[FOM] Dirichlet's theorem; boiling down proofs

[Timothy Y. Chow]

Joe Shipman wrote:
>If future mathematics changes in such a way that my thesis is frequently 
>wrong, that is not a good thing, it means that mathematics will become 
>more esoteric, fragmented, and difficult to achieve new results in, 
>because it will take longer for researchers to "reach the frontier" than 
>it historically has, and fewer will be willing to attempt it.

I believe that these remarks presuppose a common, but mythical, view of 
mathematics.

To do mathematical research today, it is not necessary to learn everything 
that is already known.  Large areas of mathematics, which were once de 
rigueur, have fallen out of fashion.  The average mathematician today has 
only the foggiest notion of what 19th century work in abelian functions 
or invariant theory was like.  Although some of that material can be 
treated more simply using modern techniques, that is beside the point.  
Whether or not that material can be simplified to the point where today's 
undergraduates can assimilate it, there is no obstacle for today's 
students, because they don't need to know any of that stuff---simplified 
or not---in order to do research today.

The other thing is that there isn't any need to understand how something 
was proved in order to use it.  It would certainly be nice if one 
understood the proofs of everything that one cited, but it's possible to 
make mathematical progress without that.  Already, we realize that it is 
absurd to demand that a mathematician understand a detailed proof of the 
classification of finite simple groups before being allowed to cite it.  
And how many people who apply the Weil conjectures know how to prove it 
from the ground up?

Mathematical progress will continue to be made regardless of whether the 
distillation of existing knowledge happens in the idealized way you 
suggest.

Tim