Can we gain anything from a maths book so technical that only experts can keep up?by Tom Kirkwood / March 20, 2004 / Leave a comment
Book: The Millennium Problems Author: Keith Devlin Price: Granta, ?20
Reading “The Millennium Problems: the Seven Greatest Unsolved Mathematical Puzzles of Our Time” could be your ticket to a prize of $7m – $1m for each of the seven problems – to say nothing of the adulation you will receive from mathematicians around the world and the likelihood of a Fields medal, the equivalent in mathematical circles to a Nobel prize.
There is, however, a catch. Most of the problems in Keith Devlin’s book are so hard that not only will you have no idea how to begin solving them unless you have a PhD in mathematics already, you will be fully stretched even to reach the base camp of understanding the problem. It is noticeable that the endorsements on the book’s cover are all from professional mathematicians. Devlin himself admits in the early pages: “How can the non-mathematical reader understand these words, when they in turn don’t link to everyday experience?… it’s not that the human mind requires time to come to terms with new levels of abstraction – that’s always been the case – rather, the degree and pace of abstraction may finally have reached a stage where only the expert can keep up.”
So is it worth struggling through a book where your grasp of the meaning behind the words is likely at times to be shaky? Well, yes. At times I felt frustration with Devlin’s treatment of his subject, not least because of his habit of referring forward to explanations that will come later. But by the end of the book I did feel that a window had been opened for me on to the frontiers of modern mathematics. As human knowledge becomes more specialised, it is important that we at least know the lie of other minds’ lands.
The seven millennium problems, in the order Devlin presents them, are these: the Riemann hypothesis; Yang-Mills theory and the mass gap hypothesis; the P versus NP problem; the Navier-Stokes equations; the Poincar? conjecture; the Birch and Swinnerton-Dyer conjecture; and the Hodge conjecture. Since Devlin takes a whole book to explain what these problems are, only the briefest description can be attempted here. Note that five of the seven are described as a “hypothesis” or “conjecture,” so the essence of the problem is to prove what someone has suggested and many believe to be true, but no…