The Parallel Postulate, from “The Unreasonable Influence of Geometry”

Part of the article “The Unreasonable Influence of Geometry” by Doug Muder

Given a line and a point not on the line, how many lines parallel to the first line can be drawn through the point? If you draw the line and the point on a piece of paper, it seems obvious that the answer is one. Euclid certainly thought so; one of his postulates – those self-evident assumptions that Descartes and Spinoza were so impressed by – said more or less exactly that. This assumption became known as the Parallel Postulate.

As far back as Roman times, mathematicians had found something unsettling about the Parallel Postulate. No one doubted that it was true, but it was just not obvious enough. It seemed somehow less self-evident than the other postulates. Time and again, mathematicians proposed some other assumption from which the Parallel Postulate could be proved. Unfortunately, none of these efforts to rewrite Euclid were any more satisfying than Euclid’s original formulation. Each involved assuming some postulate that did not seem like a postulate. In time the following fantasy developed: what if the Parallel Postulate could be proved without making any additional assumptions?

With this in mind, a wide variety of mathematicians over several centuries attempted to prove the Parallel Postulate from Euclid’s other postulates. Typically these attempts took the form of a proof by contradiction. In other words, assume that there are two lines through the point, each of which is parallel to the original line; or assume that is no line parallel to the original line through the given point. If both of these assumptions led to contradictions, then the Parallel Postulate would be proved.

And so mathematicians found themselves working in these strange logical systems, in which there were either not enough or too many parallel lines. They did not believe in the truth of these systems – quite the opposite, their purpose in constructing their arguments was to reach contradiction and show that the system was untenable. But no matter how outlandish the propositions they proved seemed, these propositions stubbornly refused to contradict each other. Shortly after the beginning of the 19th century, it began to dawn on Gauss, one of the greatest mathematicians of all time, that there was no contradiction to be found in either system. He wrote to a colleague, “The path I have chosen does not lead at all to the goal we seek. … It seems rather to compel me to doubt the truth of geometry itself.”

Fearful of the uproar that his discovery would cause – perhaps recalling what the Pythagoreans did to Hippasus – Gauss decided not to publish it in his lifetime. But non-Euclidean geometry was rediscovered three more times in the next 20 years, and by 1832 the secret was out: There was more than one consistent geometry, and reason by itself could not choose among them. Ultimately, experiments showed that the large-scale structure of the universe is better described by a non-Euclidean geometry than by Euclidean geometry.

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