[FOM] The lure of the infinite

Martin Davis martin at eipye.com
Wed Feb 15 17:11:38 EST 2006

Warnings about the dangers and contradictions lurking in reasoning about 
the infinite have abounded since the ancient Greeks. For them, the very 
word for the infinite "to apeiron" suggested an unapproachable chaotic 
domain from which our finite world emerged. Aristotle famously upheld the 
potential infinite as against efforts to deal with the absolute infinite. 
Centuries later Gauss issued a similar warning. The famous paradoxes of 
Zeno, at least in part have to do with antinomies that seem to arise in 
connection with the infinite.

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 was two 
centuries away. I found particularly revealing Paolo Mancosu's "Philosophy 
of Mathematics & Mathematical Practice in the Seventeenth Century" 
especially his discussion of the amazed reaction to the "paradoxical" 
discovery by Torricelli in 1642 that a solid that was infinite in extent 
could have a finite volume.

In a recent post, Weaver reminds us that the Russell-Cantor antinomies 
concerning the set-theoretic hierarchy are still with us. How can we have a 
notion of set such that one cannot form the set of all the entities that 
fall under it? Workers in f.o.m. are far from having a good answer to this 
question. Set theorists may speak of proper classes, but, as one of them 
asked me not so long ago: am I really comfortable with that? Aren't proper 
classes really just sets in disguise?

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.

                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
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