolve deep foundational issues, but I would guess that it is
significant.  I further doubt that Friedman is alone in understanding
his postings.

-Lucas Wiman