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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"?2011-07-17
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    The modern definition is that a line is a certain kind of subset of Euclidean space: http://en.wikipedia.org/wiki/Line_(geometry)#Euclidean_space2011-07-18
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    @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.2011-07-18

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