To understand the strength of Aristotle's arguments we must realize that
he gives the final setting to the crucial questions of "being" and
"becoming" in Greek and Middle Ages philosophy. In the earlier Greek
philosophers these themes were dealt with by principles ("archai") such
as water, air, fire, etc. and pair of 'physical' opposites such as
love/hate, cold/warm, wet/dry, etc. "Infinite" ("apeiron") was included
as a principle by many authors (Anaximander, earlier Pythagoreans, late
Plato's dialogues). In the ancient Greek culture however the meaning of
the "apeiron" was very cloudy: there was no 'size' aspect, and the word
was the primitive form of the substantive "peras" which means "limit",
but more as a "mark" than as a "border". Even in Aristotle it meant
"something that can not be passed through" (as pointed out in the e-mail
of S.Kutler). The general idea had to be something like "not determined,
not marked", so that it was often associate with the idea of "not
divisible, perfect and whole".
Another source of the "infinite" question was its role in mathematics
where it was employed both as the 'infinite size' of the world (Archytas,
as 'potential' infinite) and the 'infinite divisibility' of the continuum
(Anaxagoras, Eudoxus, even here as potential).
The new Aristotelean 'establishment' was centred on a world of
"individual objects" as first substances with their "attributes" and on
a series of "metaphysical" pairs of opposites (form/matter, act/power,
substance/accident) to deal with the real world description. There was
no room at all for the 'infinite' as a principle. In other terms the
"infinite" was completely out of the Aristotelean framework, which was
'structurally' finitist, from physical, logical and epistemological
views, and the arguments in the Aristotean texts simply made this clear.
Aristotle however was 'constrained' to accept the potential infinite
just for its role in mathematics. But even in mathematics the number
was always finite and nobody accepted the real existence of the "point",
rejected by Plato (as referred in Aristotle's texts), but even from
Eudoxus and Archimedes who managed their infinitesimal problems with
the exhaustion methods just to avoid the infinite: for a modern reader
it is noteworthy how near Archimedes was to the idea of 'limit' and
how sharp was his refusal to accept that idea. Here we find the core
of the ancient Greek mathematics: a concept of integer number as "number
of" which excluded both zero and infinite, a fuzzy idea of 'fraction',
the non-numerical nature of the idea of "geometrical magnitude".
In the following two thousand years these themes will dominate
mathematics and philosophy. Only the Christian idea of God undermined
the old establishment. Paradoxes of the infinite were discovered and
analyzed by mathematicians and philosophers till the last century, when
the problem found its modern solution. In my opinion, the real question
about the actual infinite is not to understand the Aristotelean rejection,
but to understand the Cantorian acceptance.
Luigi Borzacchini