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.]
wmconnolley: scienceblogs.com/stoat archive 
Test
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[Oh *f*ck* mt’s Captcha system. I’ve had to turn it off again; apologies, and thanks to those who told me it wasn’t working.]
I think it hinges on his definition of “indivisible”.
To build a real line from points, the definition of almost all of those points has to be infinitely long,
[I don’t know what “has to be infinitely long” means to you -W]
and there has to be more than a countably infinite number of them. I don’t think Aristotle would have recognised that as “indivisible”. After all, if the definition of a point goes on for ever,
[Again, I don’t understand you. The only thing I can think of is that you’re thinking of the decimal representation of a number (or the Dedekind cut?) as the number. But it isn’t; the number is a point; the representation in a decimal system is just a representation. And of course A wasn’t thinking of a number as either of them. But perhaps you mean something else -W]
that means that to make that point you have to keep dividing the line for ever – since you never stop dividing, then the elements are not indivisible.
Seen in this light, then his definition of “indivisible” is something akin to the rationals, and his argument is entirely correct: the real line is not composed of rationals. It’s possible that he was one step further down the rabbit hole and knew about irrational numbers – but even that doesn’t help: Cantor’s diagonal argument shows that.
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What distinguishes the density of rationals from the density of reals is that for both if you have two non-intersecting finite sets of numbers where every member of one set is larger than the every other then there is rational number between the sets; while if you have two non-intersecting infinite sets, then a real number is always between
them or on the boundary.
For example Ïâ3 sits on the boundary of sets {x â [0,1] ⣠sin(3 + x) > 0 } and {x â [0,1] ⣠sin(3 + x) ⤠0 } even though there is now a proof that no rational number can exist “between” them. (Or you can make the same argument with open sets by substituting (0,1) for [0,1] and < for
â¤.)
[This is effectively the reals defined by Dedekind cuts, I think -W]
A similar geometric distinction is the Axiom of Pasch has no
qualifications on the triangle or the points on its sides. Lines that intersect intersect in points, not between them. Tarski's axiom of continuity uses a definition of between that blurs the line between > and ⤠in a way that might be useful to address those that have other continuum problems like denying 1 = 0.999…
Metaphorically, the points spring into existence faster than you can name them, which is the poetic way of talking about an uncountable infinity.
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You have the right idea: Aristotle’s continuum was not the same as ours – his conception of a line was more than the collection of constituent points. I think that David Brightly, at the original discussion you link to, had a right idea too, which you echo by mentioning the metric: the closest thing modern maths has to Aristotle’s continuum is the standard _topological space_ on a line, which rests on open sets, not on points. If you don’t have the topology, you don’t have a continuum, you just have a set of points (on which you can impose many other topologies – such as the trivial or discrete topologies – which are _not_ the continuum). (I guess Aristotle had a metric too, so you probably need both the standard metric and standard topological spaces).
In other words, you can cast Aristotle’s argument into modern terms in such a way that he was right, but he didn’t have the necessary apparatus to do that, to the satisfaction of a modern audience. He was right, at least in part, for the wrong reasons (which of course, for a proof, amounts to being wrong), because he didn’t have the necessary notions of infinity.
[I think you need to expand on that a bit, as you’ve skipped over some stuff. “his conception of a line was more than the collection of constituent points” – I guess it must have been, since he believed he had proved it wasn’t. But was the “more than”? It wasn’t the topology – that sits “on top”, I think. You say “you can cast Aristotle’s argument into modern terms in such a way that he was right” – would you try to do that? It might clarify what you mean -W]
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(taking it as read that I am making this up as I go along…):
The key property of a continuum, to either Aristotle or modern mathematics, is continuity. A set of points has no continuity. Modern mathematics understands continuity as a topological property (and the continuity of a line as resulting from the topology induced by the standard metric). Of course, Aristotle understood continuity – and number – somewhat differently.
This looks good, at a first skim:
Click to access AristotleContinuum.pdf
[That may be what I’m looking for but will take a while to process.
