Our earlier discussion of Aristotle's argument that the continuum is not composed solely of points ('indivisibles') neatly illustrated an important philosophical principle: that the only adequate reply to a philosophical argument is to show what is wrong with it. It is no good simply saying that the conclusion is false. Nor claiming that some respected authority says it is false. Nor even stating an arguments against it (which simply shows that there are two arguments with conflicting conclusions). No: the only suitable way is to show what is wrong with the argument. And there are only two ways of doing that: either show that the argument is not valid, i.e. that the premisses can be true with the conclusion false. Or show that the argument is not sound, i.e. one or more of the premisses is false.
Now Aristotle's argument is this.
(1) If a continuum is composed solely of points, these points must be either continuous with one another, or touch, or be in succession.
(2) The points are not continuous
(3) They do not touch
(4) They are not in succession
(5) Therefore the continuum is not composed solely of points.
Clearly the argument is valid. If the four premises are true, the conclusion cannot be false. For the consequent of the implication in (1) is a disjunction. A disjunction is false when all of the disjuncts are false. Premisses (2)-(4) assert the falsity of each of the disjuncts, so if they are true, the consequent is false. If the consequent is false (and if the implication is good) the antecedent is false – consequens falsum ergo antecedens. And if the antecedent is false, the conclusion is true, for the conclusion is the opposite of the antecedent. Therefore the argument is valid.
Is the argument sound? I don't see anything wrong with premises (2)-(4), given the definitions that Aristotle supplies in the text, i.e. the definitions of continuity, contact, succession etc. (A common problem with replies to philosophical arguments is that they ignore careful definitions given in the preliminaries, and focus on something else). So the culprit is clearly (1), as is obvious after a little thought. Take any two points on the continuum you like. Then it does not have to be true that they are either 'in contact', or 'continuous' or 'in succession'. Of any two different points, it can – and indeed always is - true that there are further points in between them.
Why is the argument convincing at all? Probably because it has the appearance of rigour, and because it starts with the covert and natural assumption that each point has a successor. If it has a successor, then there can't be a point in between them (otherwise the point between would be the successor). But there must be something else in between, something that is not a point, and therefore the continuum can't consist wholly of points. But only if there is a successor, which is not necessarily true.
So Aristotle's argument is flawed. But not because Cantor was right, nor because modern mathematics is better, or clearer, or because mathematics is different from the real world (whatever the real world is). It is flawed because it has a flaw, a flaw which we can clearly demonstrate. That is where we philosophers are coming from.
24 comments:
Aristotle is talking about all points. Thus, we are dealing with a unity and not with an aggregate. Consequently, we can only meaningfully use "every" and not "any".
Where is the flaw with (1)? You have merely argued for (2), (3), and (4).
>>Where is the flaw with (1)? You have merely argued for (2), (3), and (4).
Above: "So the culprit is clearly (1), as is obvious after a little thought. Take any two points on the continuum you like. Then it does not have to be true that they are either 'in contact', or 'continuous' or 'in succession'. Of any two different points, it can – and indeed always is - true that there are further points in between them. "
I agree that (1) is false, as Jan said in the last post. It was better disguised in the original.
> that the only adequate reply to a philosophical argument is to show what is wrong with it (etc)
That I disagree with. Showing what is wrong is nice, but showing that it is wrong is sufficient, if all you care about is the truth or falsity of the conclusion (of course if all you care about is the argument, that would be a different matter). We already know that A's "the continuum is not composed solely of points" is false, without examining his argument in detail. And we know this not by authority, but because we have a model of a continuum for which it is not true. Much in the same way that we know that Euclidean geometry isn't the only geometry, because we have models of other geometries.
> and because it starts with the covert and natural assumption that each point has a successor
I don't understand why you say that. Being "in succession" is just one of 3 "or"s, and isn't assumed.
There is more: A has defined most of is terms, but not "continuum". Is that just supposed to be obvious, or is it defined elsewhere. Do the rationals fulfil his defn of continuum?
