If point is zero-dimensional, how can it form a finite one dimensional line?

Question

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?

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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

Answer 1

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 $L(point) = 0$.

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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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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.

Answer 7

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.