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Few authors are better at understanding their readers than the prolific mathematics writer Ian Stewart. In his latest title, Visions of Infinity: The Great Mathematical Problems, his target audience is clear and his aim is faultless. Anyone who has always loved math for its own sake or for the way it provides new perspectives on important real-world phenomena will find hours of brain-teasing and mind-challenging delight in the Warwick University (U.K.) emeritus professor's survey of recently answered or still open mathematical questions.
"What makes a great mathematical problem great?" Stewart asks. "Intellectual depth, combined with simplicity and elegance. Plus: it has to be hard.... Great problems are creative: they help bring new mathematics into being." [emphasis in original]
Remarkably, even the hardest of the great problems are easy to state, though some require deep mathematical knowledge to grasp the vocabulary. Others are so simple that little or no mathematical training is needed to understand them.
For example the four-color map problem asks whether four colors are enough to distinguish contiguous regions on a map from their neighbors. In practice, that seems to be true, but could it be proved? The answer turned out to be yes, but it took a mathematical tour de force and thousands of hours of computer time to do so.
Likewise Fermat's last theorem, named for the great 17th-century mathematician Pierre de Fermat, considers integer solutions to a simple extrapolation of Pythagoras' famous relationship: the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two smaller sides.
Fermat's last theorem replaces squares in that formula with larger integral powers and asks whether there are sets of three integers that satisfy the new equations. In Fermat's time, the common conjecture was that no such triplets existed for any power other than 2. Some people had proved it for certain powers, but not the general case. In the margin of one of his personal books, Fermat wrote a note that he had found a proof for that generalization; but if he had written it down, no one ever found it.
In 1995, British mathematician Andrew Wiles (1953- ), building on the efforts and false starts of many predecessors, finally cracked the problem that had entranced him since he first discovered it at age 10.
Visions of Infinity devotes a long chapter to Fermat's Last Theorem, but Stewart also uses it in an opening vignette about a remarkably popular 1996 television documentary on the subject. "Wiles's solution is much too complicated and technical for television," he notes.... "The proof does involve a nice mathematical story, as we'll see in due course, but any attempt to explain that on television would have lost most of the audience immediately. Instead, the programme sensibly concentrated on a more personal question: what is it like to tackle a notoriously difficult mathematical problem that carries a lot of historical baggage."
Stewart employs a similar strategy that is bound to please his more mathematically sophisticated audience, who will relish the breadth and depth of topics ranging from ancient attempts to square the circle to challenges as modern as the complexity of computer programs and the standard model of particle physics.
Individual readers will dig deeply into certain chapters and skim read others according to personal preference, but every one of them will be captivated by the technical achievements, loose ends, and human insights that Stewart shares on his grand mathematical tour.