Not all topics that are part of today’s elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become “elementary.” Elements of Mathematics: From Euclid to Gödel, by John Stillwell gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.
You’ve been writing math books for a long time now. What do you think is special about this one?
JS: In some ways it is a synthesis of ideas that occur fleetingly in some of my previous books: the interplay between numbers, geometry, algebra, infinity, and logic. In all my books I try to show the interaction between different fields of mathematics, but this is one more unified than any of the others. It covers some fields I have not covered before, such as probability, but also makes many connections I have not made before. I would say that it is also more reflective and philosophical—it really sums up all my experience in mathematics.
Who do you expect will enjoy reading this book?
JS: Well I hope my previous readers will still be interested! But for anyone who has not read my previous work, this might be the best place to start. It should suit anyone who is broadly interested in math, from high school to professional level. For the high school students, the book is a guide to the math they will meet in the future—they may understand only parts of it, but I think it will plant seeds for their future mathematical development. For the professors—I believe there will be many parts that are new and enlightening, judging from the number of times I have often heard “I never knew that!” when speaking on parts of the book to academic audiences.
Does the “Elements” in the title indicate that this book is elementary?
JS: I have tried to make it as simple as possible but, as Einstein is supposed to have said, “not simpler”. So, even though it is mainly about elementary mathematics it is not entirely elementary. It can’t be, because I also want to describe the limits of elementary mathematics—where and why mathematics becomes difficult. To get a realistic appreciation of math, it helps to know that some difficulties are unavoidable. Of course, for mathematicians, the difficulty of math is a big attraction.
What is novel about your approach?
JS: It tries to say something precise and rigorous about the boundaries of elementary math. There is now a field called “reverse mathematics” which aims to find exactly the right axioms to prove important theorems. For example, it has been known for a long time—possibly since Euclid—that the parallel axiom is the “right” axiom to prove the Pythagorean theorem. Much more recently, reverse mathematics has found that certain assumptions about infinity are the right axioms to prove basic theorems of analysis. This research, which has only appeared in specialist publications until now, helps explain why infinity appears so often at the boundaries of elementary math.
Does your book have real world applications?
JS: Someone always asks that question. I would say that if even one person understands mathematics better because of my book, then that is a net benefit to the world. The modern world runs on mathematics, so understanding math is necessary for anyone who wants to understand the world.
John Stillwell is professor of mathematics at the University of San Francisco. His many books include Mathematics and Its History and Roads to Infinity. His most recent book is Elements of Mathematics: From Euclid to Gödel.