[FOM] The lure of the infinite

Arnon Avron aa at tau.ac.il
Sat Feb 18 14:46:49 EST 2006

On Wed, Feb 15, 2006 at 02:11:38PM -0800, Martin Davis wrote:
> Warnings about the dangers and contradictions lurking in 
> reasoning about  the infinite have abounded since the 
> ancient Greeks. 

This for itself is a good and more sufficient reason to take ZF
(to say nothing about anything that goes beyond it) as going beyond
what can be taken as safe according to *strict mathematical*
standards. No  artificial "story" should be invented here. The
story exists, and have been repeatedly told since the time
of the Greeks. 

> Nevertheless, mathematicians have pushed ahead, knocking down 
> one barrier after another, guided by their intuitions  and 
> by a kind of overspill of  concepts and formalisms that seemed 
> to work well beyond the range for which 
> they had been devised. Berkeley's devastating critique of the nascent 
> calculus was entirely correct, but it failed to deter 
> mathematicians from 
> using and expanding it while a minimally satisfactory foundation 


> Although we may be as far from a satisfactory reply to 
> such questions as  Berkeley's contemporaries were to his, 
> one can confidently predict that so 
> long as mathematicians can obtain worthwhile results using 
> set-theoretic  methods (and the incompleteness theorem 
> does suggest that that will be the 
> case), calls for working with more "reliable" methods 
> will fall on deaf ears.

Obviously,  Berkeley's devastating critique did NOT fall
on deaf ears. Otherwise we would still teach and make proofs 
in mathematics using infinitesimals (the fact that some justification
for using them was found many years after they have been BANNED
from official mathematical proofs, though not from less
rigorous applications, is irrelevant). The fact is that 
nowadays we totally reject "proofs" that were given in
the 17th and 18th century, and also some of the "theorems"
proved then. So although mathematicians to worthwhile results using
the dubious methods, a quest for more reliable methods
did start - and even succeeded. So based on this experience
one can confidently predict that sooner or later mathematicians
will pay attention to justified criticisms and replace 
dubious set-theoretic  methods by more reliable one.

A side remark:
 Actually, I am not confident that this is what will happen.
History does not necessarily repeats itself, and
unlike Friedman, I don't have the ability to make prophecies
about what will happen at 2100 (I do predict, though, with great
certainty albeit not mathematical certainty, that none of us will
live to see).  My personal *impression*, by the way, is that 
the current core mathematicians are doing old-fashioned mathematics, 
that will be dismissed as uninteresting and unimportant by the
coming  generations, for whom problems related to computing 
science will be the main focus. This impression of mine is due
(among other things) to the fact that in Israel not only there are
far more students (including research students) in  computer
science than in the  math departments, but also most of the *better* 
students prefer theoretical CS to Math. But maybe in the
states and in Europe the situation is different (is it?).

Arnon Avron

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