Aristotle’s potential infinity

George asks:

What is meant by “being gone through” in Aristotle’s Physics Book 3, Chapter 4 (204a): What is incapable of being gone through, because it is not in its nature… Examined? Analysed?

Answer by Craig Skinner

Here “gone through” is meant literally, not metaphorically as in perused, examined, analysed. So a better translation is “What is incapable of being traversed, because it is not the kind of thing that can be traversed.” Here he speaks of infinite physical magnitude — we can set out to traverse it but the journey never ends.

The main Aristotelian view on infinity which is still relevant is that an actual infinity cant exist, only a potential one.

We can apply this both to the infinitely large and to the infinitely small.

Thus, the natural numbers are a potential infinity. No matter how many we list, there is always a next one. But we cant collect all of them at once as an actual infinity (we can of course deal with the notion of different sizes of infinity and give them symbols, as Cantor does).

As regards the infinitely small, consider division of a finite line. We can divide it in two, divide each of the halves, then each of the 4 pieces, and so on as long as we like but we never reach an end because each line segment, however small, can always be further divided. So a line is potentially infinitely divisible, but not actually. It contains an infinity of potential points. If we divide it, say, exactly in the middle, we create one actual point. We can divide it anywhere, creating as many actual points as we want, but cant divide it everywhere to produce an actual infinity of points. So the potential here cant be completely fulfilled (as in an acorn’s potential to be an oak tree) only fulfilled as completely as possible in the process of division ad infinitum.

All this is relevant to the modern view of the continuum, that a line consists of an uncountable infinity of points. But it cant: a point has no size, and no matter how many we lay down, infinite number or otherwise, the totality has no length. So, whilst a line can contain an infinity of points, it cant consist of an infinity of points. I wont go into attempts to resolve this with the notion of infinitesimals, discredited in the 18th century but respectable again these days).

This is only one of very many ways in which Aristotle’s views are highly relevant today.

2 thoughts on “Aristotle’s potential infinity

  1. The comment you refer to (Physics iii 4, 203b30) is “For eternal things there is no difference between being possible and being actual”. Here Aristotle says that given enough time anything that can possibly occur will occur, or no potential can remain unactualized for ever. It is a version of the Principle of Plenitude. The theological take on it is that god’s perfection would only be realized if he created everything that was possible. But it neednt be linked to religion, for instance Lewis’s view that all possible worlds are real. I dont think it is in conflict with Aristotle’s view on primary substances ie individual concrete things (en-formed matter) which bear qualities and undergo change.

  2. Yes thank you for your answer, and additional comments. A midpoint of course, cannot have any dimension, for then it would contain the mid point.
    A further matter: Does Aristotle’s Statement re Infinity “What can be, must be”, contradict the individuality of ‘primary substance’ ?

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