[FOM] Concerning "dam breaking" f.o.m.

[Harvey Friedman]

On Mon, Jan 15, 2018 at 12:07 PM, tchow <tchow at alum.mit.edu> wrote:
> Bill Taylor wrote:
>>
>> Could you perhaps be more specific about this coming revolution
>> especially regarding the nature of the "dam breaking", that is
>> going to focus the attention of mathematicians with great force?
>
>
> Friedman has spoken about this frequently, but let me give my paraphrase of
> his claim, in case that helps clarify.
>
> One dominant attitude today among mathematicians is that it is not important
> to think about foundational issues when doing "core" mathematics.  If you
> stick to questions that are sufficiently concrete, then questions of
> undecidability, or of which axioms are needed to prove which theorems, will
> never arise.
>
> Friedman thinks that after the "dam breaks," questions of which axioms are
> needed to prove which theorems will flood over all of mathematics.  For
> example, questions that formerly were regarded to be extremely concrete and
> therefore "safe" will be shown to require large cardinal axioms (or at least
> their 1-consistency) to resolve.  Attention about which axioms are needed
> for which theorems will not be confined to an extremely small subset of
> arcane and exceptional results, but will pervade all of mathematics.
>
I did start a series https://cs.nyu.edu/pipermail/fom/2018-January/020793.html

>From experiences with some major top core mathematicians, I have an
expectation that even just a single entirely convincing example of
where intense f.o.m. plays a provably essential role in something that
is viewed as transparent, interesting, fundamental, thematic,
perfectly natural, etcetera, is sufficient to make a deep enough
impression that the word would be spread here and there, say at least
in informal conversations, and start penetrating the general
mathematical community at least slowly at first, but with a
compounding effect - especially if it is followed up by more of the
same, building up some variety. The ingredients for this kind of
development should be well enough documented and exposited within
about one year, so that it is ready to be "taken on the road" with the
core mathematicians.

Of course, progress would be greatly accelerated by major improvements
that penetrate more deeply into details of the specific environments
that are currently widely researched by core mathematicians. But I am
expecting that to happen only evolutionarily, nibble by nibble.
Instead, I focus on compellingly transparent compellingly thematic
compellingly simple compellingly perfectly natural constructions, with
the easier material having simple and interesting proofs of an
entirely normal kind. Then I take the leap.

In many versions, things are trivial in dimension 1, and intriguing
but not too challenging in dimension 2. More dimensions becomes mind
boggling, and large cardinals are required to come to the rescue
(sometimes low dimensions are known to require them, sometimes not).
In this kind of situation, which is emerging, I expect to spend much
more time going forward, on dimension 2, building up from even easier
dim 2 statements, with almost familiar proofs readily absorbed, all
the way up to full dimension 2. This is intended to give them a very
very false of security that everything is normal and unremarkable -
before bringing in more dimensions, where all hell breaks loose.

In summary, with a truly spectacular totally unexpected monumental
breakthrough connecting smoothly with core mathematical environments,
beyond what I have been able to do in 50 years, I don't need any
profound level of exposition and fleshing out of all of the
interlocking ideas from the ground up. HOWEVER, without that, with
things that I am bringing under strong control now, a profound level
of exposition and fleshing out of all of the interlocking ideas from
the ground up is essential to have the desired effect.

Such are the multifaceted labors involved in demolishing such
ingrained, very natural and compelling - but naively wrong -
conventional wisdom about the fundamental nature of mathematical
thought.

Harvey Friedman