More: that appears to contain A’s defn of a continuum: “Since a continuum is âthat which is divisible into parts which are always further divisibleâ (Phys. 232b25),”. Of course, this is wrong. The rationals satisfy it, and I doubt he means the rationals. So does the union of disconnected line segments, and I doubt he meant that either. But again, if you take his defn literally, then you can see at once that the continuum can’t be composed of points, because points are indivisible. Depending on what he means by divisible, of course -W]
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One aspect of the different bases for reasoning about these questions is this: when Aristotle writes about composition, he seems to mean something akin to set-theoretic union. Thus: a line segment may be composed of (be the union of) two line segments. He does _not_ mean something like set membership: even if we consider a line segment to be the _set_ of points on the line, it is not the _union_ of those points – that would be a category error.
(again, I’m just making this up as I go along).
[That also seems to be somewhat of what he is saying. But in that case, too, his proof makes no sense: if he has defined a line to be something other than the points composing it, then he doesn’t need a proof of the same. If the problem is a category error, he should say so -W]
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(speaking from utter incomprehension, but I did like geometry)
> the number is a point
Was Aristotle assuming Platonic solids make up the universe? If so their lines and planes, um, exist.
We see quantum foam uncertainty. He saw geometric forms, line/angle/vector, where we (maybe) see points?
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I’m not sure that this isn’t all argument by semantics, so I hesitate to get involved. But let me attempt an argument like Aristotle in more modern terms.
Look at an interval on the real number line, say, from 0 to 1. This interval has a length. Points have no length. Therefore, no set of points can have a length. Since the interval has a length, it can’t be made only of points.
[This is definitely one of his arguments: that a sum of zero magnitudes cannot ever amount to anything other than zero. It fails because he doesn’t know how to do infinite sums, and doesn’t know that he doesn’t know (erm, slightly abusing language) -W]
In response to your claim that the real line is made up of points and therefore Aristotle is wrong, I think he would argue that you’re begging the question. The whole goal is to prove that the real line is not made only of points, so saying that everyone knows the real line is made of points is not much of a counter argument.
[You’ve missed the point: when I said “real line” I meant the real-line-as-defined-by-maths (I did say that a few times). We *know* what the “real line” is made of, because we *defined* it -W]
His conclusion is probably that the real line is made of intervals. Every interval can be subdivided into smaller intervals, but it can’t be subdivided into individual points.
[That is more his starting point that his conclusion, but otherwise I agree -W]
Aristotle is wrong, mostly because we use a different definition of continuity than he does. His definition doesn’t work, but one of the reasons we say that is because we want to be able to define the real line as a continuous set of points.
[I somewhat disagree. I don’t think he actually has a definition of continuity, and doesn’t realise it -W]
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So on one level, since matter is not infinitely divisible we have a practical solution (the line is made of a countable number of atoms, etc.)
[That is one way of looking at it. Though atoms themselves are divisible, and their components are not point objects -W]
On another level this navel gazing is damn close to some of Zeno’s paradoxes.
[Philistine -W]
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A linear continuum as defined as an ordered set that lacks both doubly open and doubly closed gaps can be considered to consist solely of points and half open gaps.
[Don’t really understand what you mean. Are you thinking of generalised continuum, or a subset of the reals? And you can’t define a set of consisting of things and gaps; the gaps, obviously, aren’t part of the set -W]
Though continuous it may contain such gaps everywhere except at its points. Into each such gap can be inserted a continuous half open interval of the correct orientation and other such intervals removed and it will remain continuous but may differ as to whether it is connected.
The two required properties, every subset with an upper bound having a least upper bound and every pair of points having an intermediate point preclude only doubly closed and doubly open gaps.
The only points it contains are indeed points and the only gaps are half open gaps. In that sense to say that a continuous but not necessarily connected subset of the reals consists solely of numbers is to say only that everything that is not a half open gap is a point and all the points are reals.
In A’s case the continuum is more generic than the reals for there is no requred metric.
[Wrong, I think. In A’s world, the existence of a metric is so basic that he hasn’t realised it is there; nonetheless, his ideas of a continuum are obviously based on lines and the idea of measure is implicit -W]
His problem may be statement (2) “The points are not continuous” in that he has defined continuous to be points that share an extremity and goes on to say that only intervals not points (which are extremities) have extremities so they cannot share that which they do not possess. However he might have seen that even in his terms, an inifinite monotonic series of points might share share its extremity with a point (e.g. if that series is bounded by the point as its closest bound). That does not contradict either statement (3) or (4) but it is not sufficient as does not provide completion.
