xii Preface Mathematics occupies a unique position among the fields of human intellectual in- quiry. It shares many of the characteristics of science: like scientists, mathematicians strive for precision, make assertions that are testable and refutable, and seek to discover universal laws. But there is at least one way in which mathematics is markedly different from most sciences, and indeed from virtually every other subject: in mathematics, it is possible to have a degree of certainty about the truth of an assertion that is usually impossible to attain in any other field. When mathematicians assert that the area of a unit circle is exactly , we know this is true as surely as we know the facts of our direct experience, such as the fact that I am currently sitting at a desk looking at a computer screen. Moreover, we know that we could calculate to far greater precision than anyone will ever be able to measure actual physical areas. Of course, this is not to claim that mathematical knowledge is absolute. No human knowledge is ever 100% certain—we all make mistakes, or we might be hallucinating or dreaming. But mathematical knowledge is as certain as anything in the human experience. By contrast, the assertions of science are always approximate and provisional: Newton’s law of universal gravitation was true enough to pass the experimental tests of his time, but it was eventually superseded by Einstein’s general theory of relativity. Einstein’s theory, in turn, has withstood most of the tests it has been subjected to, but physicists are always ready to admit that there might be a more accurate theory. How did we get here? What is it about mathematics that gives it such certainty and thus sets it apart from the natural sciences, social sciences, humanities, and arts? The answer is, above all, the concept of proof. Ever since ancient times, mathematicians have realized that it is often possible to demonstrate the truth of a mathematical statement by giving a logical argument that is so convincing and so universal that it leaves essentially no doubt. At first, these arguments, when they were found, were mostly ad hoc and depended on the reader’s willingness to accept other seemingly simpler truths: if we grant that such-and- such is true, then the Pythagorean theorem logically follows. As time progressed, more proofs were found and the arguments became more sophisticated. But the development of mathematical knowledge since ancient times has been much more than just amassing ever more convincing arguments for an ever larger list of mathematical facts. There have been two decisive turning points in the history of mathematics, which to- gether are primarily responsible for leading mathematics to the special position it occupies today. The first was the appearance, around 300 BCE, of Euclid’s Elements. In this mon- umental work, Euclid organized essentially all of the mathematical knowledge that had been developed in the Western world up to that time (mainly geometry and number theory) into a systematic logical structure. Only a few simple, seemingly self-evident facts (known as postulates or axioms) were stated without proof, and all other mathematical statements (known as propositions or theorems) were proved in a strict logical sequence, with the proofs of the simplest theorems based only on the postulates and with proofs of more com- plicated theorems based on theorems that had already been proved. This structure, known today as the axiomatic method, was so effective and so convincing that it became the model for justifying all further mathematical inquiry. But Euclid’s Eden was not without its snake. Beginning soon after Euclid’s time, mathematicians raised objections to one of Euclid’s geometric postulates (the famous fifth postulate, which we describe in Chapter 1) on the ground that it was too complicated to be considered truly self-evident in the way that his other postulates were. A consensus
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