In philosophy and theology, infinity is explored in articles under headings such as the Absolute, God, and Zeno's paradoxes.
In Greek philosophy, for example in Anaximander, 'the Boundless' is the origin of all that is. He took the beginning or first principle to be an endless, unlimited primordial mass (ἄπειρον, apeiron). The Jain metaphysics and mathematics were the first to define and delineate different "types" of infinities. The work of the mathematician Georg Cantor first placed infinity into a coherent mathematical framework. Keenly aware of his departure from traditional wisdom, Cantor also presented a comprehensive historical and philosophical discussion of infinity. In Christian theology, for example in the work of Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity.
Early thinking
Egyptian
Greek
Anaximander
An early engagement with the idea of infinity was made by Anaximander who considered infinity to be a foundational and primitive basis of reality. Anaximander was the first in the Greek philosophical tradition to propose that the universe was infinite.
Anaxagoras
Anaxagoras (500–428 BCE) was of the opinion that matter of the universe had an innate capacity for infinite division.
The Atomists
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 and after
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 the work entitled Physics, written by Aristotle, Aristotle deals with the concept of infinity in terms of his notion of actuality and of potentiality.
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, , 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:
The parts are actually there, in some sense. However, on 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 LEJ 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.
Roman
Plotinus
Plotinus considered infinity, while he was alive, during the 3rd century A.D.
Simplicius
Simplicius, alive circa 490 to 560 AD, thought the concept "Mind" was infinite.
Augustine
Augustine thought infinity to be "incomprehensible for the human mind".
Early Indian thinking
The Jain upanga āgama Surya Prajnapti (c. 400 BC) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
Enumerable: lowest, intermediate and highest
Innumerable: nearly innumerable, truly innumerable and innumerably innumerable
Infinite: nearly infinite, truly infinite, infinitely infinite
The Jains were the first to discard the idea that all infinities were the same or equal. They recognized different types of infinities: infinite in length (one dimension), infinite in area (two dimensions), infinite in volume (three dimensions), and infinite perpetually (infinite number of dimensions).