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I have extracted the below passage from the wikipedia webpage - Point (geometry):

In particular, the geometric points do not have any length, area, volume, or any other dimensional attribute.

I think the above passage imply\ies that the point is zero dimensional. If it is zero dimensional, how can it form a one dimensional line?

Physics texts sometimes talk of lines' being made up of points, planes' being made up of lines and so forth. Clearly a line segment, thought of as a connected interval of the real numbers, cannot be built as a countable union of points. What axiom systems define the building up of a line from points, or, how do we rigorously define the building of a line from points?

Links:

  1. The section one (Physical meaning of geometrical propositions) of part one of the book "Relativity: The Special and General Theory" seems to be giving Einsteins view on this matter.
  2. What was the intended utility of Euclid's definitions of lines and points?

Related: History of Euclidean and Non-Euclidean Geometry

Sensebe
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  • I would had posted this question in MSE, but the topic might get better discussions when studied along with physics ideas (physical existence, etc). – Sensebe Dec 27 '14 at 11:21
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    It might help if you told us why, exactly, you find this counterintuitive. Otherwise, we'd just have to guess. (Also, could you please clarify what specific kind of process you're thinking of, when you speak of a point "forming a line"? My initial assumption would be that you're talking about how a point, moving continuously along a line, traces out that line -- but surely that should not be so hard to accept.) – Ilmari Karonen Dec 27 '14 at 11:29
  • @llmariKoronen: Thank you for the comment. If point is zero dimensional, how much ever join them to form line, you get zero dimensions. I hope this is clear about what I am asking, if anything wrong, I will be happy to know. – Sensebe Dec 27 '14 at 11:33
  • I think @IlmariKaronen is trying to ask you for more context: where have you read about a point forming a line, can you describe the process? If so, then we have a better chance of helping you. – Selene Routley Dec 27 '14 at 12:07
  • @WetSavannaAnimalakaRodVance: Thank you for the involvement in the discussion. I have not read anywhere about how this zero dimensional point can form a finite dimensional line. I thought, the point is the building block which should form the line. It is the only participant in forming line, as it is zero dimensional, I have no idea how it forms finite line, except the method of adding them, which fails. – Sensebe Dec 27 '14 at 12:21
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    @Godparticle Well finite or countable sets of points can't be added to form a line segment, as you say. But unions over sets of cardinality of the reals can: $[0,,1]=\bigcup\limits_{x\in[0,,1]} {x}$. I know this seems a little circular, but I think you are striking your problem because you can't visualise the line as a countable union of points. This is a truly deep concept, and you now are leaping head first into the definition of the real numbers, which is done axiomatically quite differently from how you are trying to precisely because we can't think of them as being a countable union. – Selene Routley Dec 27 '14 at 12:38
  • @WetSavannaAnimalakaRodVance: "I know this seems a little circular,"- I feel the same (by replacing the word little with completely, I will be happy to know, if it is not). I want to know, say truth, are you able to visualise line as countable union of zero dimensional points? – Sensebe Dec 27 '14 at 12:48
  • @WetSavannaAnimalakaRodVance: "which is done axiomatically", Lets make a start, I will start searching those axioms. Please post any of the source of axioms you know. Thank you. – Sensebe Dec 27 '14 at 12:49
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    @Godparticle This book The cosmic sphere may help. The very first topic of Chapter 1 contains information about relation between points and lines. – Dheeraj Kumar Dec 27 '14 at 12:53
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    Begin with the "Real Number" Wiki page and the "Construction of the Real Numbers" wiki page. Also see Tarski's Axioms. The essential thing here is the notion of either metric completion, if you conceive of the reals as being equivalence classes of Cauchy sequences of rationals - essentially Weierstrass's approach - or the notion of being Dedekind complete (Dedekind's approach).... – Selene Routley Dec 27 '14 at 13:04
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    ...Both of these lead to the notion of uncountability, i.e. "bigger" infinities than the cardinality $\aleph_0$ of the integers / rationals, which I'm guessing is one of the sources of your unease. IMO the real numbers are one of the most complicated and mind-bending notions in all of science, so it's quite normal to feel a bit freaked out by their odder properties (such as, aside from a countable subset, most real numbers are undefinable!) - if you think this is easy, you're not thinking deeply enough! – Selene Routley Dec 27 '14 at 13:08
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    I should also say the Weierstrass approach - thinking of sequences of the rationals, is the easiest approach and the one to begin with. But it is also the more complicated and messy by quite a long way, so the subtler and harder to grasp, but cleaner and simpler Dedekind notion is something you should read up on later. – Selene Routley Dec 27 '14 at 13:15
  • @WetSavannaAnimal: I tried to explain to OP some time ago the concept that uncountable unions produce higher dimensions (kinda tautologically). The problem here is that Godparticle applies their intuitive ideas about "point" and "size" to the formal mathematical concepts of dimension, point and so on. – ACuriousMind Dec 27 '14 at 13:39
  • @ACuriousMind: I mentioned about our discussion in the question, but Daniel deleted it (I am not against him). Even now I look forward for your answer. – Sensebe Dec 27 '14 at 13:49
  • Dear GodParticle: I actually voted on this before our conversation and it genuinely was unclear to me what you were asking: I thought you were thinking of a process. I am definitely happy to continue this conversation, and I have flagged your question for migration to MathsSE: from what I can tell, your questions seem to be about mathematical foundations and they are deep, so MathsSE is definitely the better forum. Also, an excellent book to read here is John Stillwell's "Roads to Infinity"; he also has a new book on real analysis: I haven't read it, but I'd almost bet my last dollar ... – Selene Routley Dec 27 '14 at 22:24
  • ... that it would be worth reading (John Stillwell was my teacher a long time ago and has many books - he is a most excellent technical writer). Can you see the flag option below your question. If so, and if you want your question moved, ask a moderator to do so. I have already done so, but your flag (as the OP) would bear more weight than mine. I highly recommend migration and think you would get some better feedback there. – Selene Routley Dec 27 '14 at 22:27
  • Could I also suggest your rewording the question to something like: "Physics texts sometimes talk of lines' being made up of points, planes' being made up of lines and so forth. Clearly a line segment, thought of as a connected interval of the real numbers, cannot be built as a countable union of points. What axiom systems define the building up of a line from points, or, how do we rigorously define the building of a line from points?" – Selene Routley Dec 28 '14 at 00:08
  • @WetSavannaAnimalakaRodVance: Thank you for the time and useful comments on books. I have edited the question and even flagged it for migration. You can make modifications in the question. – Sensebe Dec 28 '14 at 00:43
  • Well, lines are one dimensional, how can they form planes? And so on. Hydrogen doesn't create energy, but when there's a big bunch of it in a small space, then a star forms. Almost the same, plus the fact that there is a *very big* bunch of points in a line. – egreg Dec 28 '14 at 20:39
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    This question already occupied the ancient Greeks, for example Aristotle: http://global.britannica.com/EBchecked/topic/34560/Aristotle/33174/The-continuum – Hans Lundmark Mar 03 '15 at 14:19
  • See this Numberphile video. – user26486 Mar 03 '15 at 15:32
  • Isn't this question pointing out the difference between countable and non-countable sets? – Broken_Window Mar 03 '15 at 17:52
  • See the difference between a mathematician and an applied mathematician. http://www.angelo.edu/faculty/kboudrea/cheap/cheap4_engineers.htm Read the paragraph that ends in: "If I can't solve the problem, I change it!" – dtmland Mar 03 '15 at 19:23

