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I have researched this question for days and can not locate a good answer. It could be a mathematical object that is defined by an axiom as Euclid or Hilbert. But if a curve is drawn between two points can it be should using only the rules of plane geometry that the curve is a "straight line"? If the curve is not a straight line then does it follow that the theorems that rely on such a construction will not necessarily be valid? BTW I am speaking of Euclidean geometry only.

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As you state yourself, it really depends on the axioms of the theory you are working within. What do you mean by "the rules of plane geometry"? – Bruno Joyal Jul 17 '11 at 21:43
The modern definition is that a line is a certain kind of subset of Euclidean space: – Samuel Jul 18 '11 at 1:14
@clkirksey: Line for the ancient Greek geometers meant what we now mean by curve. (Well, not exactly, since they did not speak English.) This usage continued for a long time. What we call a line was called a straight line. Omitting the qualifier "straight" is relatively modern. Even angle was not necessarily between straight lines. – André Nicolas Jul 18 '11 at 2:24

Euclid didn't define "line" by an axiom; in one translation, he defined it as "breadthless length," which is the kind of definition that only helps you if you already know what a line is. Isaac has it right when he suggests that what's important about a line is what it does, not what it is. An alternative approach is to found Euclidean geometry on analytic geometry. Define the real number line, then the Cartesian coordinate system, then define a line to be the set of solutions of $ax+by+c=0$ for fixed $a,b,c$ with $a$ and $b$ not both zero.

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Euclid defines a straight line as a line (meaning curve) which lies evenly with the points on itself. There is no agreement about what this is intended to mean. Not a lot of disagreement either, just a shaking of heads. – André Nicolas Jul 18 '11 at 3:03
@Andre, yes, that's better than the quotation I gave. I think we agree that with the benefit of 2,000 years of hindsight we can say it's not much of a definition. – Gerry Myerson Jul 18 '11 at 3:07
Perhaps I should have said "I shall not today attempt further to define a line; and perhaps I could never succeed in intelligibly doing so. But I know it when I see it." like Justice Potter Stewart. – Isaac Jul 18 '11 at 3:18
So how would one construct a "Cartesian coordinate system" which implies orthogonality of "straight lines"? It just seems that there is potential for circularity in the definition. – clkirksey Jul 18 '11 at 17:17
@clkirksey, a Cartesian coordinate system is just a one-one correspondence between ordered pairs of real numbers and points in the plane. Two lines are orthogonal if their slopes multiply to $-1$, the slope of $ax+by+c=0$ being $-a/b$ (with the proviso that this is undefined if $b=0$, and that lines with $b=0$ are orthogonal to lines with $a=0$). "Circularity" only comes in with quadratics, along with hyperbole and ellipsis (just kidding). – Gerry Myerson Jul 19 '11 at 0:05

I am not entirely sure what you are asking, so this is likely a partial answer at best.

In many high-school-level texts in the U.S., line is one of several basic terms that are not formally defined. For example, in UCSMP Geometry, 3rd ed. (© 2009 Wright Group/McGraw-Hill), the first lesson in the text has:

Description of a line
A line is a set of points extending in both directions containing the shortest path between any two points on it.

Shortly thereafter, the Point-Line-Plane Postulate states:

a. Unique Line Assumption
Through any two points there is exactly one line. If the two points are in a plane, the line containing them is in the plane.

b. Number Line Assumption
Every line is a set of points that can be put into a one-to-one correspondence with the real numbers, with any point on it corresponding to 0 and any other point corresponding to 1.

c. Dimension Assumption
(1) There are at least two point in space.
(2) Given a line in a plane, there is at least one point in the plane that is not on the line.
(3) Given a plane in space, there is at least one point in space that is not in the plane.

Basically, a line is described by its properties but not formally defined.

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Many (US) texts leave "point, line, and plane" as undefined terms. – The Chaz 2.0 Jul 17 '11 at 22:40

When I was in Grammar School, our mathematics teacher defined a straight line as: "The shortest connection between two points which, when extended to infinity in either direction can never intersect with itself." This definition has always worked for me.

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I like the idea that lines satisfy the triangle inequality. – Juan Sebastian Lozano Dec 16 '15 at 9:16

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