## Wednesday, April 04, 2012

### Aristotle against the continuum

Belette ponders how we could show how Aristotle's argument that the continuum can't be composed of indivisibles is wrong. For reference, the argument is in Physics book 6 at 231 a2. Thomas Aquinas' discussion of it is in his lectures on Physics 6, lecture 1 n2.

Aristotle says that two thing are 'continuous' if their extremities are one, 'in contact' if the extremities are together, and 'in succession' if there is nothing of their own kind in between them. An 'indivisible' is that which has no parts.

Thus a continuum cannot be composed of indivisibles. For such indivisibles are either continuous, or in contact, or in succession. Not continuous, for no point can have separate extremities. Not in contact, for one thing can be in contact with another only if whole is in contact with whole or part with part or part with whole. But since indivisibles have no parts, they must be in contact with one another as whole with whole. And if they are in contact with one another as whole with whole, they will not be continuous: for what is continuous has distinct parts: and these parts into which it is divisible are spatially separate. Not in succession, for things are in succession if there is nothing of their own kind intermediate between them. But there is always a line between two points. And (supplementing his argument) a line is either composed of points, in which case the points are not in succession, by definition, or it is not, in which case the continuum is not composed of indivisibles alone.

What is wrong with the argument?

Belette said...

You missed my other point, which is that Aristotle appeared to be thinking about the real world. I'll answer for the real numbers (using the standard construction), which provides mathematical exactness; you can then ponder whether that applies to the the real world, and whether that matters.

So, the standard view of the real numbers (say, those between 0 and 1, just so we don't have to think of an infinite line) is that there are a non-denumerably infinite number of points, of which the real line is composed. The real line is continuous, via the std mathematical definition, or via Aristotle's version - there are no gaps.

Any given real number doesn't have a successor (well, and neither does any given rational. Actually considering the rationals thows up some problems with his defn's I think - you wouldn't consider then "continuous", since there are real "in between", but nonetheless there are none "of their own kind" in "between").

So the std view is that the reals are continuous. Thus his "Not continuous, for no point can have separate extremities" is the argument to address. But it doesn't make any sense to me - a point doesn't have separate extremities, but so what? A point isn't "in contact" with any other point, because there are always an infinite number of reals between any two distinct reals - but again, so what?

I think (with respect) that there isn't really an argument here to address. What we have here is someone struggling to think about infinity, but not really getting anywhere, because he is using words (and worse, words that relate to the real world), not mathematics. In maths, this stuff is limpidly simple and beautiful. But as soon as you stray off the straight path into not-quite-right words it immediately falls apart into mud.

> And (supplementing his argument)

> a line is either composed of points, in which case the points are not in succession,

Yes, agreed. The points are not in succession. So what? What does that tell you about a line being composed of points?

Belette said...

Just to add: I think a problem with this is that it is completely impossible to visualise in any meaningful way; if you're used to thinking in pictures you get nowhere. I tend to think of it as two points, with a point between, and then points between that, sort of receding and greying off into the distance. But that is wrong; I don't think of it like that because it helps, but because I can't suppress the image.

Edward Ockham said...

But you haven't answered the question, "what is wrong with his argument". He gives definitions of the key terms, and there is the appearance of logic. The 'real world' stuff is irrelevant. What is wrong with his argument?

Edward Ockham said...

>>The points are not in succession. So what?

The argument is of the form "p implies A and B and C", but not A and not B and not C, therefore not p. A perfectly valid argument. 'Being in succession' = C, so not being in succession eliminates the final disjunct.

There clearly is something wrong with the argument, but what is it?

Belette said...

I was trying to say that whilst his argument appears to have a logical form, I couldn't really parse it as such. I think the appearance of logical form is spurious.

Is this bit your version? "And (supplementing his argument) a line is either composed of points, in which case the points are not in succession, by definition, or it is not, in which case the continuum is not composed of indivisibles alone." I didn't understand why it was of any relevance. I said that. My reply to that would be: Yes, a line is composed of points. No, the points are not in succession. So what? Is it supposed to prove something?

Let me try to go through his argument, then:

> Thus a continuum cannot be composed of indivisibles.

OK, this is what he is trying to show.

> For such indivisibles are either continuous, or in contact, or in succession.

The reals are continuous. No two distinct reals (a, b) are "in contact", in the sense that there is always a finite distance between them (abs(a-b) in fact). The reals are not in succession.

