What is “mathematical maturity”? While many of us use the phrase and think we know what it means — at least in general terms — it is somewhat ill-defined. The current book aims to answer the question. It also attempts to understand how the idea of mathematical maturity sets mathematics apart from other disciplines, how to recognize students who are, or are becoming, mathematically mature, and how teachers and mentors can aid the process.

For the most part, the author limits himself to mathematicians, as the title would suggest. Can non-mathematicians be mathematically mature? The author’s opinion here is not entirely clear. He writes admiringly, for example, of mathematical modeling. (But he seems rather oddly to think that it involves only physics.) He would, it appears, include “cutting edge modern physicists” in the esteemed group of the mathematically mature. In the last chapter, he suggests that the category might even be a bit wider. Yet, for the most part, the only real form of mathematical maturity that the author wants to consider is that which develops in a professional mathematician who is proceeding to a Ph.D. Much of the time he seems more interested in describing the successful progression of an academic mathematician from student to professor. He gives a considerable amount of advice in this direction. (Go to a top graduate school, find a distinguished advisor, choose fruitful research topics, publish widely, and so on.) A title for the book that better reflects its contents might be “A Guide to a Successful Academic Career in Mathematics”.

However, a reader coming to this book looking to learn more about the broader subject of mathematical maturity is likely to be disappointed, for the author’s concept of that is very narrow. The “tree of mathematical maturity” that appears at the end of the book is formed of topics that culminate in a very traditional mathematics graduate program. If it means anything, mathematical maturity has to be more than that.

The author also argues that there is something unique about mathematical maturity, something not shared by any other discipline. While there are very clear differences, there is also much in common with mature scholarship in any number of other disciplines. Common elements, I suggest, include immersion in a subject and its tools, broad exposure to its many elements, an ability to integrate and make connections across its several parts and with other disciplines, and ability to identify and focus on key ideas. While creating, reading and analyzing rigorous proofs may be unique to mathematics, the process is not intrinsically different from crafting and dissecting careful arguments in philosophy, history, or other fields.

Apart from the continuing theme of mathematical career development, the book is rather a jumble. There are the author’s thoughts on teaching and the current state of education. There is a chapter on social issues that includes discussions of math anxiety, national standards, the Myers-Briggs index, Asperger’s syndrome, women in mathematics, and the author’s view of why students drop out with “all-but-dissertation.” A companion chapter on cognitive issues has nineteen sections in twenty-five pages and takes up nature-nurture, maturity-immaturity, motivation, Piaget’s ideas, types of intelligence, and more. Too many topics, too little depth, too much amateur psychology.

Graduate students in mathematics or undergraduate mathematics majors intent on academic careers could benefit from some of the tips this book has to offer. The tone is often too preachy for my taste, but the advice is generally good. Anyone wanting to learn about mathematical maturity is advised to look elsewhere. I found a far more useful discussion on Wikipeida and in the links it provides.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

## Comments

## Annie Selden

This small book contains a very personal view of mathematical maturity, described in 79 small subsections that are distributed across seven chapter with titles such as "Teaching Techniques," "Social Issues," and "Cognitive Issues." In the preface, Krantz states that mathematical maturity consists of the ability to

So far, so good. However, at the end of this little book (on p. 119), Krantz provides "The Tree of Mathematical Maturity" in the form of a hierarchical chart going from integer arithmetic and fractions to theory of functions, linear algebra, and differential equations, and on to qualifying exam courses and seminar courses. This tree, in contrast to the list in the preface, is about topics, sometimes even courses, in mathematics.

The book is written in typical Krantz style, with no holds barred. Krantz lets the reader know, early on and in no uncertain terms, that he considers himself mathematically mature because he has 165 research papers and 65 books to his credit, and that he is "a well-known and accomplished scholar." (p. 17) However, he does allow that there are "people who are more mathematically mature than I", such as Fields Medalists.

There are sentences in the book that I would not expect of such a distinguished mathematician, who is also editor of the

Notices of the American Mathematical Society. For example, on page 66, in showing that mathematicians, in general, do not have "tunnel vision," but can also be accomplished artists and musicians, etc., Krantz recounts the following anecdote. He was at a dinner at another university where he had given a talk, when a woman remarked to him, "So I guess you don't do anything but mathematics? You have no other interests." Of this Krantz writes, "Naturally I wanted to smack her. But I instead gave some polite answer and then turned to talk to the person on the other side of me." This may well be how Krantz felt, but letting on that he wanted to "smack her" in a book on mathematical maturity does not seem appropriate, and lowered my opinion of Krantz. Surely such a reaction was not his intent in writing a book on mathematical maturity. Where was the editor of the Spectrum Series?