Induction and Analogy in Mathematics

"This is one of the books that has a deep personal meaning to me. In some ways it is the most challenging of the books I have picked, because there is a lot of real maths in it. George Polya, a professor at Stanford University, was a fantastic maths teacher and a real genius. He has a way of taking you gently through the process, and making you feel as if you can do it. I received this book as a prize when I was in high school, for the top graduating maths student. I said “thank you” and then put the book aside, and didn’t even look at it for about two years, until I was in college. Then I picked it up one day and started reading it, and I was absolutely flabbergasted. The particular thing I was captivated by was a passage that Polya wrote about a discovery by an 18th century mathematician called Leonhard Euler, who is often considered one of the greatest mathematicians of all time. Polya wrote about how Euler solved a sum which no one else had been able to figure out. He realised that the sum was pi squared over six, pi being the famous number that gives the ratio of a circle’s circumference to its diameter. The first amazing thing is that this problem does not seem to have anything to do with circles. In fact, it has to do with squares and what happens when you add up their reciprocals. How on earth do these squares turn into circles? Polya takes us through this discovery process, in which Euler violates just about every rule your high school maths teacher taught you. He says, in effect: “Let’s pretend that the sine function is a polynomial. What polynomial would it be?” Sines and polynomials are two kinds of functions that are central to mathematics, from high school on. So he takes two things that everyone thought were different, and finds a connection between them. That is exactly what great scientists do – not just mathematicians. This is something that maths books, especially maths textbooks, consistently get wrong. They do not teach students to discover things. At best they teach students how to prove things, but most of them don’t even do that. Polya teaches you how to discover. It’s easy to think that this is just a book about maths, but it’s not. When you read Polya’s book you will learn about science in general. He does an amazing deconstruction of scientific psychology. He talks about making analogies. And he is fond of guessing, which is something you’re not supposed to do in school mathematics. But Polya realises that to discover things, you have got to make leaps of intuition. You need to guess what the answer is. They won’t tell you that in school, but it’s true."