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. Dheeraj Kumar has given a link to a book Cosmic Sphere which speaks about the same complexity. I don’t think (as of now) the book really solves the problem. But it gave some very good points like line can’t even have thickness, as points are zero dimensional, etc.
“Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World” by Amir Alexander is the wonderful book on the matter of the question.
The section one (Physical meaning of geometrical propositions) of part one of the book “Relativity: The Special and General Theory” gives the Einsteins view on this matter.
First few chapters of the book “Euclidean and Non-Euclidean Geometries-Development and History” by Marvin J Greenberg might be of some taste to few.
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.
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.
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.
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.