Math, around every corner, beneath every façade

kw: book reviews, nonfiction, mathematics

I remember realizing that Geometry and Algebra are the same thing, just using different (very different!) notations. The feeling was one of deep beauty. My lifelong interest in languages, which turned mainly toward formal languages as I matured, was kicked into high gear. Geometry and Algebra are formal languages that happen to have a great overlap of vocabulary; you just need to learn the translations.

But this isn't like English vs French; it is more like written Chinese vs written Spanish. One is entirely pictorial, with nearly no phonetics, and the other is the most phonetic of Western languages...and we won't get into their very different grammatical rules!

As in any pair of languages, Geometry and Algebra have their unique strengths. For example, the geometric proof that one cannot (truly, impossibly can not) trisect an angle using only a compass and unmarked straight edge is rather involved. Recasting it in Algebraic notation, we find that all the operations that are possible with those tools are solutions to linear or quadratic equations...that all means every single one! So geometrically, taking a square root is easy, nearly trivial. But no higher roots can be extracted using an unmarked straight edge. Algebraically, we find that trisecting an angle requires the extraction of a cube root. QED: no geometric trisection is possible.

Later, much later, once I'd (mostly) mastered "college calculus" and (finally) passed the required course in Differential Equations, I realized that I was conversant in four notations for Differential Calculus. These are very consonant, like dialects rather than distinct languages. But each has its strengths also. I prefer "prime" notation to perform implicit differentiation, for example, although I know how to do so in all four.

Fortunately, along the way, I got fully decompressed from high school and college freshman levels of calculus, and realized that the real power of calculus, and of all mathematics, is in learning how to pose a question. A properly posed question literally drags you towards its appropriate solution.

In Letters to a Young Mathematician, Ian Stewart, FRS and a professor of mathematics, drops all of the above bombshells and a great many more, in a most engaging way. His stated intention is "to give an inside view of the mathematical enterprise, and to explain what it is really like to be a mathematician." The work I do requires me to be a working mathematician, sort of a mathematical journeyman. Author Stewart hits every nail square on the head. I kept saying as I read, "Exactly!"

The scope of the book is really remarkable. The author doesn't go over the mathematical landscape in the ordered way we learned as kids: numbers, fractions, ...algebra ... calculus ... and on to things whose name we can never remember. He likens it instead to an inverted pyramid, a 3-dimensional structure, then pops all over the place, including several of those disciplines whose name I forget before the end of the sentence.

Fortunately, he doesn't mentions such things just to impress or awe, but to show connectedness; and he always has a bit of explanation so we at least find out what it's about.

This isn't just a book for mathematicians. I think it especially useful for young people of the "I HATE Math!" variety, to open for them a window onto the breadth and beauty of mathematics. Math (Maths if you're from the UK) is (are) all about patterns, and the fundamental mathematical activity is to operate on some entity to transform it into an item that is more useful or easier to understand. In other words, mathematical operators are Verbs, and the Nouns they consume and produce are numbers, functions and formulas. Thus, learning and performing music are mathematical activities, as is learning the grammar of a new language or following a cooking recipe.

Mathematics takes place in the background, but it underlies everything. Math is how the universe works. Just as we aren't all auto mechanics, but most of us drive cars, so few of us use math in an explicit way, but we do mathematical things all the time without realizing it. Professor Stewart, great mentor that he is, shows us just how much that really is.