Book Review … a small idea of what it is I do all day… Introductions to Mathematics Günter M. Ziegler What Is Mathematics? An Elementary Approach 4. What are the most important problems in math- to Ideas and Methods ematics? And why don’t all mathematicians Richard Courant and Herbert Robbins work on these? 5. What does it mean to a mathematician that Second edition, revised by Ian Stewart something is proved? And what does it mean if Oxford University Press, 1996 he believes a proof? Paperback, 592 pages, $21.50 6. How and why is mathematics relevant for “the ISBN 0-19-510519-2 real world”? The Heart of Mathematics: An Invitation to 7. Why is mathematics so unreasonably effective Effective Thinking in some applications? Edward B. Burger and Michael Starbird 8. Why is mathematics so hard to understand? Key College Publishing (Springer-Verlag), 2000 And why is most of it so hard to explain? Hardcover, 646 pages, $69.95 All these are valid questions. There are classical an- ISBN 0-555953-407-9 swers: G. H. Hardy’s A Mathematician’s Apology Mathematics: A Very Short Introduction probably contains the most famous and the most Timothy Gowers controversial ones. But none of them are eternal or universally valid. As mathematics and the na- Oxford University Press, 2002 ture of mathematics develop and change, we have Paperback, 144 pages, $9.95 to come up with new explanations again and again. ISBN 0-19-285361-9 One may also try to come up with answers that are 1089 and All That. A Journey into Mathematics not even intended to be universally valid, with per- David Acheson sonal views of the world of mathematics. I believe Oxford University Press, 2002 that these are equally useful and important. So, Hardcover, 178 pages, £9.95 what is mathematics to you? ISBN 0-19-851623-1 Suggested Answers One reason why you should look for, and provide, What Is Mathematics? your own answers to such questions is that you Here are some questions for you: won’t like many of the answers that you’ll get from 1. What is mathematics? others. 2. What are you doing when you are “doing math- For example: Why is it that these days we start ematics”? with applications whenever we try to present math- 3. Mathematical research—how does that hap- ematics to the public? Perhaps this is the easiest pen? Can it be planned? Can it be cast into “pro- and the most effective way, or the most convinc- jects”? ing? The one that even politicians might under- stand? My own experience certainly supports this Günter M. Ziegler is professor of mathematics at Technische impression. I can’t explain the high-dimensional Universität Berlin. His email address is ziegler@math. geometry I do to nonmathematician friends at night tu-berlin.de. at a bar, but I can tell them about mathematicians’ FEBRUARY 2005 NOTICES OF THE AMS 221 work on bus schedules and on hyperthermia can- Courant was perhaps one of the most influen- cer therapy (to mention two Berlin examples), and tial “applied mathematicians” of the twentieth they’ll be suitably impressed. However, although ap- century. But there’s no “applied math” in his book! plications answers may be interesting and easy However, there is a lot of mathematics, just very and effective, they shouldn’t be the only ones! little about mathematics. Courant and Robbins’ Another example: What kind of pictures of answer to his title question is to develop and “what mathematics is like” do your kids bring home explain mathematics, a wealth of very classical from school? Do these pictures have anything to and fundamental topics: numbers and number do with what we believe they should be? Aren’t their theory, geometry, topology, calculus. The last views of mathematics very lopsided and incomplete, sentence of the introduction reads at best? Do they represent what mathematics “really is about”? Do kids experience the power of For scholars and laymen alike it is not mathematics, the virtues of precision and ab- philosophy but active experience in straction, mathematical creativity and mathemat- mathematics itself that alone can an- ical ideas? Do they learn about this in a way that swer the question: What is mathematics? we think is fair, colorful, multifaceted, exciting, and inviting? Do they get to see mathematics as New Answers we would view it? How do we view it? What is mathematics? Is it changing? Of course it Mathematics According to Courant is, and so we have to ask for, and attempt to give, new answers to the old question. What Is Mathematics? is the title of a classic book The Heart of Mathematics: An Invitation to by Richard Courant and Herbert Robbins from Effective Thinking is the title of one such answer, 1941, published by Oxford University Press, with a book