Mathematics: The New Golden Age

Keith Devlin
Pages: 336
https://www.jstor.org/stable/10.7312/devl11638

1. Front Matter
(pp. i-iv)
(pp. v-vi)
3. Preface
(pp. vii-x)
Keith Devlin
4. Acknowledgments
(pp. xi-xii)
K. D.
5. 1 Prime Numbers, Factoring, and Secret Codes
(pp. 1-36)

The biggest (known) prime number* in the world is a giant that requires 909,526 digits to write out in standard decimal format. Printing out the entire number in a book like this would require about 500 pages. Using exponential (or power) notation, however, the number has a more manageable form:

23021377 — 1.

That is, you get the number by multiplying 2 by itself 3,021,376 times and then subtracting 1 from the answer.

Numbers that can be obtained by raising 2 to a power and then subtracting 1 are called Mersenne numbers, for reasons I will give later in this chapter....

6. 2 Sets, Infinity, and the Undecidable
(pp. 37-60)

Sometimes the solution of a long-standing problem marks the end (or the beginning of the end) of a mathematical era or field, the culmination of years of effort. On other occasions, it may open up an entire new area of research, possibly previously undreamed of. Such was the case with the discovery in 1963, by the 29-year-old Stanford University mathematician Paul Cohen, of the solution to Cantor’s continuum problem. Not only was the nature of the solution itself revolutionary, but also the methods Cohen developed to obtain it were new. These methods were soon found to have a wide range...

7. 3 Number Systems and the Class Number Problem
(pp. 61-84)

In 1983, Don Zagier of the University of Maryland and the Max Planck Institute in Bonn, and Benedict Gross of Brown University, Providence, Rhode Island, announced that they had solved the class number problem, a famous (among mathematicians) problem posed by Gauss in 1801. Although their proof was by no means the longest in mathematics (chapter 5 deals with that), at 300 pages it was longer than most. But it is not the length of the proof that mathematicians found so fascinating; it was its nature. It was very indirect and linked two seemingly unrelated areas of mathematics in quite...

8. 4 Beauty from Chaos
(pp. 85-110)

Bertrand Russell wrote in his 1918 book Mysticism and Logic that “mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture.”

Another famous British mathematician, G. H. Hardy, wrote in his book A Mathematician’s Apology (1940):

The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful, the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test; there is no permanent place in the world for ugly mathematics. . . . It may be very hard to define mathematical beauty,...

9. 5 Simple Groups
(pp. 111-142)

Some time during the summer of 1980, Ohio State University mathematician Ronald Solomon put down his pen after solving a technical problem in algebra and, with that one simple action, ended a quest that had begun in the 1940s and involved more than a hundred mathematicians from the United States, Britain, Germany, Australia, Canada, and Japan. What Solomon had done was to fill in the last piece of an enormous and highly complex puzzle: the classification of the finite simple groups.*

The classification theorem is by far mathematics’ biggest theorem. The initial proof ran to nearly 15,000 pages spread across...

10. 6 Hilbert’s Tenth Problem
(pp. 143-162)

In August 1900, the world’s best mathematicians gathered in Paris for the Second International Congress of Mathematicians (an event that, except for periods of war, has continued to be held every four years at different venues around the globe). Among them was a 38-year-old professor from the University of Göttingen, David Hilbert. As one of the leading mathematicians of the time, Hilbert was scheduled to deliver one of the keynote addresses to the meeting. The day fixed for his lecture was August 8.

Because the meeting was being held on the eve of the twentieth century (indeed, it had been...

11. 7 The Four-Color Problem
(pp. 163-192)

In 1976, Kenneth Appel and Wolfgang Haken, two mathematicians at the University of Illinois, announced that they had solved a century-old problem concerning the coloring of maps. They had, they said, proved the four-color conjecture. This in itself was a newsworthy event. At that time, the four-color problem was, after Fermat’s last theorem (see chapter 10), probably the second most famous unsolved problem in mathematics. But for mathematicians, the most dramatic aspect of the whole affair was the way that Appel and Haken had achieved their proof. Large and crucial parts of their argument were carried out by a computer,...

12. 8 Hard Problems About Complex Numbers
(pp. 193-222)

For many readers, this will be the most difficult chapter in the book. Not because the mathematics is intrinsically any harder than in other chapters, but because of the degree of abstraction involved. True enough, numbers—both natural and complex—form the core of the subject. But the essential task of complex analysis (the term complex function theory is used almost synonymously) and the closely allied field of analytic number theory (the application of the results and techniques of complex analysis to the study of the natural numbers) is to root out and exploit the deep structure and interconnections that...

13. 9 Knots, Topology, and the Universe
(pp. 223-262)

How can you tell a reef knot from a granny knot? Most Boy Scouts would have no difficulty in answering, but can a mathematician make the distinction? In 1984, some significant developments affected this question.

Do physicists use the right kind of mathematics to study the four-dimensional space–time universe we live in? Until 1982, every mathematician would have said, Yes, of course, it is the only kind. But now we know better. There are other, quite different kinds of mathematics that apply to a four-dimensional universe—but only to a four-dimensional universe; no such special treatment is required for...

14. 10 Fermat’s Last Theorem
(pp. 263-290)

In October 1994, a British mathematician named Andrew Wiles announced that he had solved a problem that had resisted all attempts at a solution for more than three hundred years, a problem that had grown to become the most famous unsolved problem in mathematics: Fermat’s last theorem. With the subsequent agreement among mathematicians that his proof is correct, Wiles brought to an end not only a three-hundred-year saga but also a nine-month period of uncertainty that had begun when a flaw had been found in an earlier proof he had announced in June 1993.

The fame of the problem was...

15. 11 The Efficiency of Algorithms
(pp. 291-310)

The concept of an algorithm has already played a prominent role in this book, in chapter 6. (I shall assume from now on that you have read that chapter.) With Hilbert’s tenth problem (considered there), the issue was whether or not a particular problem could be solved by an algorithm, with the stress on the “could be.” Pure existence was the order of the day—there was no question about whether the algorithms being discussed were at all practicable. In the context of Hilbert’s question, this was, of course, in order. But for problems arising in the real world around...

16. Index
(pp. 311-320)