[FOM] foundational crisis??

[Harvey Friedman]

This is a copy of an email sent to another email list.

Dear Pen,

I looked a bit at your RSL paper with interest - linked at your website. A
rather startling evolution over your career!

One thing that I see in all your writings is a blackboxing of mathematical
activity - which can be construed as an uncritical acceptance of the
mathematical community to pursue what it feels is important. You know what
I said about this earlier - foundational exposition is critically needed,
and the like.

In light of your "conversion", I would like to get your take on whether we
are witnessing an emerging "foundational crisis" or whether this is best
viewed as ordinary business as usual.

With all of the blockbuster incompleteness of Goedel and later Cohen, there
was a sense among some (and probably a very faint sense among many) that
there may have been emerging a kind of "foundational crisis", whereby the
idea of "right" or "wrong" in mathematics was being destroyed. To this day,
I have met many mathematicians who view the "right or wrong" nature of
mathematics as a principal component in their attraction to the subject.
They say that they would be stunned - even deeply wounded - by the right
kind of incompleteness.

Because the mathematicians find it so easy and natural to sidestep the
classical incompleteness that we have, there has not been any real
"foundational crisis" - whatever that means. Instinctively they have
already warned themselves about statements about formal systems, and
statements with an unusual set theoretic component. Occasionally they don't
se it coming - but after the fact they quickly adjust to it without feeling
stunned or deeply wounded.

Sol's "inherently vague" attribute is very relevant here. It seems that
mathematiciansmay  buy into this "inherent vagueness" idea in a nonverbal
implicit way when they are confronted with the usual incompleteness (not
Con(ZFC), where the defense is on subject matter  - logic, not
mathematics). Another factor is that they may be making a "subject matter"
defense, although that can easily get awkward and tortured.

Now enter the emerging "perfect Pi01 incompleteness". There is no longer
any ontologically based defense, and a subject matter defense becomes
increasingly awkward and tortured.

Now the question is: at what point, if any, as "perfect Pi01
incompleteness" unfolds into myriad finitary contexts, with ever increasing
simplicity and clarity, with growing attractive background materials doable
with normal methods, does it rise to the level of a "foundational crisis"?

One severe answer is this: only when it actually reaches in detail almost
exactly what mathematicians are actually doing research on - and "highly
regarded" mathematicians at that. My own opinion is that for a substantial
minority of mathematicians, this is not required - it is enough to have a
diverse plate of perfect Pi01 incompleteness.

An even more severe answer is this: mathematicians will simply drop their
original feeling of "right" or "wrong", and accept consistency statements
as part and parcel of mathematics, and accept the idea that there is a
profound irremovable incomprehensibility about mathematics - roughly akin
to the bizarre incomprehensibilities in quantum phenomena. And then simply
move on.

After the dust settles with perfect Pi01 incompleteness some more, I am
going to try to get a variety of mathematicians interviewed (not my me) to
see how they think - if at all - about the developments.

Harvey
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