[He is unable to see any such thing, because he won’t accept infinity -W]
Alternatively he is correct in the narrow sense that it is not possible to construct the continuum from points, if by construction one means by some procedure of accumulation. He states the opposite but equivalent impossibility of deconstructing a continuum into points. A continuum, no matter how minute, has to be constructed whole in one action. Specifically it is the completion of a set by the addition of the limit points that must be done at a stroke as they are uncountable lest the set already contain continuous intervals. He realises that to construct a continuum in a stepwise fashion requires not just points but continuous elements that are divisible and hence not points.
Put that way a real connected interval may consist solely of points but is not so much constructed as commanded into existence. It may consist of points but cannot be composed from points in the common meaning of that verb.
He might think this to be very unsatisfactory and well he might. One might pointwise reduce a real continuum to the degree wherein no remaining residue is continuous, but that residue cannot be pointwise reduced further in the sense that its measure be reduced below its initial value. Such requires the removal of almost every point and to do so without any systematic means of achievement. He got some things very right, e.g. that there can be no sucession of points in the continuum and that there is no way to compose the continuum from its points, and he consistered points and continuous intervals to differ in quality which is also correct. We might casually shrug off the reals as the completion of the rationals by the addition of all limit points and when asked how it is to be done we say it is best done instantly.
Alex
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William,
Thank you for your response. I included a preamble as I always suspect some fundamental disagreement that will colour the rest unintelligable.
“Don’t really understand what you mean. Are you thinking of generalised continuum, or a subset of the reals?”
A generalised continuum. I stated it in terms of gaps but it is the same as the two requirements of the definition given two paragraphs on.
“And you can’t define a set of consisting of things and gaps; the gaps, obviously, aren’t part of the set”
It is not obvious at all. The linear continuum is an ordered set that lacks certain classes of gaps. Gaps that are consequential on the arrangement of, or patterns in, the points. The problem is to construct an arrangement of points that excludes such gaps.
[I think you’re either wrong, or aren’t defining your terms carefully enough. the reals have no gaps. The integers, considered as themselves, have no gaps. In order to talk properly about “gaps”, you have to define them in terms of the set itself, not in terms of an enclosing set -W]
The gaps arise as a consequence of the ordering property on the set.
[No. Or at least, incomplete. Consider the set [0,1) U [2,3]. This would normally be considered to have a “gap”. But the *ordering* of the set is exactly the same as that of [0,1) U [1,2] which has no gap -W]
You may consider whether an ordered set is more than the sum of its tangible members. That it consists of points and their determined gaps. Without acknolwedging the gaps the set would appear to be trivially continuous so I think we must recognise the gaps when considering the continuity of the set.
A finite set of ordered points is a succession thus having a class of gaps that renders it discontinuous between every point. Any arrangement of points isomorphic to the rationals has no successor points and that class of gap is excluded yet it determines an infinity of gaps of the class where bounded subsets have no closest bound. To go further requires a jump in the cardinality which unlike that which removed successors precludes a stepwise approach as you surely know.
I find the he notion of the continuum to be deep and have already alluded to some of the issues such as construction by stating a truth about a set which differs from mundane construction by infinite composition due to the lack of process. The nature of the gap in cardinality from the arrangement of points in the rationals to that of the reals cannot be decided upon so I think we already travelled far. The simple act of imposing an ordering scheme on a set leads quickly to the beguilling.
Alex
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And this is why I think you’re arguing by semantics. Yes, the modern math definition of the real line is that it’s a set of points. Aristotle is making an argument for rejecting that definition.
[I don’t think this is semantics. I used the example of the real line, because at least then we know what we’re talking about, and it provides a useful example of somewhere that we *know* A’s argument fails. A isn’t making “an argument for rejecting that definition” – he has never heard of that defn. It would never have occurred to him to define the line that way – its completely foreign to his thinking. A doesn’t really have a defn of continuum – or rather, he has at least one, but its wrong, in that it is also satisfied by the rationals, or by the Cantor set -W]
Also, Aristotle doesn’t understand infinity, but that by itself does not prove him wrong. The rational numbers is an infinite set which is not continuous and which has no length.
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Hi again,
“The integers, considered as themselves, have no gaps.”
That is the point, they do even when considered in isolation.
[No, they don’t, unless you’re using a rather odd defn of gap -W]
To consider otherwise would be to say that they are a continuum which they are not.
Between each successive integer there is no point, there is a gap.
[Again, no.]
This is definitive. The set is either continuous between two points or it isn’t. The relevant continuum requirement here is that for all points in the set there must be a point between the one that is less and the one that is greater.
