Actual Infinity - History

History

The ancient Greek term for the potential or improper infinite was apeiron (unlimited or indefinite), in contrast to the actual or proper infinite aphorismenon. Apeiron stands opposed to that which has a peras (limit). These notions are today denoted by potentially infinite and actually infinite, respectively.

Anaximander (610-546 BC) held that the apeiron was the principle or main element composing all things. Clearly, the 'apeiron' was some sort of basic substance. Plato's notion of the apeiron is more abstract, having to do with indefinite variability. The main dialogues where Plato discusses the 'apeiron' are the late dialogues Parmenides and the Philebus .

Aristotle sums up the views of his predecessors on infinity thus:

Only the Pythagoreans place the infinite among the objects of sense (they do not regard number as separable from these), and assert that what is outside the heaven is infinite. Plato, on the other hand, holds that there is no body outside (the Forms are not outside because they are nowhere), yet that the infinite is present not only in the objects of sense but in the Forms also. (Aristotle)

The theme was brought forward by Aristotle's consideration of the apeiron in the context of mathematics and physics (the study of nature).

Infinity turns out to be the opposite of what people say it is. It is not 'that which has nothing beyond itself' that is infinite, but 'that which always has something beyond itself'. (Aristotle )

Belief in the existence of the infinite comes mainly from five considerations:

1. From the nature of time - for it is infinite.
2. From the division of magnitudes - for the mathematicians also use the notion of the infinite.
3. If coming to be and passing away do not give out, it is only because that from which things come to be is infinite.
4. Because the limited always finds its limit in something, so that there must be no limit, if everything is always limited by something different from itself.
5. Most of all, a reason which is peculiarly appropriate and presents the difficulty that is felt by everybody - not only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite because they never give out in our thought. (Aristotle )

With magnitudes the contrary holds. What is continuous is divided ad infinitum, but there is no infinite in the direction of increase. For the size which it can potentially be, it can also actually be. Hence since no sensible magnitude is infinite, it is impossible to exceed every assigned magnitude; for if it were possible there would be something bigger than the heavens. (Aristotle )

Our account does not rob the mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraversable. In point of fact they do not need the infinite and do not use it. They postulate only that the finite straight line may be produced as far as they wish. (Aristotle )