[Aethaeryn]

  > A zero-length line segment is a point.

The Wikipedia definition of a line segment says that it is bound by two endpoints.[1] It provides a reference to Planet Math that goes into specifics.[2] In this page, it is made clear that the two endpoints cannot be equal. Planet Math provides an equation for a closed[3] line segment:

  L = {a + tb | t in [0, 1]}

This means that a line segment can be expressed as all of the points a + tb, where t is the range [0, 1] (which contains 0, 1 and all the points between them). It also limits a, b as real or complex numbers with b != 0. In other words, any line segment is just "morphing" the basic 0 to 1 range, with a shifting it and b scaling it.

Now, since b can't be 0, and 0 != 1, you're not going to get the endpoints of the range 0 and 1 to equal each other no matter how you scale them with b or shift them with a.[4] In other words, line segments will always have length.

Because you cannot get the endpoints to equal each other, a point cannot be thought of as a line segment under what appears to be the common definition.

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[1] https://en.wikipedia.org/wiki/Line_segment

[2] http://planetmath.org/encyclopedia/LineSegment.html

[3] Open is the exact same thing, except with (0, 1) instead of [0, 1] and it doesn't include the points 0 and 1.

[4] Of course, if b = 0 was allowed, you could just say that points are where b = 0 for all a.

[lmarinho]

I'd also refer to Hilbert's Axioms[1], which are the basis for the modern treatment of Euclidean Geometry.

[1] http://www.gutenberg.org/files/17384/17384-pdf.pdf

[Sniffnoy]

No, the basis for the modern treatment of Euclidean geometry is the explicit construction of the plane as R^2. Axiomatization is reserved for things like set theory and elementary number theory. Although Hilbert's axioms aren't exactly even a proper axiomatization anyway, seeing as it's a second-order axiomatization, and thus requires some ambient set theory. But if you've got set theory, you may as well construct it. You could think of the axioms as just conditions, of course, but then you need an existence proof, which is provided by the explicit construction as R^2, so...

[lmarinho]

Good point, my statement was poorly formulated. While I agree that analytic geometry and other developments in the field could be more fundamental, Hilbert's formalization can still be useful to express more precisely what one means by "points" and "lines", being, I think, applicable to the current discussion.

[Sniffnoy]

On the contrary, it's an axiomatization, and thus only describes the relation between them. If by "what one means by them" you mean a definition, then you'll need a construction for that. Of course, for the most part, the relation between them is what we mean by them; but Hilbert's formalism doesn't seem to be a good way to address such statements as the one that started this, "a point is a line segment of zero length".

[colanderman]

Ah good I was looking for the restriction b ≠ 0 but couldn't find in on the Wikipedia page. Forgot to check out PlanetMath, thanks.

[Aethaeryn]

I'm actually surprised by the lack of rigor in many of the sources I checked. It's obvious that b != 0 is necessary because otherwise you can't get results like line segments having an infinite number of points, and hence equal to the amount of points in a line.[1] The thing is, a lot of common references seem to leave out that the endpoints can't be equal.[2] It's so obvious that it's just implied, but it's dangerous to treat mathematics like that!

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[1] http://mathworld.wolfram.com/LineSegment.html

[2] Just saying 'two distinct endpoints' instead of 'two endpoints' would work, so it's literally just one word that makes a big difference!