Greek
Anaximander was an early thinker who engaged with the idea of infinity, considering it a foundational and primitive basis of reality. Anaximander was the first in the Greek philosophical tradition to propose that the universe is infinite.
Anaxagoras (500–428 BCE) believed that the matter in the universe has an innate capacity for infinite division.
A group of thinkers of ancient Greece (later identified as the Atomists) all similarly considered matter to be made of an infinite number of structures as considered by imagining dividing or separating matter from itself an infinite number of times.
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 Book 3 of his work entitled Physics, Aristotle deals with the concept of infinity in terms of his notion of actuality and of potentiality.
... It is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence the infinite is potential, never actual; the number of parts that can be taken always surpasses any assigned number.
This is often called potential infinity; however, there are two ideas mixed up with this. One is that it is always possible to find a number of things that surpasses any given number, even if there are not actually such things. The other is that we may quantify over infinite sets without restriction. For example,
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{\displaystyle \forall n\in \mathbb {Z} (\exists m\in \mathbb {Z} [m>n\wedge P(m)])}
, which reads, "for any integer n, there exists an integer m > n such that P(m)". The second view is found in a clearer form by medieval writers such as William of Ockham:
Sed omne continuum est actualiter existens. Igitur quaelibet pars sua est vere existens in rerum natura. Sed partes continui sunt infinitae quia non tot quin plures, igitur partes infinitae sunt actualiter existentes.But every continuum is actually existent. Therefore any of its parts is really existent in nature. But the parts of the continuum are infinite because there are not so many that there are not more, and therefore the infinite parts are actually existent.
The parts are actually there, in some sense. However, in this view, no infinite magnitude can have a number, for whatever number we can imagine, there is always a larger one: "There are not so many (in number) that there are no more."
Aristotle's views on the continuum foreshadow some topological aspects of modern mathematical theories of the continuum. Aristotle's emphasis on the connectedness of the continuum may have inspired—in different ways—modern philosophers and mathematicians such as Charles Sanders Peirce, Cantor, and L. E. J. Brouwer.
Among the scholastics, Aquinas also argued against the idea that infinity could be in any sense complete or a totality.
Aristotle deals with infinity in the context of the prime mover, in Book 7 of the same work, the reasoning of which was later studied and commented on by Simplicius.
