[FOM] Reply to Zeilberger

Alasdair Urquhart urquhart at cs.toronto.edu
Wed Nov 12 10:17:57 EST 2003

I am afraid we are arguing at cross
purposes.  There are two questions here:

1. Does Hardy say the things you attribute
to him?  The  answer is "no".  Hardy does
not say a word about "core" or "mainstream"
mathematics.  He does have a detailed discussion
of what we mean by "serious" mathematics.  
You take his critical remarks about chess 
problems to be an attack on discrete, finitary
mathematics.  However, he says:

	No chess problem has ever affected
	the general development of scientific
	thought; Pythagoras, Newton, Einstein
	have in their times changed its whole

Can you honestly quarrel with this?

Having said all this, I will admit that Hardy had some
blind spots.  See, for example, Weil's discussion 
(in his Collected Papers Vol. III, p. 281) of
Hardy's comment on Ramanujan's work on infinite series
arising in the theory of modular functions:
	"We may seem to be straying into one of the
	back-waters of mathematics, but the genesis of
	\tau(n) as a coefficient in so fundamental
	a function compels us to treat it with respect."
			("Ramanujan", p. 161)

On the other hand, Hardy does give a detailed discussion
of Ramanujan's work and is far from dismissing it.
Incidentally, I just noticed that Weil significantly
misquotes Hardy; he quotes Hardy as saying:

	"We seem to have drifted into one of the back-waters
	of mathematics,"

which has a much more dismissive tone that Hardy's original

By the way, was Hardy uninterested in computational mathematics?
Certainly not!  Look at the delightful chapter entitled
"Asymptotic theory of partitions" in his "Ramanujan".
It contains the following wonderful passage:

	"At this point we might have stopped had it not been
	for Major MacMahon's love of calculation.  MacMahon
	was a practised and enthusiastic computer, and made	
	us a table of p(n) up to n = 200."
				("Ramanujan", p. 119)

So Hardy certainly was a fan of computer-based mathematics!

2.  Is discrete, finitary mathematics important and interesting?
Of course!  The prejudice directed towards the subject is just
silly, and I certainly wasn't identifying with it in any way.
I wouldn't have spent the last 20 years working on the complexity
of propositional proofs and similar things if I had thought
that such problems were trivial and uninteresting.

Alasdair Urquhart

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