[Ah, you *do* mean something different by “gap”. That was very confusing – you can’t expect people to understand you if you use words that don’t work. Also, your definition makes the rationals continuous -W]
No point may have a successor, the set of integers being a succession is nowhere continuous. This is not due to an embedding in a greater set nor to any metric on the set it is a property of the arrangement of points. You may decide upon a different word to describe the lack of a continuum between two points I have used gap it is short but you may insert “lack of a continuum” if you wish.
[I could try, I suppose. You said “The linear continuum is an ordered set that lacks certain classes of gaps”. That would translate, with your proposed wording, into “The linear continuum is an ordered set that lacks certain classes of lack of a continuum”. Which is slightly ungrammatical, and vacuousuly true -W]
“Consider the set [0,1) U [2,3]. This would normally be considered to have a “gap”. But the *ordering* of the set is exactly the same as that of [0,1) U [1,2] which has no gap”
The set [0,1) U [2,3) is continous by definition but not connected.
[Not quite. Its continuous, but not by defn. You might call it continuous by construction, perhaps, although even that would be odd -W]
Continuity does not depend on the absence of all gaps e.g. the half open [1,2). This is because it is determined by the arrangement of points, not by reference to some greater set [Ah, good, we agree there -W], not by the labelling of the points, only on the ordering of the points.
I will try some form of illustration.
It might be thought that the rationals are incomplete due to certain equations (e.g. surds) based on the number value of the points, e.g. SQRT(1/2) not being points of the set. That is so but following the removal of those value labels the set is still incomplete in just the same way. It is discontinuous in just the same way.
Consider the binaries:
(0,1)
(0,1/2,1)
(0,1/4,1/2,3/4,1)
(0,1/8,1/4,3/8,1/2,5/8,3/4,7/8,1)
Start at 0 and choose to the right 1/2
From 1/2 choose to the left 1/4
From 1/4 choose to the right 3/8, etc.
The series converges to a point not contained in the set i.e. 1/3. It is an “unbinary” point by analogy to “irrational” point.
Consider the Farey sequence
(0/1,1/1)
(0/1,1/2,1/1)
(0/1,1/3,1/2,2/3,1/1)
(0/1,1/4,1/3,2/5,1/2,3/5,2/3,3/4,1/1)
Start at 0/1 and choose to the right 1/2
From 1/2 choose to the left 1/3
From 1/3 choose to the right 2/5, etc.
The sequence converges to the golden mean again outside the set which is the rationals.
The two sets have the same arrangement of points with a different naming convention.
Rename the points as you may or remove the labels altogether the same sequences defined by a pattern of left and right turns will converge outside of the set. This is a property of the arrangement of points not their names, nor values, nor location in some greater set, nor any metric.
Stripped of those attributes we can ask only certain questions but importantly we can ask whether for every pair of distinct points of the set there is a point in between them and whether every (non-empty) subset with an upper bound has a least upper bound, as both are determined solely by the arrangement or ordering of the points. By this method we can determine whether the set is continuous.
You seem to be thinking of continuous as being more akin to connectedness than I think is warranted for the topic and is of a different character to what determines a continuum. I think the definition I am using is well recognised and I shall state it yet another way:
“The linear continuum is a dense ordered set that has the least upper bound or supremum property.”
That may not be where your intuition has lead you but it is one of several equivalent definitions commonly given.
Alex
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Hi again,
I don’t think you have understood how the continuum is defined. When you do, you may see that you are wrong about what you are saying.
[I think that, instead of being patronising, you’d do better to address the errors and vacuousness I’ve pointed out in what you’ve said -W]
To motivate I will describe what is meant by the continuum and then how this fits with a more formal definition.
[Wouldn’t you be better off just reading, and then pointing to, http://en.wikipedia.org/wiki/Linear_continuum ? -W]
The objective is to ensure that the set does not have certain classes of gaps.
These I will name according to what is missing from the continuum or needed to be added to the set to ensure a continuum.
Open gaps, the gaps between successive points or the set that need to be filled with an open isomorph of the continuum. These occur whenever there are successive points.
Closed gaps, the gaps that occur due to failing to meet the least upper bound property and need to be filled with either a point or an isomorph of the continuum that has been closed above and below.
Half open gaps are permitted, this is necessary in order to enable the continuum to be defined solely as a property of the ordered set. They are not detectable in terms of the set alone so they could not be excluded based on the set alone.