> Not continuous, for no point can have separate extremities.

I'm not sure how to interpret this. A point doesn't have separate extremities, but so what? This appears to be his disproof that the reals are continuous. If so, it is where his argument falls over. It doesn't prove anything, although presumably A thought it did.

> Not in contact, for one thing can be in contact with another only if whole is in contact with whole or part with part or part with whole.

Irrelevant now, since we've agreed the reals aren't "in contact", assuming I've interpreted his "in contact" properly.

> But since indivisibles have no parts, they must be in contact with one another as whole with whole.

No. Why should this be true?

> And if they are in contact with one another as whole with whole, they will not be continuous: for what is continuous has distinct parts: and these parts into which it is divisible are spatially separate.

Irrelevant, since they aren't in contact.

> Not in succession, for things are in succession if there is nothing of their own kind intermediate between them.

This seems to be garbled. The reals are not in succession, but there is nothing else in between. This, I think, might be A not knowing about non-denumberable infinity. I'm not sure it matters, though.

> But there is always a line between two points.

Is that him or you? I'm not sure. There is indeed always a finite difference in between two successive points.

Edward Ockham said...

>>Is this bit your version? "And (supplementing his argument) a line is either composed of points, in which case the points are not in succession, by definition, or it is not, in which case the continuum is not composed of indivisibles alone." I didn't understand why it was of any relevance. I said that. My reply to that would be: Yes, a line is composed of points. No, the points are not in succession. So what? Is it supposed to prove something?
<<

I did add it, because it is required. Note Aquinas in the version linked-to above does exactly the same. We need it because of the definitions Aristotle gives. He says that two things are 'in succession' if there is nothing of their own kind in between them. He wants to prove that the points are not 'in succession'. It is not enough to say that there is a line between them, because a line is not of the 'own kind' of a point. So we have to supplement his argument by adding a further disjunction: either the line is composed of points, or not. If not, he has want he wants: the continuum includes lines as well as points. If it is, the points are not 'in succession' as he has defined it. Clearer?

As to " So what? Is it supposed to prove something?", clearly it does, as I said. Aristotle's argument is of the form 'if p, then A or B or C', and C is the final part of the disjunction he wants to prove.

>> I'm not sure how to interpret this. A point doesn't have separate extremities, but so what? This appears to be his disproof that the reals are continuous. If so, it is where his argument falls over. It doesn't prove anything, although presumably A thought it did.
<<

No, you haven't read his definition of 'continuous', which I included in the post above.

>> But since indivisibles have no parts, they must be in contact with one another as whole with whole.
>>>>No. Why should this be true?

True by his definition of 'in contact'. He gives careful definitions of each of the terms he is using.

>> But there is always a line between two points.
>>>> Is that him or you?

Him. I'll post on this tomorrow, but his argument is as follows: "If a continuum is composed solely of points, they must be either continuous with one another, or touch, or be in succession."

This has the logical form 'if p then A or B or C'. His aim is to show that A and B and C are false, therefore not-p, i.e. the continuum is not composed solely of points. If the argument is sound, then (1) the initial implication must hold, i.e. if the continuum is composed solely of points, then they are neither continuous with one another, by his definition of continuous, nor touch, by his definition, nor be in succession, by his definition. And (2) each of A B and C must be false.

There is an obvious flaw in his argument, and you are close to it, but you aren't really there.

Jan said...

This is a very nice argument and, contrary to one of the previous comments, it is evident it comes from a first class mathematical mind. In fact, it might be possible to convert it into a valid (modern) mathematical argument by substituting 'extremity' with 'topological boundary' (and making some other formalizations).

That being said, the crucial first premise:

P1. Indivisibles are either continuous, or in contact, or in succession

is false. Indeed, the negation of (P1) can almost be taken as a definition of continuity. If by the hypothetical indivisibles we take points, then continuity and being in contact as defined by Aristotle are trivial and ~P1 reduces to

D. For every two points x, y of the set S ⊂ R there exists a third point z ∈ S between x and y.

(D) is equivalent to saying that S is dense in line segment of R. In particular, the rationals, irrationals, rationals from the segment [0, 1] satisfy (D) and thus constitute counterexamples to (P1).

Edward Ockham said...

>>That being said, the crucial first premise .. is false

Yes precisely.