FOM: Re: Grothendieck and Friedman

[Harvey Friedman]

Reply to McLarty 2:26PM 4/6/99:

>        Both you and Grothendieck oppose unecessary abstraction.

I am extremely interested in the necessary use of abstraction. I oppose
abstraction when it is not needed whenever I feel that it weakens what you
can prove. But I like elegance as much as anyone if the elegance is
obtained by universally understood abstractions.

For example, I do not oppose the definition of propositional formulas as
the least class of strings satisfying the obvious conditions. That is
unecessary abstraction. I have even taught this definition. However, if I
am in a context where I am considering what principles are needed in order
to do what, then this definition is anathema. One instead gives the
finitary treatment of the inductive definition involved.

Grothendieck's work is, from what I understand, full of abstraction that is
well known not to be necessary. I assume Grothendieck knew that it was not
necessary, also.

Friedman wrote:

>>1. Mathematicians are very fond of using relatively abstract machinery to
>>solve relatively concrete problems. E.g., use of complex analysis to prove
>>number theory.
>
>        They use the methods they can. Finding an elementary proof of the
>prime number theorem was no easy thing, even given analytic proofs. And
>analytic questions related to this, notably to the Riemann Hypothesis, are
>still important today.

What you know I meant is that a lot of extra fuss is made over the
productive use of machinery. We all know of cases where equally striking
results are obtained without any use of machinery, and the attention paid
is comparatively minimal.

>        Browsing the web I find a proof that if the Generalized Riemann
>Hypothesis holds, every odd number (greater than 5) can be written as the
>sum of three primes. This "odd Goldbach theorem" has been unconditionally
>proved for odd numbers below 10^20 or above 10^46,000. One of the authors,
>Gove Effinger, tells me he sees no way to prove it for all odd numbers
>without proving the whole Generalized Riemann Hypothesis.
>
>        If you can find an elementary proof go right ahead. But so far,
>analytic methods still yield results that people cannot find in other ways.
>The announcement is at
>http://webdoc.sub.gwdg.de/edoc/e/EMIS/journals/ERA-AMS/1997-01-015/1997-01-0
>15.html

What you write is incoherent. The GRH is open, and so the "odd Goldbach
theorem" is open. So the result you cite mentions complex analysis. It is
not interesting to ask whether complex analysis can be removed in the proof
of a statement involving complex analysis. The answer is no.

>        They know how to eliminate Grothendieck universes--and they do not
>eliminate them in practice, because loss of clarity can well mean the loss
>of actual discoveries even if it serves some foundational goal.
>
>        These people do not WANT to find necessary uses for abstract
>machinery.

Bombieri, I think, explicitly said he wanted this.

>        Grothendieck universes were developed for a different goal: to
>actually solve certain problems that people had pursued for decades (and
>now, with Fermat, for centuries) without success. This is just a different
>project from yours.

Universes of course turned out to have nothing to do with, say, Fermat.

>Mathematicians differ, and sometimes even argue in
>print, over how much of Grothendieck's methods to use (toposes, derived
>categories, and more...). But I have never seen a textbook or research
>publication on cohomological methods in number theory that did not use
>Grothendieck universes (i.e. that did not quantify over universe-sized sets).

This is in direct contradiction with my expert source.

NOTE: The first serious application of Universes to the integers is going
to be in my numbered series of postings on the FOM. And these will be
demonstrably unremovable.