More formally this is stated as two requirements on the ordered set.
That there must be no succession, that is for any two points there must be a point greater than the lesser point and less than the greater point.
That every non-empty subset with an upper bound has a least upper bound.
The second requirement is why the rationals do not consitute a continuum.
It should be clear by this definition that any isomorph of the continuum remains an isomorph of the continuum after the removal of half open invervals. Neither requirement prohibits the removal of such intervals.
This may enable you to see where you are in error.
The real line has properties beyond what is required for the continuum. Thinking in terms of the real line seems likely to lead to mistaken reasoning about what the continuum is and isn’t.
You don’t seem to have grasped the essence of the continuum. It stems from the definition, apply the definition (given here or if you like look one up) and you should be able to distinquish between when a set has the continuum property (meets the defined criteria) and when it doesn’t.
Is there a definition of the continuum that you are working from or is this intuitive feel?
Alex
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Planck length if you wish. Same point, and yes, Eli is descended from a long line of Phil S. Steins
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I feel compelled to try one more time. I have two points.
1. Definition of “gaps”. From the Wikipedia page on linear continuum, which you linked to, a set is continuous if it has two properties.
a. Between any two points, there exists another point. (i.e. if a and b are points, then there must exist a point x such that a<x<b).
b. Every subset has a least upper bound.
The rationals are not continuous because they fail the least upper bound property. The integers are not continuous because they fail the between property.
[Yes, agreed. Of course 🙂 -W]
The way I, and I think Alex, would define "gap" is any place where a set is not continuous because it fails either property. The rational numbers have gaps at every irrational number, because of the least upper bound property. The integers have gaps between every pair of adjacent integers, because of the between property. How do you define "gaps"?
[Yes, this is what I'm complaining about. You can't define "gaps" as places where the set isn't, because, errm, those places aren't in the set so aren't a gap in the set. Its largely terminological -W]
2. Back to Aristotle, let me attempt to recast his definition in modern terms. According to Aristotle, two sets are continuous if
a. There does not exist a set between them.
b. The distance between the two sets is zero.
Now, this definition is not the same as our definition. (His first property is the opposite of ours.) So arguing that he's wrong because a continuous set according to our definition violates his definition seems to be missing the point.
[I'm arguing that his definition is wrong, because *as stated* it simply doesn't work. If you substitute a different definition then you may well end up with a different result -W]
However, his definition is a reasonable starting point for defining connected sets. Reading his definition as a definition of "connected" rather than "continuous", he argues that two sets, each consisting of a single point, cannot be connected. And he is correct. Indeed, no set constructed of a finite number of points, or even a countably infinite number of points, can be connected. And I can't blame him for not being familiar with uncountably infinite sets.
[But unfortunately he wasn't trying to define connected. I suspect he would have felt it too obvious to need definition -W]
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The following is cribbed from another blog post on the topic of Planck’s length.
I was going to fashion a response along the same lines but found that I hadn’t the time or initiative to better this one.
“Current established theories neither require nor propose a minimal length. However, tentative models proposed for quantum gravity sometimes assume some kind of discreteness or granularity of space-time at the Planck scale. Thus, the nature of the Plank length as the smallest measurable one should be considered as one of the hypotheses assumed by these models.
While there are arguments for the validity of this hypothesis, like all others, it must be subject to experimental verification.”-John MS
[Fair enough: we don’t know. But even if we did accept something like a Planck length being a minimum, it still wouldn’t imply quantisation of space. That may just be a limitation of present theories; but at present, the equations of QM are continuous, and require continuous wavefunctions -W]
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I wonder if Aristotle was thinking in terms of something like atomless gunk when he had the continuum in mind:
http://en.wikipedia.org/wiki/Gunk_%28mereology%29
From the link, “gunk” is such that it every part is divisible to more parts, and lacks atoms/points. Essentially, it appears to follow up on his idea that the continuum is “potentially divisible but not actually divided into infinite parts”. From my quick read on it, it may be an alternative conception of the continuum that Aristotle may opt for.
[Thanks for the link. I think the concept is right, for A’s thinking, but that page (I think, I haven’t checked the refs) makes a number of philosophical mistakes because of the same failures that A made (nature of inifinity, etc.) without his excuse of not having had the theory developed. The division of meaning between the real and the maths world needs to be explored, too -W]
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