[FOM] Predicative foundations

José Félix Costa fgc at math.ist.utl.pt
Mon Feb 20 06:55:48 EST 2006

Bill Taylor question on Zeno's phenomenon:

Zeno presented four paradoxes, one for each partition of space x time.
Suppose we were discussing the dicotomy paradox for continuous space and a
discrete time: in each division of space we get always the same quantum of
time --- the amount of time is then infinite to cross the distance from A to
B, and, consequently motion can not be. But if, e.g., space is discrete and
time continuous, then I will present you the arrow paradox... and motion can
not be. And so on.

The four paradoxes make one paradox to show that motion can not be.

The usual account of such paradoxes is to isolate one (first error), e.g.,
the paradox of dicotomy, introducing it as Zeno's paradox (second error),
and then show how the magic of the limit process can explain that motion can
be (by introducing the concept of velocity). Zeno was smarter than this. We
can call this «the use of reason to confuse reason».

In fact, infinite is encoded in Zeno's paradox(es) in a subtle way. Koyré
explains it quite well.

When I discovered about the true Zeno's paradox(es), I was puzzled by the
role of infinite, in the same vein I was astonished when I discovered, many
years ago, that the elements of the greeks, like «water», are not exactly
the substances of their names.

I have been observing FOM for many years, and I realize that Phylosophy,
Phylosophy of Science, History of Science, made also part of FOM.

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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