» Letters to a Young Mathematician (Art of Mentoring)

Letters to a Young Mathematician (Art of Mentoring) Reviews

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Summary: A convincing description of the mathematical facts of life
Comment: The life of a professional mathematician is one that requires a great deal of preparation. While learning mathematics is of course essential, there are many other facets to making a living in mathematics. Depending on the job, there can be any combination of conducting research, teaching, industrial applications, committee work, writing grant applications, writing grant reports, presenting results, writing and defending papers, and other interactions with colleagues.
In this book, Stewart writes a series of letters to Meg, a fictitious person, and they give advice to her that covers the complete spectrum. Everything from how to prepare for the study of mathematics before you start college to how to conduct yourself after you have published quality research is discussed. As Stewart himself acknowledges in the preface, he was strongly influenced by Godfrey Harold Hardy's classic book "A Mathematician's Apology."
Mathematics is one area of life where you can find everything from absolute known truth to absolute unknown truth. With such a rich range of experiences, it is one of the most rewarding occupations there is. Stewart is convincing when he is telling his young protégé the facts of life as they apply to a professional mathematician.

Published in Journal of Recreational Mathematics, reprinted with permission

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Customer Rating:
Summary: Great Insight into a Rarefied World
Comment: I have read and enjoyed books by Prof. Stewart in the past so I was happy to see that he was selected to write this book. This is the second book in "The Art of Mentoring" series that I've read. It's an interesting concept--an experienced expert in a field writes "letters" to a person just starting out, giving them information and advice. It doesn't always seem to come off perfectly but, in my so far limited experience, these brief books certainly seem to be worth the effort.

In this volume, Prof. Stewart gives us insight into the world of mathematics. As a long-time teacher of math and physics, I am always looking for things that will open my students' eyes to the wonders of the subject and help them understand the "mathematical perspective." Fortunately, there are a number of excellent passages here that do just that. I am particularly pleased by his descriptions of the thrill mathematicians have when they solve a challenging problem--something I've tried to share with my students from my own experience. I also find his discussions of proof interesting and something I will find useful for my students.

Of course, the assumption of this book is that the reader is already interested in math, so some of it is really beyond what would likely interest a casual reader. Still, I found many of these even more rarified letters hitting close to home. For example, his discussion of the difference between pure and applied math, something I came across a lot in my physics studies. I also have to give Prof. Stewart credit for a writing style that is much more accessible than that of many mathematicians. He hooked me right away by addressing his letters to Meg (pulled from one of my favorite novels as a kid, A Wrinkle in Time).

But it must also be said that there is much here, particularly as the book goes on, that is going to be less interesting for those not familiar with or considering a career in mathematics. The details and experiences he describes, while well-done, are simply not going to be close to the experience of the average reader. That doesn't mean it's not valuable stuff, but a potential reader should consider himself or herself forewarned.

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Customer Rating:
Summary: Valuable though a bit theory-biased
Comment: The declared goal is "to give an inside view of the mathematical enterprise, and explain what it is really like to be a mathematician". Half the book is popular mathematics, and the other half is remarks on "being a mathematician". The book is well written, easy and fun to read, sensible throughout, and prospective mathematicians could only benefit from reading it. But I was disappointed (as a professional mathematician) to find few particularly new or individualistic insights in either half. The popular mathematics half relies on well-worn topics like "Platonism versus realism" and "can't computers solve everything?". And only the following three thoughts on being a mathematician struck me as particularly insightful.

(i) [as conclusion of "what is mathematics/mathematicians?" discussion] A mathematician is someone who sees opportunities for doing mathematics.

(ii) Being a research mathematician is akin to being a writer or an artist; any glamour that's apparent to outsiders fades quickly in the face of [the reality]. Your satisfaction must come from the high you get when you suddenly, for the first time, understand the problem you're working on .....

(iii) As your career develops, the worldwide mathematical community will be increasingly important to you. You will become part of it, and then you will have a home in every city on Earth.

A nice introductory chapter extols the breadth of uses of mathematics, and could be used by Math Dept web sites to recruit majors. There is also a well-balanced "pure or applied" chapter. But the rest of the book identifies "mathematics" as "theorem-proof mathematics", as illustrated by quotes such as

(i) There has been a spate of popular math books in recent years ...... there are even books on the applications of mathematics.

(ii) The true mathematician is not satisfied until the statement is proved.

This attitude is irritating to those of us who regard this identification as akin to identifying "visual art" with oil painting. What matters isn't the tools, it's how competently and creatively you use them.

A final irreverent thought: instead of A Mathematician's Apology (Canto), wouldn't The Screwtape Letters have made a more intriguing model?

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Customer Rating:
Summary: Excellent mentoring guide
Comment: While the book certainly does an excellent job encouraging the study of mathematics I found the insights in how best to encourage a study of such a subject an excellent mentoring guide for other subjects as well. I am thinking specifially of logic but feel certain there would be a range of subjects these great stories and insights would apply to. So it is not just a good book for those young people thinking of studying math. It is also a good read for those who would encourage young people to study and be serious in their studies. It helps show just how important such seriousness is, while at the same time find the work enjoyable. This is a great read for young people who have not even deciced what they want to be yet as well as those who are interested in careers with emphasis on logic and mathemtaics.

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Customer Rating:
Summary: Adds Up to a Good Read
Comment: Stewart writes a series of letters to a (probably) fictitious young person to inform each step of that student's journey along the path to becoming a mathematician - starting with grade school questions about "What's math good for?"- and going through advice on how to negotiate office politics in post-doctoral academia.

Each of these chronological letters contain practical advice as well as interesting insights into scholarly subjects, not limited to the field of mathematics.

For example, Stewart presents some of the crispest, most comprehensible definitions I've run across. He defines postmodernism as "the belief that everything is social convention." Other people have spent whole windy pages and chapters trying to define that term, and only succeeded in putting me in a greater fog than before I started.

Stewart defines fractals as patterns that "exist in a fractional number of dimensions." And Godel's theorem gets intelligibly summarized as "the theory that there will be statements that can never be proved either true or false, but that can be used as axioms of equations either way - without creating inconsistency."

Along the way, Stewart renders some wise advice on how to live life in general. He points out how often the theory that people struggle for decades to prove or disprove (like Fermat's Last Theorem) is not important in and of itself. But the process of trying to prove it often opens whole new fields of mathematics. That's another take on the old truism - It's the journey not the destination.

Stewart made only one remark that I didn't think was totally reliable in this neat little book of essays. He says that a primary pleasure to be found in the practice of mathematics is that there is only one correct answer. Once that correct answer is found and proved, all squabbling ceases. Everyone unites and moves on. Well, I don't know about that. From what I've heard, mathematicians can squabble indefinitely over an issue and divide into standing camps - precisely because so many equations don't have just one correct answer. What about fuzzy math - equations that allow a smear of different answers over a range? Stewart doesn't mention this field at all, and so omits consideration of what many are beginning to feel might be a paradigm for how the human brain works. Very little is yes or no. Most things are entertained as different strengths of "maybe's."

Aside from that one lapse into unwarranted optimism about the ultimate pacifism of life in the mathematical community, Stewart does a fine job presenting the joys and challenges to be expected pursuing a career as an academic mathematician. And he makes a lot of math's most interesting precepts accessible to even the most math-phobic layman.

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