Monday, April 1, 2013

Ian Stewart's "Visions of Infinity"

Ian Stewart is Emeritus Professor of Mathematics and active researcher at the University of Warwick. The author of many books on mathematics, he lives in Coventry, England.

Stewart applied the “Page 99 Test” to his latest book, Visions of Infinity: The Great Mathematical Problems [UK title: The Great Mathematical Problems], and reported the following:
From Page 99:
There is a price to pay for this approach: you have to solve about 100,000 linear programming problems. The calculations are lengthy, but well within the capabilities of today’s computers. When Hales and Ferguson prepared their work for publication it ran to about 250 pages of mathematics, plus 3 gigabytes of computer files.

In 1999 Hales submitted the proof [of the Kepler conjecture] to the Annals of Mathematics, and the journal chose a panel of twelve expert referees. By 2003, the panel declared itself ‘99 per cent certain’ that the proof was correct. The remaining uncertainty concerned the computer calculations; the panel had repeated many of them, and otherwise checked the way the proof was organised and programmed, but they were unable to verify some aspects. After a delay, the journal published the paper. Hales recognised that this approach to the proof would probably never be certified 100 per cent correct, so in 2003 he announced that he was starting a project to recast the proof in a form that could be verified by a computer using standard automated proof-checking software.
Page 99 of Visions of Infinity is about the Kepler conjecture, a baffling mathematical enigma that remained unsolved from the time Johannes Kepler first posed it in 1611, in a book about snowflakes, until Thomas Hales and his student Samuel Ferguson solved it 387 years later. Is the most efficient way to pack spheres together the one that greengrocers use to stack oranges? The answer surely has to be ‘yes’, but proving it is another matter entirely.

The book as a whole is about the great mathematical problems, the ones that every mathematician would dearly love to solve. The obstacles that hold up mathematical progress, often for centuries. Whenever some genius answers one of these problems, the media start paying attention to mathematics.

However, media reports sometimes give the impression that finally, after decades of inactivity, mathematicians have produced something new. One of my aims in writing the book is to make it clear that modern mathematics is a hotbed of creativity, with new and important mathematics constantly being brought into existence. In fact, so creative are today’s mathematicians that most of the great problems of the past have crumbled under the onslaught of fresh insights.

However, the more we learn, the more we become aware of that we don’t know. So new great problems are arising all the time. The book is therefore a mixture of solved and unsolved problems. Each gets a chapter to itself, explaining how the problem arose, what mathematicians have done to tackle it, and either how someone eventually solved it, or why it remains as baffling as ever.

I have covered most of the famous problems, and all of the Clay Institute Millennium Prize Problems, with a million dollars on offer for each. I’ve included familiar problems like the four colour conjecture and Fermat’s last theorem, and others less well known, such as the P/NP problem: do genuinely hard mathematical questions exist?

My main aims in writing the book were to collect together in one place accessible descriptions of a selection of the greatest mathematical problems ever posed, and to demonstrate the ever-growing power and creativity of modern mathematics. The great mathematical problems open a fascinating window into the otherwise obscure workings of great mathematical minds.
Learn more about the book at the author's website.

See Ian Stewart's top ten popular mathematics books.

The Page 99 Test: Why Beauty Is Truth.

The Page 99 Test: In Pursuit of the Unknown.

--Marshal Zeringue