Mathematical Pamphlet

I read a lot, perhaps to the detriment of my eventual graduation plans. Recently, I've been enjoying books of "recreational mathematics." This is a combined review of all such books I've read recently.

You might wonder what I mean by "recreational mathematics" since clearly all math is fun . Well, I'm talking about math that is intended to be fun before being instructive. It usually mixes in a bit of history and description of famous math personalities for added interest, and has much much MUCH more motivation than your average math textbook; I find that this helps me immensely in remembering the math. However, I am not talking about books only about the people, places, and times of mathematics. All the books in this review have, to some degree, real and rigorous mathematical content. But none of them are a "pure" exposition of consequences from axioms. It'll be plenty clear if you take the time to find and read some of these wonderful books.

---

Journey Through Genius by William Dunham

If you read just one book, read Journey Through Genius. It hits many mathematicians and topics, and is a great overview.

Each chapter is centered on one "great" theorem, but is arranged beautifully such that the theorem is merely the peak of the story arc. For example, the first chapter on Hippocrates' quadrature of the lune gives 17 pages to classical constructive geometry, discussing the disconnect between "number" and "length" and showing the quadrature of various polygons. The proof of the great theorem is then about one full page. After the main result, an epilogue eases you out with a discussion of some following results, in this example about 7 pages, finishing with the impossibility of squaring the circle with straightedge and compass.

Perhaps my favorite thing about the book is that it really feels like it is about math, and the artistry with which great theorems are proved. People, places, and the passage of time appropriately spice up the story; reading Newton or Euler's work doesn't seem like a disjoint concern from that of Archimedes, but rather like a change of camera angle in a (highly metaphorical) movie.

Journey through Genius ends with Cantor's theorem, so don't expect to gain new appreciation for modern abstract nonsense - I suspect it will be a long time before a recreational math book tries to tackle what they teach in grad school today.

---

Yearning for the Impossible by John Stillwell

This book is definitely my second favorite next to Journey Through Genius. Most books are themed by mathematical content or a famous mathematician, or deliver a historical overview, but this one manages to truly have an emotional theme just as implied by the title. I literally felt the suspense of wondering what would be discovered next as the mathematicians in the book are practically forced to discover irrational numbers, complex numbers, projective geometry, infinitesimals, hyperbolic space, quaternions, prime ideals, and uncountability.

The treatment of prime ideals especially struck me, since I've never seen an ounce of motivation elsewhere, and certainly nothing as down-to-earth as providing unique factorization for Gaussian integers. I'm sure more sophisticated uses abound, since they bother teaching it in the first quarter of Algebra, but since I study programming languages my grad math experience is somewhat ad hoc… I learn math for kicks, and count on motivation and formalism to be presented together, which is very uncouth in modern math culture! Anyhow, I sure wish I had read this book years ago.

---

Four Colors Suffice by Robin Wilson

This one I just got for Christmas, and makes it to number three in my list because it treats a different sort of problem that has a very different sort of proof. The subject, of course, is the Four Color Theorem. In order to create an entire book for the history of this theorem, Wilson adopts the interesting approach of describing many false starts and negative results - of course, they really do form a major part of the story, and the eventual correct proof relies on the techniques of an early incorrect proof. Also nice are the numerous primary source documents, such as the first document where DeMorgan communicates the problem to Hamilton, and where Story says (translating into modern Santa Cruz English) "Dude, this problem is gnarly!" I hadn't realized that this problem moved through such prestigious hands as DeMorgan, Hamilton, Peirce, Euler, Cayley, and Kempe (apologies to the other names who will simply have to wait 100 years to achieve proper prestige - I visited the Boston museum of science recently and they have this wall with a chronology of math where pre-1900 there are perhaps 1-4 famous people alive at a time and post-1900 it explodes to fill the entire wall vertically, so I can't be expected to remember them all )

Anyhow, as the book progresses, results build up - but not results that aid the proof, rather they are results that prove how large any counterexample must be, and how many special cases must be considered. And then there is the moment where over a thousand cases are checked by computer algorithm, with the expected reactionary backlash against such a proof. Amusingly, the proof starts with a ton of tedious work by hand yet it is the computer part that people didn't trust! Subsequent work automates the first part of the proof so now a computer can replay the proof at home in a short while. Especially in the context of mechanizing the metatheory for programming languages, this is an interesting issue.

---

Euler: Master of us All by William Dunham

This book has a very similar feel to Journey Through Genius, but is focused entirely on Euler. The same good intro/climax/epilogue format is followed for each chapter, and the same wonderful style that is distinctly Dunham. Added to that, Euler's math is truly artistic; I feel like yelling "BAM!" at the end of each proof. I absolutely loved this one too, but don't read it right after Journey Through Genius or the tone will be getting old.

There is much discussion of rigor issues with Euler's approach, which are certainly valid in the absolute sense that he was working with terms and notations that weren't clearly defined. But in my opinion it isn't reasonable to retroactively apply rigorous reinventions of terminology.

---

Imagining Numbers by Barry Mazur

This one was a gift that got me started on this train of books. It is OK, but in retrospect not as good as the others, and complex numbers make a prominent appearance in many of these books, particularly Yearning for the Impossible. On the upside, it is short and nice, and has poetic and philosophical interludes, making it a unique experience.

---

Fearless Symmetry by Avner Ash and Robert Gross

I read this because it had a catchy title . I recall it was pretty good, and had content and motivation I hadn't seen before, since I haven't yet had time to reach Galois theory in my sampling of math courses. It has been a while since I read it, and it has been returned to the library, so I can't say anything terribly specific about its content or presentation, but I certainly recommend reading or re-reading it to anyone engaged in group theory or Galois theory in particular. The more of it you already know, the quicker you will blaze through it anyhow. Apropos a comment I made in the review of Journey Through Genius, this book is probably the closest to actually touching abstract and somewhat difficult modern topics.

---

The Pea and the Sun by Leonard Wapner

This one is all about the Banach-Tarski "paradox" so it has a natural advantage because of the curiosity of the subject matter. Unfortunately, it doesn't really take advantage of it. Before getting to the main theorem, it develops many analogous smaller versions of the paradox to get the reader situated, and I found it rather dragging. Simpler versions of the paradox do make great party tricks, if you know them well enough to deliver the proof quickly and with flare, and if you go to the kind of parties that I do…

---

Unknown Quantity by John Derbyshire

This one is really popular and well-reviewed, so I'm almost afraid to say it was my least favorite of all I've read. It was just too heavy on the history, and not exciting enough to keep my interest. The book is also quite long, while I like knowledge in bite sized chunks… I'll stick to mathematical pamphlets!

Filed under: Mathematics, Reading by Kenn