Also: so, we agree A is wrong. How long (you're the historian...) did it take people to realise this? My guess is that it wasn't until the work of Cantor etc provided a proper model for the continuum that people actually understood the sequencing, and that A's stuff played no part in this. In other words, it was a sterile error.
"Take any two points on the continuum you like. Then it does not have to be true that they are either 'in contact', or 'continuous' or 'in succession'. Of any two different points, it can – and indeed always is - true that there are further points in between them."
(1) says "If a continuum is composed solely of points, these points must be either continuous with one another, or touch, or be in succession."
Your argument denies the consequent, but it does not affirm the antecedent. Your argument rests on the hidden premise that the continuum is composed solely of points.
>>Your argument denies the consequent, but it does not affirm the antecedent.
I'm not with you. The consequent is false, therefore also the antecedent.
I am not arguing against Aristotle, but against Edward. Specifically, his "Take any two points on the continuum you like. Then it does not have to be true that they are either 'in contact', or 'continuous' or 'in succession'. Of any two different points, it can – and indeed always is - true that there are further points in between them."
The issue I have is with his use of 'any'.
The problem here is that his use of 'any' presupposes the continuum to be an aggregate (of points) such as to allow picking out non-adjacent points. However, a continuum is not an aggregate, but a unity and the only way to get at a point is to arbitrarily divide the continuum into two parts, giving us two adjacent points (one on each side of the divide).
>> The problem here is that his use of 'any' presupposes the continuum to be an aggregate (of points) such as to allow picking out non-adjacent points.
I was simply pointing out the logical flaw in Aristotle's argument, and the presuppositions contained there.
>> However, a continuum is not an aggregate, but a unity and the only way to get at a point is to arbitrarily divide the continuum into two parts, giving us two adjacent points (one on each side of the divide).
A number of questionable assumptions here.
>>Showing what is wrong is nice, but showing that it is wrong is sufficient,
I don't see the difference between these (emphasis on word 'showing').
"The consequent is false, therefore also the antecedent."
If the consequent of (1), "these points must be either continuous with one another, or touch, or be in succession", is false, and therefore the antecedent of (1), "a continuum is composed solely of points", is false, then what's the problem with (1)?
You have claimed that (1) is clearly flawed, and then your argument that (1) is flawed is to deny the consequent of (1) ("Take any two points on the continuum you like. Then it does not have to be true that they are either 'in contact', or 'continuous' or 'in succession'. Of any two different points, it can – and indeed always is - true that there are further points in between them."). Your argument doesn't even touch the antecedent ("a continuum is composed solely of points"). So you have done nothing to show that (1) is clearly flawed. Your argument that (1) is flawed rests upon the readers acceptance of "a continuum is composed solely of points", which you have done nothing to show true.
Also, I think a less confusing translation for (1) would be "If a line is composed solely of points", not "If a continuum is composed solely of points". The word continuum carries with it a lot of extra baggage, and it doesn't seem to appear in the argument you link to.
It is also important to remember that Aristotle is talking here about the mathematics which pertains to the actual physical world, not the imaginations of present-day mathematicians.
I'm wholly not with you here. (1) is a material implication and to show it is false it is enough to show that the antecedent can be true and the consequent false. Thus it can be true that a continuum is composed solely of points, but false that these points must be either continuous with one another, or touch, or be in succession. Thus the implication is false.
This has nothing to do with 'denying the consequent' which I mention earlier in the post, though not here. 'Denying the consequent' means that if the implication is true (which in fact it isn't), the falsity of the consequent implies the falsity of the antecedent. You are confusing two entirely separate parts of the argument although I am at a loss to see how you could have confused them at all. The first part of my post, which I thought I had signposted pretty well, was to show that Aristotle's argument is valid. That's where 'denying the consequent' comes in. Note that I have a separate para beginning 'clearly the argument is valid'. The second part was to show that the argument is not sound, i.e. one of the premisses is false. I then go through the premisses one by one to see which it is. I conclude that it is (1), by showing that the antecedent can be true but the consequent false. This is the bit which you confused with the first part about validity. I separated these so clearly and carefully that I am, as I say, at a loss to see how you got it wrong.
