Talk:Line segment

The reinstated parenthetical about the real line is to prepare the next sentence, which talks about line segments of the real line--the very common and very basic concept of an interval. This is the categorical example of a line segment, Ed. In order to talk about line segments, you need to talk about finite length, and the only way you're going to do that is by assigning real number coordinates to your line. In other words, you have to identify your line with the real line to talk about length in any sensible way. Once you have made this identification, then the line segments correspond exactly to intervals. This definition does not confuse plane geometry with something else--it's exactly the definition you need for plane geometry. -Foxtrot 22:31, 20 August 2008 (EDT)

If this is an addition to the plane geometry I learned in high school, you could have provided a source. Our math and science articles (among other topics) are being systematically sabotaged by people who want our website to present incorrect or useless information. I don't know you, but your work so far has not inspired trust. From now on, the burden of proof is upon you to convince me that what you write on math or science is correct. --Ed Poor ^Talk 06:26, 21 August 2008 (EDT)

"I don't know you, but your work so far has not inspired trust."

I find this statement hard to fathom. I have inspired enough trust to have earned blocking, night editing and uploading rights from Andy last month. In my time with CP since January 2008 I have been one of the most prolific mathematics contributors on the site. I have also been one of a handful editors to remain very active during the course of the summer. That you have not noticed my presence, let alone the quality of my contributions, is hard to believe. But so be it, now you know who I am.

With regard to the mathematical content of the article, here are sources which support my point. Mathworld's definition of line segment is: A closed interval corresponding to a finite portion of an infinite line. The third word is "interval", a word which I asserted above was important but which you proceeded to remove. Examining Mathworld's definition of "interval", we find: An interval is a connected portion of the real line. So there is your connection to the real line. If you'd like another reliable source, PlanetMath's definition of line segment explicitly invokes the real numbers in making its definition. At heart, you need the real line to talk about finite length and connectedness (="unbroken" in the CP article).

In the CP article, you left two definitions of line segment. The first, "a finite, unbroken portion of a line", refers to finite length and connectedness, which I have already argued needs the real line to make sense. The second definition, "the set of all points that lie on the line between two given points", needs the concept of "betweenness"--what does it mean for a point to be between two other points on a line? This may sound like a dumb question, but if you are going to have solid definitions for mathematical concepts, it becomes a serious question. Hilbert (see the intro paragraph here) spent some time coming up with a rigorous definition of "between" to work for geometry (his Foundations of Geometry put the subject on sound footing). The notion of "betweenness" begets an ordering of the points of your line (even Merriam-Webster's dictionary agrees on that point), and then to talk about connectedness (="unbroken"), you once again have produced enough conditions on your definition to need to invoke the real line to properly define the term.

I cannot know for sure what definition you learned in plane geometry in your high school, but I can address two likely objections. Objection 1: the geometric concept of line does not need the real numbers. I agree -- lines and points are primitive geometric concepts that exist in both Euclidean (e.g. plane) and non-Euclidean geometry. That's why you can talk about lines in geometries like the Fano plane. However, you can't talk sensibly about things like "line segment" in such a geometry, because the concept you're seeking involves a finite length, "unbroken" portion of an infinite line. All three of these conditions - finite length, "unbrokenness" and infinitude (and density) of the line imply that the line looks like the real line. Even though "line" is not limited to real geometry, the concept of "line segment" really is.

Objection 2: In high school, a teacher may "define" a line segment by drawing a straight dash on the blackboard. However, a drawing is not a definition; it is at best an example. It's great for intuition, but the responsibility of an encyclopedia is to have proper definitions and furthermore, mathematics demands that concepts are not defined vaguely. The intuitive notion is good to include in the article, but the proper definition needs to be there too. The involvement of the real line is fundamental to the definition and furthermore uses only the high school concepts of real line and interval (following the guidelines for math articles).

Most importantly (this is the reason I'm writing such a lengthy reply), it is not an attempt to insert erroneous information, so please lay your suspicions of sabotage to rest. Regards. -Foxtrot 04:13, 28 August 2008 (EDT)

I'll accept Wolfram as a source. Let's work together to define the basic concepts of plane geometry (the "visual" sort of math). We can also work on computational math: arithmetic, algebra, trig, etc.

As for number theory, I'd prefer to postpone that until we have a something to theorize about. If you really like the real line, you can mention it in the real numbers article; but not in the first place a homeschooled teen looks when seeking a trustworthy definition of a line segment. --Ed Poor ^Talk 05:41, 29 August 2008 (EDT)