Aristotle and the continuum

And now for something almost completely different. This all begins at History of the infinite where I said it might be fun to work out why his [Aristotle’s] “proof” that the continuum can’t be composed of indivisibles is wrong. This lead on to Aristotle against the continuum – reply (wherein you will find Aristotle’s proof that the continuum cannot be composed just of points, laid out reasonably comprehensibly). There is also The history of the continuum and Another argument against indivisibles (which is a less viable attempt via addition).

Anyway, to summarise: thinking about infinity is hard. Suppose we abstract A’s argument away from “the continuum” (whatever that is) to the real line (which is at least clearly defined) – and let’s say, just the real numbers between 0 and 1 (I’m using “real” in the mathematical sense of “real number“, not in the sense of belonging-to-the-real-world, of course. The first sentence of that linked article is a bit rubbish, though. Sigh). Then to restate A’s argument, we’re obliged to say “the real line is not made up of just numbers” (numbers == points). This is self-evident twaddle (how can the real be made up of anything other than numbers? It is them, by definition. Although if you want to be pedantic it also has an ordering and a metric), so the argument collapses in a heap (although it took me a while to realise this). If you want to, you can try to read through A’s original argument without the hints, and see where his argument falls down, but it isn’t necessary to do that in order to see that it is wrong.

Indeed the problem I’m having now is to see how his argument can ever have been believed, by him or by anyone else. It doesn’t help that from “continuum” I automatically go to “real line”, where his stuff falls over without you pushing it. So we have to try to think like him, and I think the key is to think geometrically not numerically (incidentally, I think the issue of rationals vs irrationals, or countability, is irrelevant here; A postdates the proof of irrationals, if that helps). And also you need to blur the line between the real-world and the maths-world; he is thinking, I think, largely in terms of the real world, albeit a slightly idealised real world. So he is used to thinking of lines, and of line segments, and of geometrical proofs in which those lines are marked by a few points. So he thinks of the line as a thing, to which you can add a few points, and then a few more, but obviously never by that process make the whole line.

If anyone out there has a way of stating his thinking in a way that makes any kind of sense, do please comment (I believe I may have turned on Captchas, don’t let that put you off).

[Update: NB found me http://web.maths.unsw.edu.au/~jim/AristotleContinuum.pdf, and I think that essentially resolves the problem, with:

points only come into (actual) existence for Aristotle when a division is made between two line segments

That sounds correct, and explains the problem (together with his dislike of actual, as opposed to potential, infinities). So if you’re A, then given a line segment between two points, you can keep cutting it and keep finding points, none of which (of course) touch. And in your mind, therefore, you have a series of line segments spearated by points. What you can’t do is consider all possible cuts, because that kind of realised infinity is foreign to his way of thinking.

In which case, the final step is to go back and say, given that definition / idea, is his original proof valid? I think that, given that, his original result is valid, but vacuously so: he refuses to consider completed infinities, and a line, to be made of points, needs an infinite number of points, which he has ruled out, therefore a line isn’t made of an infinite number of points. But only because of his artifical restriction on the meaning of infinity.]