Letters to a Young Mathematician Read Online Free PDF Page B

between mathematicians and everyone else. What is mathematics? It is the shared social construct created by people who are aware of certain opportunities, and we call those people mathematicians. The logic is still slightly circular, but mathematicians can always recognize a fellow spirit. Find out what that fellow spirit does; it will be one more aspect of our shared social construct.
    Welcome to the club.

4
Hasn’t It All Been Done ?
    Dear Meg,
    In your last letter you asked me about the extent to which mathematics at university can go beyond what you have already done at school. No one wants to spend three or four years going over the same ideas, even if they are studied in greater depth. Now, looking ahead, you are also right to worry about the scope that exists for creating new mathematics. If others have already explored such a huge territory, how can you ever find your way to the frontier? Is there even any frontier left?
    For once, my task is simple. I can set you at ease on both counts. If anything, you should worry about the exact opposite: that people are creating too much new mathematics, and that the scope for new research is so gigantic that it will be difficult to decide where to start or in which direction to proceed. Math is not a robotic way of replacing thought by rigid ritual. It is the most creative activity on the planet.
    These statements will be news to many people, possibly including some of your teachers. It always astonishes me that so many people seem to believe that mathematics is limited to what they were taught at school, so that basically “it’s all been done.” Even more astonishing is the assumption that because “the answers are all in the back of the book,” there is no scope for creativity, and no questions remain unanswered. Why do so many people think that their school textbook contains every possible question?
    This failure of imagination would amount to deplorable ignorance, were it not for two factors that together go a long way to explain it.
    The first is that many students quickly come to dislike mathematics as they pass through the school system. They find it rigid, boring, repetitive, and, worst of all, difficult. Answers are either right or wrong, and no amount of clever verbal jousting with the teacher can convert a wrong answer into a correct one. Mathematics is a very unforgiving subject. Having developed this negative attitude, the last thing the student wants to hear is that there is more mathematics, going beyond the already daunting contents of the set text. Most people want all the answers to be at the back of the book, because otherwise they can’t look them up.
    Dame Kathleen Ollerenshaw, one of Britain’s most distinguished mathematicians and educators, who continues to do research at the age of ninety, makes exactlythis point in her autobiography To Talk of Many Things .
(Do read it, Meg; it’s inspirational, and very wise.) “When I told a teenage friend that I was doing mathematical research, her reply was, ‘Why do that? We have enough mathematics to cope with already—we don’t want any more.’”
    The assumptions behind that statement bear examination, but I content myself with just one. Why did Kathleen’s friend assume that any newly invented mathematics would automatically appear in school texts? Again we encounter the same belief, that the math you are taught at school is the entire universe of mathematics. But no one thinks that about physics, or chemistry, or biology, or even French or economics. We all know that what we are taught at school is just a tiny part of what is currently known.
    I sometimes wish schools would go back to using words like “arithmetic” to describe the content of “math” courses. Calling them “mathematics” debases the currency of mathematical thought; it’s a bit like using the term “composing” to describe routine exercises in playing musical scales. However, I lack the power to change the name, and