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:
- 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.
- What was the intended utility of Euclid's definitions of lines and points?
Related: History of Euclidean and Non-Euclidean Geometry
Best Answer
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.