So if you're A[ristotle], then given a line segment between two points, you can keep cutting it and keep finding points, none of with (of course) touch. And in your mind, therefore, you have a series of line segments separated 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.I think the idea of points 'appearing' when you divide the continuous is foreign to Aristotle's intention (at least at Physics 231a21). Rather, you divide the continuous and you get more continuous, period. You don't 'find' any points after a finite number of division, for the 'points' could only appear when the process of division is complete, which (for Aristotle) can never happen.
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.
Remember that Aristotle doesn't talk about 'points'. He talks about the 'indivisible'. You start with the idea of a continuous thing as something which when divided gives two continuous things. It follows logically from this that the continuous is not indivisible, since it is part of of its definition that it can always be divided. It also follows that no finite process of division will yield anything that cannot be divided further.
If we then define 'composed of' as the relation between the continuous and any set of parts that result from any process of division, it follows that the continuous is composed solely of parts which are continuous when the process is finite. I.e. no points, no 'indivisibles' at all. Just many bits of continuous. Now add the assumption, which Aristotle thinks is impossible, that the process of division can be completed, and by definition (a) the process cannot be finite, from our original definition (b) what is left over will be indivisible, otherwise the process would not be complete and (c) the original continuous thing will be 'composed' of these indivisible thingies, from our definition of 'composed'.
That is, it’s not that the points start appearing as soon as you start splitting the marble. Rather, you only get more bits of marble. But if you keep bashing away hard enough so as to get millions of tiny grains of marble, a heap of fine sand, you can visualise where the process is heading – do this infinitely many times and those little grainy atoms as it were turn into real atom which cannot be further subdivided. Then, and only then, do the points appear. For points are indivisible.
On William's claim that Aristotle has an 'artifical restriction on the meaning of infinity' that's completely wrong. Aristotle understands the same as we do: an infinite process is one that cannot be completed in a finite number of steps. But he also holds that such a process cannot be completed at all, because it is infinite.
6 comments:
> But he also holds that such a process cannot be completed at all, because it is infinite.
Yes, that is his artifical restriction. Modern maths is richer, because it does not impose that artifical restriction.
> You start with the idea of a continuous thing as something which when divided gives two continuous things
Agreed. But this too makes his original proof vacuous. He is trying to show that "Thus a continuum cannot be composed of indivisibles" (from your first post). And his defn of the continuum is "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". Therefore, from his defns, you can immeadiately infer his result. The "proof" he gives is pointless (ha ha).
EO,
If you checked the mathematical definition of continuity that I linked earlier, you might note that it does not invoke "indivisibles." Rather, any point or element of a continuum must be within an "episolon neighborhood" of another element. If so, it is continuous. They phrase it this way, btw, because they do not what to "spatialize" the concept of continuity.
Consider the family of open intervals defined recursively as follows:
i) (0,1) is in the family
ii) if (a,b) is in the family then so are (a,m) and (m,b) where m=(a+b)/2, ie, m is the midpoint of the interval (a,b).
The family consists of all open intervals of the form (n/2**m, (n+1)/2**m) for m>=0 and 0<=n<2**m. That is, the original unit interval, the two half intervals, the four quarter intervals, the eight eighth intervals, and so on.
This family of sets forms a model for the Aristotelian discussion of infinite divisibility. Every element is a continuous that not only can but has been divided into continuouses. So in a sense the division process has run to completion. But the family still contains no 'indivisibles'.
>>But the family still contains no 'indivisibles'.
I was puzzling about that. What if we reject any part that can be further subdivided as not being an 'ultimate part' of the parent member of the family (0,1). Then is something left over after the infinite division process, or nothing?
That there is something left over: if we expand pi to any degree of accuracy, we can always define two numbers which pi must be between. Eg
3.141592
3.141593
It seems like the process must be 'heading somewhere'. Indeed, heading towards the indivisible pi itself. Yet the family of intervals consists only of intervals, not 'indivisibles'.
My head explodes.
>>If you checked the mathematical definition of continuity that I linked earlier, you might note that it does not invoke "indivisibles."
Yes I read that but I didn't understand the concept of 'neighbourhood'. That whole website seems to consist of links to other definitions, with no end in sight.
>> Then is something left over after the infinite division process, or nothing? <<
Since we can't get there it's a question best not asked. But a modern mathematician would differ from Aristotle, I think, in saying that we can conceive of the totality of possible divisions that the process produces, and it doesn't include any indivisibles.
>> It seems like the process must be 'heading somewhere'. Indeed, heading towards the indivisible pi itself. Yet the family of intervals consists only of intervals, not 'indivisibles'. <<
Yes, it would seem that {pi} is the limit of this sequence of intervals but it doesn't occur in the sequence itself. In the paper that Belette recommended (well worth a look) Newstead says that Aristotle appreciates two out of three of Cantor's criteria for a continuum, viz, denseness and connectedness, but isn't so hot on closure. That is, the continuum contains the limit of every convergent sequence. This might seem a bit surprising as the ancients knew about irrational numbers. But they probably saw them geometrically rather than arithmetically---they didn't have our concept of an infinite decimal number, say, which is essentially a sequence of rationals---and didn't see them as limits. We had to wait until the mid-nineteenth century arithmetisation of the calculus (Weierstrass, et al) for this.
>> That whole subject seems to consist of links to other definitions, with no end in sight.
My head explodes. <<
Welcome to Math101 ;-)
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