[FOM] Current issue of Philosophia Mathematica devoted to mathematical depth

[Martin Davis]

Based on a conference at UC Irvine, the issue has a number of interesting
essays. The one by Alasdair Urquhart (one of FOM's editors) surveys what a
number of mathematicians have said about what makes a theorem or proof
"deep".

I want to note his quoting G.H. Hardy's famous "Mathematician's Apology" to
the effect that the prime number theorem is deep because its proof uses
"the most powerful methods of the modern theory of functions". This has to
do with Riemann's zeta function and its simple pole at z=1. One can only
speculate about whether Hardy would have changed his mind had he known the
Erdös-Selberg proof of the theorem that does not use these methods at all.
It has been argued that the classical proof is still in many ways more
interesting.

Martin
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