# Visions of Infinity

## The Great Mathematical Problems

Book - 2013
"It is one of the wonders of mathematics that, for every problem mathematicians solve, another awaits to perplex and galvanise them. Some of these problems are new, while others have puzzled and bewitched thinkers across the ages. Such challenges offer a tantalising glimpse of the field's unlimited potential, and keep mathematicians looking toward the horizons of intellectual possibility. In this book, celebrated mathematician Ian Stewart provides a fascinating overview of the most formidable problems mathematicians have vanquished, and those that vex them still. He explains why these problems exist, what drives mathematicians to solve them, and why their efforts matter in the context of science as a whole. The three-century effort to prove Fermat's last theorem --first posited in 1630, and finally solved by Andrew Wiles in 1995 --led to the creation of algebraic number theory and complex analysis. The Poincaré conjecture, which was cracked in 2002 by the eccentric genius Grigori Perelman, has become fundamental to mathematicians' understanding of three-dimensional shapes. While mathematicians have made enormous advances in recent years, some problems continue to baffle us. Indeed, the Riemann hypothesis, which Stewart refers to as the 'Holy Grail of pure mathematics', and the P/NP problem, which straddles mathematics and computer science, could easily remain unproved for another hundred years."--Dust jacket.

Publisher:
New York : Basic Books, c2013

Description:
x, 340 p. : ill. ; 24 cm

ISBN:
9780465022403

0465022405

0465022405

Branch Call Number:
510 Ste

## Comment

Add a Commenthas chapters on many mathematical classical and current (CLAY) problems. Author connects these chapters together and build a coherent picture by going back and forth between the chapters. Focus is not on infinity but on various conjectures that have been and have not been resolved. Even a good chapter on P vs NP and something from physics, the Quantum Mass Gap.

As with all his books, he presents a very clear description of the topics he deals with. Here for the most part the Millennium problems the Clay group has set-up prizes for.

A very good read.

Ian Stewart, an emeritus professor of mathematics at the University of Warwick, makes a valiant attempt to explain fourteen of the most important mathematical problems to a lay audience. For the most part, he succeeds. However, some of the problems are sufficiently abstract (particularly, the Hodge Conjecture) that even a gifted expositor such as Stewart cannot state them in terms a lay audience can understand. However, to his credit, he does try. Some of the problems such as the Goldbach conjecture that every even number larger than 2 can be expressed as the sum of two primes are readily understood, if difficult to solve. Others, such as the Hodge Conjecture, can only be understood by a specialist. In explaining these problems, Stewart gives the reader a sense of what it is that mathematicians do. Specifically, he demonstrates how mathematicians accumulate evidence that a conjecture is true, and illustrates that crucial insights sometimes come from drawing upon seemingly unrelated branches of mathematics (such as the use of number theory in constructing regular polygons). He also conveys why the problems discussed are difficult, while not always managing to explain their importance. Readers with mathematical training will find the descriptions in the book maddeningly vague, and will, no doubt, notice errors in the text (k^2 + k + 41 is not prime if k = 40) and imprecise definitions in the glossary (he fails to specify that the integers in a Pythagorean triple must be nonzero). There is also a particularly unfortunate analogy between Ernest Rutherford's determining the shape of an atom by bombarding it with alpha particles (nuclei of helium atoms) and shooting bullets into a dark field to see what is there.