Preface xiii developed among mathematicians who made a serious study of geometry that the fifth postulate was really a theorem masquerading as a postulate and that Euclid had simply failed to find the proper proof of it. So for about two thousand years, the study of geometry was largely dominated by attempts to find a proof of Euclid’s fifth postulate based only on the other four postulates. Many proofs were offered, but all were eventually found to be fatally flawed, usually because they implicitly assumed some other fact that was also not justified by the first four postulates and thus required a proof of its own. The second decisive turning point in mathematics history occurred in the first half of the nineteenth century, when an insight occurred simultaneously and independently to three different mathematicians (Nikolai Lobachevsky, Janos ´ Bolyai, and Carl Friedrich Gauss). There was a good reason nobody had succeeded in proving that the fifth postulate followed from the other four: such a proof is logically impossible! The insight that struck these three mathematicians was that it is possible to construct an entirely self-consistent kind of geometry, now called non-Euclidean geometry, in which the first four postulates are true but the fifth is false. This geometry did not seem to describe the physical world we live in, but the fact that it is logically just as consistent as Euclidean geometry forced mathematicians to undertake a radical reevaluation of the very nature of mathematical truth and the meaning of the axiomatic method. Once that insight had been achieved, things progressed rapidly. A new paradigm of the axiomatic method arose, in which the axioms were no longer thought of as statements of self-evident truths from which reliable conclusions could be drawn, but rather as arbitrary statements that were to be accepted as truths in a given mathematical context, so that the theorems following from them must be true if the axioms are true. Paradoxically, this act of unmooring the axiomatic method from any preconceived no- tions of absolute truth eventually allowed mathematics to achieve previously undreamed of levels of certainty. The reason for this paradox is not hard to discern: once mathematicians realized that axioms are more or less arbitrary assumptions rather than self-evident truths, it became clear that proofs based on the axioms can use only those facts that have been explicitly stated in the axioms or previously proved, and the steps of such proofs must be based on clear and incontrovertible principles of logic. Thus it is no longer permissible to base arguments on geometric or numerical intuition about how things behave in the “real world.” If it happens that the axioms are deemed to be a useful and accurate description of something, such as a scientific phenomenon or a class of mathematical objects, then every conclusion proved within that axiomatic system is exactly as certain and as accurate as the axioms themselves. For example, we can all agree that Euclid’s geometric postulates are extremely accurate descriptions of the geometry of a building on the scale at which humans experience it, so the conclusions of Euclidean geometry can be relied upon to describe that geometry as accurately as we might wish. On the other hand, it might turn out to be the case that the axioms of non-Euclidean geometry are much more accurate descriptions of the geometry of the universe as a whole (on a scale at which galaxies can be treated as uni- formly scattered dust), which would mean that the theorems of non-Euclidean geometry could be treated as highly accurate descriptions of the cosmos. If mathematics occupies a unique position among fields of human inquiry, then ge- ometry can be said to occupy a similarly unique position among subfields of mathematics. Geometry was the first subject to which Euclid applied his logical analysis, and it has been taught to students for more than two millennia as a model of logical thought. It was the

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