[FOM] The lure of the infinite

José Félix Costa fgc at math.ist.utl.pt
Thu Feb 16 13:34:07 EST 2006

Martin Davis: «Warnings about the dangers and contradictions lurking in
reasoning about the infinite have abounded since the ancient Greeks.»

In fact, and unfortunately for the western culture, Zeno's paradoxes are not
well explained in books. E.g., in philosophy books. In fact, the four
paradoxes have much to do with the partition (a) space continuous & time
continuous, (b) space continuous & time discrete, (c) space discrete & time
continuous, and (d) space discrete & time discrete. One paradox for each

As Professor Martin Davis states, some of them (in the continuous projecton)
have to do with the concept of infinite.

I think that the most interesting account on Zeno's paradoxes is given by
our great master Alexandre Koyré in his «Studies on the history of
philosophical thinking». In his article, Koyré covers the definite
hypothesis of Evellin, Noel's criticism, Bergson interpretation, the
infinite of Descartes, Bolzano's paradoxes, etc.

It is a pity that all these concepts are given in most books without an
in-depth analysis. The same as the concept of «water» being first principle
being confused with the water that flows when we turn on the tap.

J. Felix Costa
Departamento de Matematica
Instituto Superior Tecnico
Av. Rovisco Pais, 1049-001 Lisboa, PORTUGAL
tel:      351 - 21 - 841 71 45
fax:     351 - 21 - 841 75 98
e-mail:   fgc at math.ist.utl.pt
www:    http://fgc.math.ist.utl.pt/jfc.htm

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