The Universe in Zero Words by Mackenzie Dana

Author:Mackenzie, Dana
Language: eng
Format: epub, pdf
Publisher: Princeton University Press
Published: 2013-06-25T16:00:00+00:00

15

the geometry of whales and ants non-euclidean geometry

At the same time that a revolution was going on in algebra, similar events were taking place in geometry. Two millennia earlier, Euclid had written down a short set of axioms from which, supposedly, all of geometry could be derived. These axioms were intended to be self-evident truths that did not require any proof.

For centuries Euclid’s Geometry was considered the ne plus ultra of deductive reasoning. The eighteenth-century philosopher Immanuel Kant built up a theory of knowledge, in which he cited Euclid’s geometry as an example of “synthetic a priori” truth—in other words, infallible knowledge about the universe that is derived from pure reason rather than observation.

However, one axiom had always appeared a little bit clumsier than the others. The axiom in question is the “Parallel Postulate,” which Euclid does not use until late in his first book: “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 produced indefinitely, meet on that side on which are the angles less than two right angles.” This assumption is used, for example, to prove that the sum of the angles of a triangle equals 180 degrees.

Many mathematicians felt the Parallel Postulate was true but far from self-evident, and thus a flaw in Euclid’s otherwise sterling system of axioms. They took up the challenge of proving it from the other axioms that Euclid had provided. This mathematical grail quest lured the famous and obscure alike. Legendre (whom we have met already) believed that he had proved it. So, at one time or another, did less-famous mathematicians like John Wallis, John Playfair, Girolamo Saccheri, Johann Lambert, and Wolfgang Bolyai. In all cases, they made hidden assumptions that, under the harsh light of scrutiny by other mathematicians, were no better motivated than Euclid’s postulate.

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