I agree that the flaw with Aristotle's argument lies with (1) but for me the difficulty is in understanding why he should put this 'definition' forward in the first place! What can he have in mind that leads him to break down continuity into these three cases? Can we reconstruct his thought?
Here is a tentative suggestion. First of all, given his aversion to the actual infinite, the thought that a continuum could be 'composed' of countably many, let alone uncountably many, discrete points is simply beyond him. We really do have to wait for Cantor and Dedekind before we can see how to make sense of this. His idea of a 'part' of a continuum, say a line interval, seems to be a subinterval---something with distinct extremities or endpoints. Now, if we think of subintervals as possible parts of an interval, then any pair of such subintervals either (1) overlap, (2) touch, or (3) are separated. So we find three cases here. But though the second of Aristotle's cases seems to be about touching, and the third seems to admit the possibility of separation in its definition, it's hard to fit overlapping into his first case. Indeed, all three of his cases seem to come down to contiguity, ie, touching. Can anyone explain the three-way distinction that Aristotle is drawing here?
My guess is that he is groping towards an understanding of what we would now call 'connectedness'. A line interval is connected in the sense that, if it is decomposed into it least two subintervals, then for any subinterval there must be another subinterval contiguous with it---a successor (or predecessor) subinterval, in Aristotle's terminology. Although this captures, I think, a necessary property of a continuum, it says nothing about its ultimate parts.* To say that an interval is subintervals 'all the way down', which I think is what Aristotle is doing here, is to beg the question against other possibilities which we can conceive now, but were unimaginable for Aristotle.
But I haven't convinced myself.
* This is also true of the modern topological notion of connectedness, which rests on the fundamental idea of 'open set'.
>>Here is a tentative suggestion. First of all, given his aversion to the actual infinite, the thought that a continuum could be 'composed' of countably many, let alone uncountably many, discrete points is simply beyond him. We really do have to wait for Cantor and Dedekind before we can see how to make sense of this.
<<
Argh. There seems to be another argument lurking underneath this. The argument is that a point has no length, no dimension, so however many of them we add together we can only get another point, even with infinitely many of them. But, lo, along comes Cantor to say that there are more than a countably infinite number of points, which explains the 'missing space' as it were.
Is that the argument? I can remember hearing something similar years ago.
>> Is that the argument? <<
Not one I'd make. Note that the Cantor set has uncountably many points but measure (length) zero. So it's not a question of 'adding together' points (compare with weighing up particles of zero mass), it's where they are that counts.
The word 'composed' doesn't help. One invariably falls back on intuitions of material stuff. The archetypical continuum, empty space, is composed of nothing at all. It's better to think in terms of places. Cantor's diagonal argument shows there are uncountably many places between any two distinct places on a line segment. And adding together places makes no sense.
"(1) is a material implication and to show it is false it is enough to show that the antecedent can be true and the consequent false."
Well, can we at least agree that, for the statement "a continuum is composed solely of points", it either is true or is false, and there's no "can be" other than what "is"?
"Thus it can be true that a continuum is composed solely of points..."
You have not shown that. You have merely implicitly assumed that.
"This has nothing to do with 'denying the consequent'"
It has nothing to do with the fallacy of denying the consequent. But in your argument, you certainly have denied the consequent.
In terms of fallacies, I'd say you're more guilty of begging the question.
"You are confusing two entirely separate parts of the argument although I am at a loss to see how you could have confused them at all."
Recheck that premise, then. To show (1) is flawed you must show that the consequent is false (i.e. you must deny the consequent) and that the antecedent is true (i.e. you must affirm the antecedent). You have done the former, but not the latter.
It is as though I said "if I had ate a cheeseburger, then I wouldn't be hungry", and you responded with "wrong - you're hungry".