7 Answers7

16

It's a good question. Here's one approach that is broadly consistent with modern measure theory:

Start with a line segment of length $1$. If we halve its length $n$ times, then the resulting line segment has length of $1/2^n$ and that is always greater than the length of a point in the line. Write $L(point)$ for that quantity, $L$ for Length.

Then whatever $L(point)$ is (and assuming it is defined), we have

$$0 \leq L(point) < \frac{1}{2^n}$$

As $n$ is arbitrary, we can make $1/2^n$ as small as we like. The only viable conclusion is that $L(point) = 0$.

Building up the other way from the point to a line segment is problematic. How can we multiply zero by anything and get something greater than zero? We can't without throwing out the real numbers as we understand them. That is too high a price. This is why the argument starts with non-zero quantities and goes to down zero.

Simon S
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    The same argument shows that the Rationals form a line. This is a bit like the old question of whether 0.999.. = 1. You can't really argue about the properties of Reals unless you know how the Reals are constructed, and if you knew that, you wouldn't be asking the question. – Peter Webb Mar 03 '15 at 13:31
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    Your answer makes pretty good job at explaining why a point must have zero length, but it's not what the question asked. It's still unclear how we get non-zero length whole from zero length parts. – KarmaPeasant Sep 27 '18 at 15:37
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The trick is that there's more to a line than just being made up of points -- the line is also known to live in some sort of topological space or some richer structure. e.g. the axioms of Euclidean geometry talk not just of points lying on lines, but that one point on a line may be between others, that line segments might be congruent, and other stuff.

This other stuff is important to the "lineness" of a line.