by Edward B. Burger (Williams College) and successful translations into German, Italian, and Michael Starbird (University of Texas at Austin). It Russian. Apparently Courant was quite hesitant is a large volume (650 pages in the first edition), about the title and thought that Mathematical Discussions of Basic Elementary Problems for the with ample four-color illustrations throughout, General Public might be more accurate, but also and a bit of arrogance in the title. What a contrast “a little bit boring.” to Courant’s book, in style, but also in contents. This Constance Reid tells the story that at a dinner is a textbook for a “math appreciation” under- in Princeton Courant talked to Thomas Mann, who graduate class, but also fun reading and viewing told him about one of his books, a little novel for anyone else. The choice of topics is guided by whose German title was Lotte in Weimar. Mann what is or should be interesting to an audience of thought that the same title might work in English, non-science majors. So some classical topics appear; whereas his publisher Alfred Knopf (or rather after a lot of introductory drumming the book Knopf’s wife) had suggested The Beloved Returns, starts with the pigeon-hole principle and Fibonacci saying that Lotte in Weimar would sell 10,000 or numbers. But this is not part of a systematic de- perhaps 20,000 copies, but The Beloved Returns velopment. There are rope tricks, games, aperiodic might sell 100,000—with the corresponding au- tilings, fractals, lots of “modern mathematics”, all thors’ royalties. So Thomas Mann went for The of this embedded into lots of motivational talk, “fun Beloved Returns, and Richard Courant went for and games”, “mindscapes”, “creating new ideas”, What Is Mathematics? Courant sold more than “invitations to further thought”, and “lessons for 100,000 copies. I don’t know about Mann’s book. life”. It’s well done, I think it is good to have, it is 222 NOTICES OF THE AMS VOLUME 52, NUMBER 2 fun, but it is not my style. At forty-one, I may be of mathematics”, without having to do any difficult too old or too old-fashioned for this. math. He speaks clearly and concretely about the Personally, I prefer the small and modest attempts role of models and about abstractions, concluding to answer the old question. One recent such attempt “Once one has learned to think abstractly, it can is Mathematics: A Very Short Introduction by Tim be exhilarating, a bit like suddenly being able to ride Gowers (of 1998 Fields Medal fame). It appeared in a bicycle without having to worry about keeping an Oxford University Press series of “Very Short one’s balance.” He talks about limits and infinity Introductions” that treats an extensive list of topics and about dimension, and at this point even a frac- concerning nearly everything under the sun, such tal curve comes up, including a picture! And he ex- as religion, philosophy, and history, but also animal plains why most mathematicians don’t worry about rights, cryptography, evolution, Heidegger, Hin- foundational questions such as “whether numbers duism, and linguistics. There’s even “Schizophre- exist” or about “infinity”. nia: A Very Short Introduction”. Gowers’ last chapter, “Some Frequently Asked Questions”, treats mathematics as a human An Author Needs to Be Convinced endeavor. Here’s his list of questions: Let’s just imagine one of the friendly Oxford Uni- 1. Is it true that mathematicians are past it by the versity Press mathematics editors knocking at time they are thirty? Professor Gowers’s office door. She would introduce 2. Why are there so few women mathematicians? herself, and say “Well, you know, we have this lit- 3. Do mathematics and music go together? tle book series … it is really quite successful … this, 4. Why do so many people positively dislike math- for example, is the little volume on atheism … it ematics? has just appeared … couldn’t you, for example, 5. Do mathematicians use computers in their work? write one for us … about mathematics …??” And 6. How is research in mathematics possible? Professor Gowers is quite thoughtful and a bit 7. Are famous mathematical problems ever solved skeptical, and he doesn’t want to say no right away, by amateurs? because somehow he is intrigued by the challenge, 8. Why do mathematicians refer to some theorems so he just promises “… to think about it …” And and proofs as beautiful? then it comes as a considerable surprise (first to Of course I won’t give you his answers here. You him, then to his editor) that he does get started on should look at his volume: He certainly does not the impossible task, takes on the challenge. And give “the only possible correct answers”, but con- he succeeds, with style.