"I conclude that it is (1), by showing that the antecedent can be true but the consequent false."
You have not shown that the antecedent can be true. That's my point.
"First of all, given his aversion to the actual infinite, the thought that a continuum could be 'composed' of countably many, let alone uncountably many, discrete points is simply beyond him."
Yes, and I think we have to keep in mind that he is talking here about nature. He is talking about physics. He is talking about the real world. He is talking about, for example, intervals of time, or intervals of space.
He is most certainly not talking about theoretical mathematics of a type which does not bear a resemblance to the actual world.
Interestingly, in the actual physical world (which is the world Aristotle was talking about, and in fact the only world which exists), as we approach the very small, distances do become discrete.
"The archetypical continuum, empty space, is composed of nothing at all. It's better to think in terms of places."
That term also implies that one is talking about the physical world, albeit in a sense that has been disproven (there is no absolute frame of reference -> there are no "places", only "distances").
>>Showing what is wrong is nice, but showing that it is wrong is sufficient,
> I don't see the difference between these (emphasis on word 'showing').
A thinks he has an argument that demonstrates that "the continuum" (whatever that might be) or "a line" (whatever that might be) cannot be composed of indivisibles.
We have a model of something (the real line) that we believe corresponds to his idea of "continuum". We know the real line is composed only of points (by definition). Thus we know his argument must be wrong, without inquiring into the details of exactly where it is wrong.
But (having pondered this over the weekend) the problem, really, is even worse than this. Once we know that we're talking about the real line, then we know its made of points, because that is how we made it. How could it be otherwise? Simply transferring the argument from "the continuum" to discussing numbers makes it obvious (errm, in the mathematical sense :-) that his argument must be wrong.
The only way you can say, with a straight face, "the continuum cannot be composed of points" is to make sure that you don't think of the continuum as the real line, and/or as numbers. You have to make sure you keep thinking of it geometrically, in some sense. Probably as line segments, as DB seems to imply.
As I said earlier: A has no definition of "continuum"; or if he has, it hasn't yet been quoted.
> Can anyone explain the three-way distinction that Aristotle is drawing here?
No; I agree what he is saying about those doesn't really make any sense.
I'm sure I posted something here, or intended to. Never mind, I think it was:
1. A never defines continuum - or if he does, you haven't posted it. Without a definition, you can't discuss its properties, if you're talking about something in the mathematical world. Indeed, i think it is unclear whether A is talking about something in the mathematical or real world.
2. Once you decide (as we have?) that you're talking about the mathematical world of the real line - then none of this discussion makes any sense. If you're in that world, you already know that the real line "is composed of" points, because you've defined it that way.
3. As to:
>> Showing what is wrong is nice, but showing that it is wrong is sufficient,
> I don't see the difference between these (emphasis on word 'showing').
The distinction seems obvious to me. Showing that a model of the continuum (the real line) is composed of points refutes A's argument that it cannot be so composed, but without pointing out exactly what is wrong with the argument. I hope you're not making any spurious reservations about relying on Cantor's authority or somesuch, since we aren't.
The reference was here: http://www.logicmuseum.com/authors/aristotle/physics/physics.htm#bk231a21
The word "continuum" only appears in the table of contents.
Before you criticize it, it would be best to read it, in its entirety.
All emphasis is mine.
a) Aristotle: "The 'continuous' is a subdivision of the contiguous: things are called continuous when the touching limits of each become one and the same and are, as the word implies, contained in each other: continuity is impossible if these extremities are two"
b) Aristotle: "A thing that is in succession and touches is 'contiguous'."
So, by definition (a), the continuous consists of things that are contiguous, which by definition (b) means that they touch and are in succession.
So what Ed labels as (1) is true by definition, if "a continuum" = "the continuous".
Aristotle: "nothing that is continuous can be composed 'of indivisibles'"
As has been noted, "continuum" is undefined. As is "touching" or "contiguous."
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