As the title suggests, this book is written in the form of letters to a "young mathematician", offering advice and generally discussing what mathematics is and what it means to be a mathematician. The back cover promises that it "tells readers what world renowned mathematician Ian Stewart wishes he had known when he was a student", and I was intrigued because I had a mixed experience with mathematics in university and always wondered what I could have done differently.

I have to say, my experience with this book was also mixed. First of all, lest the title mislead, I think it's actually more suited for a general non-mathematical audience than for someone who's actually in the process of becoming a mathematician--which wasn't a problem for me, since I read it for general interest and not as an aspiring mathematician.

Stewart raises a lot of interesting points here, but he often doesn't discuss them in as much detail as I would have liked. I think this is part of a series, and it was probably a requirement that the book be barely 200 pages long, but there are times when a bit more depth would have been nice. It was good in a way that the book left me wanting more, and it did include references to other interesting reading throughout, but it wasn't quite satisfying enough in itself. Many people will probably appreciate the brevity, though.

A more serious concern for me was the fact that I just didn't agree with some of his more important points. I know, he's the mathematician and I'm not, but still. The book got off to a bad start, in my mind, with the initial chapter entitled "Why Do Math?". Stewart's reason, briefly, is that math is everywhere. There's a bit too much focus on the utility and not enough on the beauty of it. The same is true even in his chapter on proofs: "it would be silly to suggest that this agreement [with experiment to an accuracy of nine decimal places] is an accident, and that no physical principle is involved.... [but] it is equally silly not to try to find out the deep logic that justifies the calculation. Such understanding... will surely advance mathematics." So even proofs are presented not as necessary, but merely as useful (a view that he steps back from slightly in the following chapter, but to my mind, the damage was done).

Another point that left me wondering is his claim that not everyone can be a mathematician, because "originality is one of those things that you either have or you don't." I really appreciate the fact that he expressly disagrees with the politically-correct view that everyone can be good at everything, but I think he simplifies the issue a bit too much. In particular, I'm just not convinced that originality is the key. He compares it to the attempt of a tone-deaf person to become a great musician, but there's nothing in there to make me think that creativity is the problem rather than some more basic musical/mathematical aptitude. His explanations about how he excelled at math without effort from the time he was 10 didn't convince me either; the mathematics that one does at age 10 is so different from "real" mathematics that it can barely be called the same subject. Creativity certainly isn't the main factor there. I wonder whether Stewart's various unconvincing arguments are a direct result of the book's extreme brevity; maybe I would have been more convinced if he had elaborated his position further.

On the other hand, I think the fact that I have so much to say about this book speaks in its favour. Regardless of whether I agree with everything Stewart says, he makes plenty of interesting and provocative points in a short space. And there's a fair bit of humour, too, including a whole chapter on what not to do as a tenure-track professor that consists mainly of humorous anecdotes about various mishaps ("I was once abandoned inside a Dutch mathematics building when my hosts went off to the parking garage to go to a restaurant. I had to make my escape through a window, setting off a burglar alarm.") So in the end, I think I would cautiously recommend this book, as long as you don't take Stewart's views as the be-all-and-end-all when it comes to what mathematics is. ( )

This is a short, quick read with many interesting ideas. Ian Stewart wrote a series of letters to his niece, Meg, as she grew from a high school student to a tenured professor of mathematics. The book is his "attempt to bring some parts of A Mathematician's 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 book kept my interest because Stewart scatters his ideas about teachers throughout it. Otherwise I may have stopped reading. There simply is not enough mathematics of substance in it for me. Except in chapter five, Surrounded by Math, where he discusses "bird crystals" and about which I posted earlier (http://meeyauw.blogspot.com/2007/12/p...). And except in what I call the Doublets chapter (chapter eight: Fear of Proofs).

Lewis Carroll invented the game of doublets in which you take a word (such as WARM) and change it, one letter at a time, to another word (such as COLD). Each time you change a letter you must have a real word. Stewart proves that at some stage you must have a word that contains exactly two vowels. There was a time when I played doublets, so this proof interested me. You have to consider W and Y to always be vowels in order for this proof to be valid. What interests me, and this is something Stewart never addressed, is that in all of his examples, the vowels are double vowels (OO, AA, etc). I spent some time finding doublets games on the Internet and while all the words in each game have two vowels, they are not necessarily double vowels. For example, WARM — WORM — WORD — CORD — COLD: if you accept that W is a vowel (see page seventy-three for Stewart's explanation), then WARM and WORM do have two vowels.

