Stewart - Letters to a Young Mathematician

• Concepts of Modern Mathematics
  Game, Set, and Math
  Does God Play Dice?
  Another Fine Math Youve Got Me Into
  Fearful Symmetry
  Natures Numbers
  From Here to Infinity
  The Magical Maze
  Lifes Other Secret
  Flatterland
  What Shape Is a Snowflake?
  The Annotated Flatland
  Math Hysteria
  The Mayor of Uglyvilles Dilemma
  
• with Jack Cohen
  The Collapse of Chaos
  Figments of Reality
  What Does a Martian Look Like?
  Wheelers (science fiction)
  Heaven (science fiction)
  
• with Terry Pratchett and Jack Cohen
  The Science of Discworld
  The Science of Discworld II: The Globe
  The Science of Discworld III: Darwins Watch
  

It is a melancholy experience for a professional mathematician to find himself writing about mathematics. So the great English mathematician Godfrey Harold Hardy, of the University of Cambridge, opened his 1940 classic A Mathematicians Apology.

Attitudes change. No longer do mathematicians believe that they owe the world an apology. And many are now convinced that writing about mathematics is at least as valuable as writing mathematics, by which Hardy meant new mathematics, new research, new theorems. In fact, many of us feel that it is pointless for mathematicians to invent new theorems unless the public gets to hear of them. Not the details, of course, but the general nature of the enterprise. In particular, that new mathematics is constantly being created, and what it is used for.

The world has changed, too, since Hardys time. A typical day for Hardy consisted in a maximum of four hours of intensive thought about research problems; the rest of the day was then occupied watching the game of cricket, his great nonmathematical passion, and reading the newspapers. He must have fitted in some time for the occasional research student as well, but he was reticent about personal matters. A typical day for the modern academic is ten or twelve hours long, with teaching commitments, research grants to pursue, research to be carried out, and liberal doses of pointless bureaucracy to get in the way of anything creative.

Hardy was typical of a certain kind of English academic. He set himself high but narrow standards. He valued his chosen field for its own internal elegance and logic, not for its external uses. He was proud that none of his work could have any possible use in warfare, a position with which most of us can sympathize, especially bearing in mind that his book was published in the opening years of World War II.

He would be disappointed in the extreme to be resurrected today and to learn that on the contrary, his beloved theory of numbers plays an essential role in the mathematical theory of cryptography, with evident military uses. The movie Enigma paints a romanticized view of the period when this connection first began to emerge, in the vital wartime work of the code breakers at Bletchley Park. Prominent among them was the tragic figure of Alan Turingpure mathematician, applied mathematician, and pioneering computer scientistwho committed suicide because he was persecuted for being a homosexual, a sexual orientation that was then illegal and considered shameful. Social mores change, too.

Hardys classic little gem sheds a great deal of light on how academic mathematicians viewed themselves and their subject in 1940. It contains important lessons for any would-be young mathematician, but some of these are obscured by the books outdated attitudes, such as its default assumption that mathematics is strictly a male preserve. It is still worth reading, but only if its opinions are seen in their historical context, and it is not assumed that they all remain valid today.

Letters to a Young Mathematician is my attempt to bring some parts of A Mathematicians Apology up to date, namely, those parts that might influence the decisions of a young person contemplating a degree in mathematics and a possible career in the subject. The letters, addressed to Meg, follow her career in roughly chronological order, from high school through to a tenured position in a university. They discuss a variety of topics, ranging from basic career decisions to the working philosophy of professional mathematicians and the nature of their subject. The intention is not merely to offer practical advice, but to give an inside view of the mathematical enterprise, and to explain what it is really like to be a mathematician.