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Aristotle postulated that an actual infinity was impossible, because if it were possible, then something would have attained infinite magnitude, and would be "bigger than the heavens." However, he said, mathematics relating to infinity was not deprived of its applicability by this impossibility, because mathematicians did not need the infinite ...
Aristotle, alive for the period 384–322 BCE, is credited with being the root of a field of thought, in his influence of succeeding thinking for a period spanning more than one subsequent millennium, by his rejection of the idea of actual infinity.
In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object. The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets.
But since Aristotle holds that such treatments of infinity are impossible and ridiculous, the world cannot have existed for infinite time. The most sophisticated medieval arguments against an infinite past were later developed by the early Muslim philosopher , Al-Kindi (Alkindus); the Jewish philosopher , Saadia Gaon (Saadia ben Joseph); and ...
Initially, Aristotle's interpretation, suggesting a potential rather than actual infinity, was widely accepted. [1] However, modern solutions leveraging the mathematical framework of calculus have provided a different perspective, highlighting Zeno's significant early insight into the complexities of infinity and continuous motion. [1]
Aristotle proves that both length and time are infinitely divisible, refuting atomism. [3] Andrew Pyle gives a lucid account of infinite divisibility in the first few pages of his Atomism and its Critics .
For an infinite regress argument to be successful, it has to show that the involved regress is vicious. [3] A non-vicious regress is called virtuous or benign. [5] Traditionally, it was often assumed without much argument that each infinite regress is vicious but this assumption has been put into question in contemporary philosophy.
Aristotle especially promoted the potential infinity as a middle option between strict finitism and actual infinity (the latter being an actualization of something never-ending in nature, in contrast with the Cantorist actual infinity consisting of the transfinite cardinal and ordinal numbers, which have nothing to do with the things in nature):