Thursday, April 12, 2012

Connolley on the continuum

Bill Connolley has post at Stoat about Aristotle and the continuum, and I think I finally see what his problem is.  (and it's also my problem). Is Aristotle's notion of the continuum roughly congruous with the modern notion, and did Aristotle simply get it wrong? In which case, how on earth could he have got it so wrong?
.. the problem I'm having now is to see how his argument can ever have been believed, by him or by anyone else
Or was Aristotle's notion something quite different, such that his view that 'it' is not composed of indivisibles is perfectly consistent. In which case,  what on earth was his notion?

I think I see a way out (noting carefully that I am not a mathematician, and this is just my two cents).  Connolley starts with the idea that the continuum is just the real numbers between two points (say 0 and 1).  If that's what the continuum is, i.e. if it is just those numbers, then it's surreal to ask whether it is composed solely of indivisibles, i.e. composed of numbers. If that's how you define it, it's an absurd question. And even more absurd to argue that is isn't composed of numbers at all. That would be like concluding that bachelors are married men.

But we don't have to start with that idea at all. Suppose we characterise what-is-continuous as that which is divisible into parts, and which after any finite number of such divisions leaves parts which are continuous themselves.  Then it is an open question whether such divisibility could be completed or not.  Clearly there would have to be an infinite number of such divisions, since by definition any finite division leaves continuous parts which can be further subdivided. And if that were possible, i.e. if it were possible to complete the process, then by definition of 'complete', what was left over would be indivisible.

So perhaps it is coherent to hold that  Aristotle agrees with the moderns in a defining characteristic of the continuum (i.e. infinite divisibility into parts), but disagrees over the accidental property of whether the process of divisibility can be completed.  And disagrees, of course, that it is an accidental property at all, for he holds that the impossibility of completion can be proved by logical means, and is thus an essential property.

10 comments:

Belette said...

William, if you please.

I agree that the main problem is trying to understand what he is talking about; or relating his ideas to ours. I still think this largely amounts to understanding what he meant by continuum. You can try to do that by reading his definitions, as you've just done, but (whilst instructive) this doesn't work. I tried to hint earlier about rational numbers, but I'll have to be more explicit.

> Suppose we characterise what-is-continuous as that which is divisible into parts, and which after any finite number of such divisions leaves parts which are continuous themselves

This is one of the defns that A gives. But it doesn't work (or at least, it doesn't produce what we would call a continuum). The set of rationals satisfies this property, but isn't "the continuum". Still worse, consider the Cantor set (http://en.wikipedia.org/wiki/Cantor_set). Because you've essentially provided a recursive defn of "continuous", what you end up with is something-like a fractal.

What this demonstrates, I think, is that A has a idea of "the continuous" and he provides things that he calls definitions that attempt to hea you in the correct direction, but which actually aren't definitions at all - they are hints, ideas, directions.

My current opinion (which this post doesn't alter) is that the best impression of what A was thinking is given by the update to my post http://scienceblogs.com/stoat/2012/04/aristotle_and_the_continuum.php.

David Brightly said...

Belette is probably right that Aristotle's infinite divisibility criterion doesn't exactly pin down what we now regard as a continuum. It isn't sufficient. But it, or something like it, may well be necessary. Aristotle's original argument, as expounded by Ed, says that if anything is a continuum, then its parts must fit together in certain ways. Points don't fit together in these ways. Ergo, points cannot be parts of a continuum. What we have to do is decide whether Aristotle's 'parts fitting together' criterion is indeed a necessary condition of being a continuum, ie, follows from being that kind of thing, and we can do this for both our modern idea of continuum and for a reconstruction of Aristotle's conception, if we can achieve this. In so far as I understand Aristotle's condition, it seems to be couched in terms that are applicable only to intervals and therefore begs the question against points. I'm trying to explain what I mean by 'terms applicable only to intervals' elsewhere.

Ed's suggestion that the ancient and modern views can be reconciled by seeing the modern view as accepting a completed infinity of divisions, anathema to Aristotle, is ingenious, and has to be part of the solution, I think. My feeling is that Aristotle is guided by his intuitions regarding physical matter. He sees that there is a place where a division may be made and all the material falls to one side or the other. The modern view sees the place as a thing in itself. Thus

(0,1) = (0,1/2) ⋃ {1/2} ⋃ (1/2,1)

We might say that Aristotle would see things as

(0,1) = (0,1/2) ⋃ (1/2,1)

Edward Ockham said...

I tried to hint earlier about rational numbers, but I'll have to be more explicit.

I don't see why this is such a big deal. Aristotle knew about irrationals (I think). All it amounts to, AFAICS, is being able to divide the continuous in different ways. We we can reach both a rational and non-rational point by a systematic division of a certain kind. For example, I take a line that is 10 units long and cut off a bit 3 units long. Then cut off a bit 0.1 units long. Then another bit 0.04 units and so on in the sequence
3.141592653 .... and see where that gets us. That is a method of division different from simply cutting the line into two bits of 5 units, but not essentially different.

The key is that Aristotle thought that such an infinite process could not be completed. He might have been wrong as a matter of actual fact (or not - it's moot) but it doesn't prove that he was conceptually confused.

Belette said...

> I don't see why this is such a big deal

Well DB got it :-). The point is that the rationals (or the Cantor set) satisfy what A gives as the criterion for a continuum. And yet the rationals are not a continuum. Ergo, A's defn is wrong (or incomplete, if you prefer).

> Aristotle knew about irrationals (I think)

You really ought to read the PDF I ref'd (http://web.maths.unsw.edu.au/~jim/AristotleContinuum.pdf). Yes, he did; he was usually thinking about sqrt(2) as his example.

> We might say that Aristotle would see things as (0,1) = (0,1/2) ⋃ (1/2,1)

There is some interesting discussion of whether A understood this kind of half-open interval stuff in that PDF.

> we can reach both a rational and non-rational point by a systematic division of a certain kind...

Why go about it in such a long-winded way? Start with a line 10 units long and cut off 3.141592653... if that is where you want to get to. Unless there is some reason you want to limit yourself to only cutting in rational increments. But A made no such limitation.

> The key is that Aristotle thought that such an infinite process could not be completed. He might have been wrong as a matter of actual fact

Agreed, sort-of. In the real world, of course, you can't. In the maths world, again, its a matter of your defns and axioms. If you say you can do an infinite numbers of things, then you can. Otherwise you can't.

Edward Ockham said...

>> The point is that the rationals (or the Cantor set) satisfy what A gives as the criterion for a continuum.

That's what I'm not seeing. Why should they satisfy it, if the continuum can be carved up in additional ways?

Imagine we take the segment 3.1 to 3.2, then 3.14 to 3.15 and so on, so that each segment gets smaller and smaller and as it were approaches the point pi. Why does Aristotle's method rule out getting to pi?

Edward Ockham said...

>>In the real world, of course, you can't.

There you go again. Perhaps by 'real world' you mean 'physical world'? Obviously there are some real things that are not physical. Or could be - angels for instance.

Edward Ockham said...

>>If you say you can do an infinite numbers of things, then you can

And there you go again. You can only do p if you can do p, period. Saying you can do p is not enough. Plenty of people say all sorts of things, that they actually can't do.

Anthony said...

"Obviously there are some real things that are not physical. Or could be - angels for instance."

Huh? Can you elaborate on that?

Edward Ockham said...

>>"Obviously there are some real things that are not physical. Or could be - angels for instance."

>Huh? Can you elaborate on that?

Angels are not material. They have no physical bodies.

Anthony said...

>> Angels are not material. They have no physical bodies.

Angels are not real, either.