Mathematicians Reconsider Euclid's Parallel Postulate
Overview
Ever since the time of Euclid, mathematicians have felt that Euclid's fifth postulate, which lets only one straight line be drawn through a given point parallel to a given line, was a somewhat unnatural addition to the other, more intuitively appealing, postulates. Eighteenth-century mathematicians attempted to remove the problem either by deriving the postulate from the others, thus making it a theorem, or by replacing it with a simpler statement. Nineteenth-century mathematicians would change the postulate to generate logically consistent non-Euclidean geometries, which twentieth-century physicists would in turn propose as the true geometry of space and time.
Background
In his Elements of Geometry, the great Greek mathematician Euclid (335-270 B.C.) was forced to adopt a rather awkwardly worded fifth and final postulate:
If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which the angles are less than the two right angles.
It is clear that Euclid was not entirely comfortable with this postulate, as he postponed its use in proving theorems as long as he could. Over the centuries mathematicians speculated that the postulate could actually be proved as a theorem from the other axioms and postulates or, failing that, that it could be reformulated in a far simpler way as befitted a self-evident truth about the nature of space.
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