*Encyclopedia of Philosophy*. Copyright © 2001-2006 by Macmillan Reference USA, an imprint of the Gale Group. All rights reserved.

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## Counting Beyond the Finite

Broadly speaking there were two reasons for repudiating of Aristotle's prohibition of the actual infinite. First as mathematics became vastly more general, finitistic techniques came to be seen as confining. The ancient Greeks had a marvelously sophisticated theory of polygons and conic sections, but a fully general theory of shapes requires such techniques as approximating an unruly curve by an infinite sequence of curves that are better behaved.

The second reason was the so-called arithmetization of geometry, brought about by the investigation of alternatives to Euclid's axiom that, given a line and a point not on the line, there is on their plane exactly one line through the point that never intersects the given line, no matter how far the two lines are extended. Once alternatives to Euclidean geometry emerged, one could no longer be fully confident that Euclid's axioms correctly described the world...

This section contains 8,711 words(approx. 30 pages at 300 words per page) |