[FOM] The lure of the infinite
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.
Visiting Scholar UC Berkeley
Professor Emeritus, NYU
martin at eipye.com
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