[FOM] The gold standard and FLT

[Harvey Friedman]

On Wed, Jan 10, 2018 at 5:12 PM, Arnon Avron <aa at tau.ac.il> wrote:

>...David Brenton wrote:
>
>> I think you're getting this backwards. Math has not always been
>> this "paradise of rigor", which has just stopped being so in recent times.
>> On the contrary, ... I'd say that being careful about axiom systems, and
>> what assumptions are being made, and about completely rigorous
>> proofs, and the like, is more the exception, rather than the rule.
>
> And Timothy  Chow wrote:
>
>> Idealism is one thing, but historical inaccuracy is another.
>> "Making explicit in a very precise way what are the assumptions that
>> underlie the alleged proof" is almost never done in mathematics.
>>
> I was really amazed to read this kind of defense of the
> current alleged proof, and especially of the attitude, of those who
> claim to have proved FLT.
...

I think that there is a very interesting very revealing and crucially
important phenomena here that cuts across probably all deeply intense
super highly skilled lifetime preoccupations, which clearly includes
pure mathematics. But also a lot more. I am thinking of far removed
areas like professional classical piano performance (of course other
such intense musical areas, but I am only really familiar with this
one).

Most of the top people in these areas have risen to the top by extreme
focus on a kind of like minded way of looking at the matters at hand,
largely to the exclusion (at least while they are working on the
matters at hand) of all extraneous matters that might interfere with
that focus - matters that do not, in their judgement, further the
continuously hard fought progress, but rather represent a distraction.

Accordingly, a kind of universal groupthink sets in whereby directions
and considerations are nearly universally judged as non productive
distractions. PURPOSEFUL IGNORING of these "unproductive distractions"
is used by the group as a kind of PROTEST against the idea that these
non productive distractions are worthy of any consideration or are
even LEGITIMATE.

This goes very deep at and near the top - not ALL of the top, but just
the preponderance of the top. Any idea that foundations or ZFC in
particular represents a legitimate form of mathematical thinking is
anathema and must be resisted. Who do a bunch of silly so called
logicians who never did any kind of deep mathematics think they are
when they tell us how to write or judge rigorous proofs that they do
not even understand?

The utter lack of consideration or recognition of any of the deep
critical foundational issues raised by the FLT saga is really a deep
PROTEST against the idea that mathematicians themselves and their ways
of working, communicating, and thinking, are somehow insufficient to
fully control any issues that might arise.

And until this "inner world" comes crashing down in clearly
demonstrated crisis, this is not going to change - with very few
exceptions. There are some exceptions, or at least partial exceptions
at the very very top, but this is rare. This happens only because a
few at the very very top are deeply interested in communicating with
the wider intellectual community, going way beyond the mathematical
community.

NOW MY main point is that this is ABSOLUTELY NO DIFFERENT everywhere.
The same thing is true with professional piano performers, and I think
professional classical music performers generally. The very idea that
the scientifically measurable physical processes involved in piano
performance should pollute their performing life and teaching life is
complete anathema - even if they get absolutely wedded to completely
wrong ideas about these physical processes - and negatively impact
their teaching and, heaven forbid, their performances. But that must
at all costs not even be contemplated as a possibility, let alone
acknowledged.

All of these elite super competitive super intense super skilled
inbred communities are ISOMORPHIC. They have more than enough on their
plates that the idea of bringing in anything that even has the feel of
being unproductive distractions is more than they are willing to
countenance.

In various senses and at various levels and aspects, days of reckoning
are upon us this Century in both of these areas - and probably many
such areas. Exactly what form this takes, exactly how long it takes
before the crash becomes generally apparent, is of course very
unclear.

I and many of us want to pull the various gii (general intellectual
interest) and foundational levers. How do we know that our way of
operating is sound, what are the gold standards, what if we have been
systematically wrong, how do we explain what we do in general terms
that can be understand by "everybody", etcetera. All of this has and
will continue to pass over deaf ears in such communities - until there
is a generally recognized crisis, which some of us are trying to lay
the groundwork for.

Harvey Friedman