FOM: Grothendieck and Friedman

[Colin McLarty]

Harvey Friedman <friedman at math.ohio-state.edu>  wrote

>I wish to thank McLarty for drawing similarities between aspects of
>Grothendieck's work and mine. But I still think that the analogies are
>somewhat strained.

        Both you and Grothendieck oppose unecessary abstraction. I like your
comments on the difference between yourself and mathematicians but let me
speak some in favor of the foundational methods of the mathematical project.

>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. 

        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

>2. Generally, the mathematicians know how to eliminate the machinery -
>perhaps at the cost of losing the clarity of the proofs. However,
>mathematicians  don't have any demonstrably necessary uses of relatively
>abstract machinery. They don't know how to formulate such a result.

        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. Grothendieck is much against abstraction for its own sake. Number
theorists love simple proofs. They give very abstract ones when that is all
they can find, and then set about simplifying with hindsight. As I
mentioned, one goal of elementary category theory is to make Grothendieck's
own ideas work more simply.    
>3. I have a series of demonstrably necessary uses of extremely abstract
>machinery for discrete/finite mathematics. The results are clearly and
>robustly formulated using modern mathematical logic.

        Yes, and I admire those results. I got on FOM to hear more about them.

        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. 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).