Within the context of a topological space, one can give a complete description of any shape in that space by specifying which points are in the shape. Thus, the habit of describing shapes in terms of sets of points.

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It depends on your definition of "line" and "point" as Hurkyl mentioned. In pure Euclidean geometry with only the geometric axioms you can't talk about dimension at all. If you add the Cantor-Dedekind axiom, then Euclidean geometry can be embedded in $\mathbb{R}^3$, and then you can talk about dimension, which is simply the size of the basis for $\mathbb{R}^3$ as a vector space over $\mathbb{R}$. There is then no problem with a line being 1-dimensional while a point being 0-dimensional. It just follows from definition, and also corresponds to the intuition. There are 0 degrees of freedom in a point, which says that you cannot move in any direction from any point in it while remaining in it. There is 1 degree of freedom in a line, which can be represented by the distance you are from a particular point on it when measured along 1 vector. There are 2 degrees of freedom in a plane, which can be represented with a fixed point in it and two fixed vectors by 2 coordinates telling you how much you have to go along one vector and how much along the other to get from that fixed point to a point in the plane.

Note that in the universe both a point and a line are in the same 'space', and if this space is a usual Euclidean space, their dimensions have nothing to do with the dimension of the whole space in which they are. This may be the real issue behind your question. Note also that in $\mathbb{R}^n$ any point by itself is a vector space of dimension 0 over $\mathbb{R}$, regardless of $n$. Same for a line, which is of dimension 1 over $\mathbb{R}$. In general, isomorphic vector spaces have the same dimension regardless of what they are embedded in.

Now we know that the universe isn't Euclidean, but if we can continuously parametrize an object in the universe by $n$ real numbers we could define the dimension of that object over $\mathbb{R}$ to be $n$. Then the dimension of any object in the universe has nothing to do with anything except where its points are in the universe. In particular it has nothing to do with the dimension of any other object containing it, including the universe itself. So a point is 0-dimensional by definition. Any path is 1-dimensional, straight or not, would be 1-dimensional since it is parametrized by a single real parameter. Any surface like that of a smooth object would be 2-dimensional. Note that some objects won't have a dimension under this definition, such as fractals. There are various possible different definitions for fractional dimensions to deal with that but I won't go into it.

user21820
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The simplest answer I believe is that a line is not "made up" of points, in the sense that a chain is made of links. Rather, points are best thought of as positions, and positions have no size. It is possible to define a set of possible positions in such a way that one can derive from this set all the properties that a line would have, and thus the two are logically equivalent: we might as well say that this set of points is a line. But the construction of this set is a logical procedure and not a physical process; there is no starting with one point and then adding "the next one", because of course there is no next point: between any two points on line lie an infinity of other points. By the way, the idea of a unique physical position is somewhat problematic in physics. The uncertainty principle and the problem of "singularities" come into play here - another topic.

James Herndon
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I assume you mean a line segment, not a line.

A line segment is not a "set of points". Euclid defines a line segment as a length without width. In other words, a line segment is defined as its length, not as a set of points.

Tyler Durden
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I'm none too bright, but I like Euclid, and was of the opinion that points passing through themselves generate lines, lines through themselves areas, and areas through themselves volumes (in mathematical imagination land, where reality lives). Because of this I am starting to think the unit represents the magnitude of the dimensional shift itself from zero into whichever dimension in question, not the object it describes (as in a magnitude 1 dimensional shift into 3 dimensions arranged as a cube or a sphere or a kitten etc.). This would be why square units refer not to units bearing the properties of a square, but to those of which the second dimension is comprised, square referring here to exponent two. What I mean is that the unit's incommensurability with it's latent transformations reflects the artificial division of dimensions implicit in these sciences. What is divisible is composed of indivisibles, points are indivisible, and anything divisible is one of itself, so they speak of a point generating a line like 9s rolling in from the furthest decimal place. Alternatively, all of these objects are equally empty space delineated by mental projections of symmetry finding expression in the unit. That is, you have to apply a line of symmetry to the object you construct, they're not just floating through space. I apologize if I am way off base and misleading, I know my language is improper, just curious how incorrect my understanding is.

Agora345
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The short, pithy answer is: there are more points in a line then there are natural numbers. The mathematical term for this is "uncountably infinite", and the idea is that there are so many points in a line that we can't give them all a unique natural number label. Any attempt to do so would fall afoul of Cantor's diagonalization argument. This is what allows a zero dimensional object to make up a one dimensional object, a 1D one to make up a 2D one, and so on from dimension to dimension. So in math, you can get something from nothing, if you have enough nothing.

No Name
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