I found other examples where the vowels are not positioned next to each other:

GIVE to TAKE: GIVE — GAVE — RAVE — RAKE — TAKE

Strictly speaking, these are examples of Stewart's proof. But he never offered one of these as an example. Perhaps he should have.

Stewart writes about teachers: "The best teachers will occasionally, perhaps more than occasionally, make you feel a bit stupid." I am not sure, despite reading his rationale, that this is true. Other statements that he makes concerning teaching ring very true with me: "You'll find that teaching math to others improves your own understanding. But it's only natural to be a little nervous, and I'm not surprised that you think you are 'not at all prepared' for your teaching responsibilities. . . But the nerves will vanish as soon as you get started." I experience that "stage-fright" at the beginning of every college semester and public school academic year.

One reason that I enjoy teaching developmental mathematics at the community college is that I may be able to help someone discover that she can succeed at something she used to think beyond her ability. Stewart points out that we need to put ourselves in the student's position and help her understand the material. "[W]hat seems perfectly obvious and transparent to you may be mysterious and opaque to someone who has not encountered the ideas before." I have learned, through experience, that mathematics instruction must be kept simple: " Stick to the main points, and try not to digress if doing so requires the students to understand new ideas that are not in the syllabus, however fascinating and illuminating they may seem to you." That has been a difficult lesson for me to internalize over the years. There seem to be thousands of interesting side roads that we can take in every lesson. Keeping our course objectives in front of us at all times can prevent us from straying off course and bewildering our students.

Validation by Stewart of what I have learned about teaching was not reason enough for me to read this book. It is math-lite. I suggest that if you are interested in it, that you wait for your library to purchase it. ( )

For the "non-Mathmo", this book will provide a fascinating insight into the career arc of a Mathematician from high school beginnings right up to post-graduate University and research positions. However, it is so much more than that: as is now the norm for Stewart, the work is generously peppered with accessible, entertaining anecdotes which serve as appropriate interludes at any point he is in danger of entering 'yawn booorrring' territory. These anecdotes also elucidate exactly what it is that Mathematicians do and go some way to explaining to the layman just what the appeal of the subject is. Particularly worthy of praise is the fact that Stewart has not relied on the clichés of popular Mathematics which we have all seen before (The Death of Evariste Galois, Sophie Germain etc.) and instead injects some new, lesser known stories into the mix.

Much more narratively interesting than his other works, the apparently ageless Stewart corresponds with a young Mathematician giving advice as she rises, over the course of a decade or so, up the Maths student path eventually to the career ladder. The nature of their correspondence at first has a sinister edge, as if he is grooming her, but any such suspicions are swiftly dropped as it becomes more apparent that the letter format is merely a tool Stewart is employing to spread his message about the world of Mathematics. The personal touch gives Stewart's page-voice a warmth that successfully dampens the often smug and arrogant tone present in his other works.

To the Mathematics student, this work is both an invaluable motivational tool and a useful tome of advice for any career or study choices that the student finds themselves having to make. As a Mathematics student myself I often felt while reading the book that I could slip into the role of Meg (the eponymous "young Mathematican") and feel the wisdom was being addressed to me personally. Chapter 14's anthropological approach to the academic environment (a clever, almost tribal analysis of "gift giving" and "tribute") is essential reading for anyone in the academic field intending to manipulate the machine in their favour. Academia is a highly politicised field and Stewart's insights will minimise frustrations if carefully considered.

While it is a perpetual problem for the popular Mathematician to dumb down the subject enough to make it accessible, Stewart has achieved something truly praiseworthy here, a triumph of style that manages to address two audiences on two separate levels in one single voice. If before his work and the work of his popular Mathematics writing peers (especially Keith Devlin) was really only suitable for Mathematicans and students in Mathematical Sciences then Letters to a Young Mathematician marks an important turning point in contemporary popular Mathematical literature for being a work that can be appreciated truly outside of the academic context. Stewart is finally taking Mathematical literature to the places that Russell Stannard did with his Uncle Albert books and Jostein Gaarder did with Sophie's World and for this he can only be praised.

Math as an educational and career specialty.