Don Hosek - Past reading - Mathematics

An assortment of different levels of material. If you're looking for good reading, the best books are usually the cheapest. Imagine that.

What I've been read in the past - Mathematics
DateAuthorTitle
What Number is God? by Sarah Voss
[Finished 10 March 2015] See my review at dahosek.com

Everything and More: A Compact History of Infinity by David Foster Wallace
[Finished 19 September 2014] Wallace’s “compact” history is actually fairly involved. It’s engagingly written, although oddly targeted, apparently at math-English double majors or somesuch. I found a lot of the discussion elucidating although I continue to be unclear on what keeps the rational line from being a continuous space or the reason why c is not equal to א1 (and his distinction between “number line” and “real line” seems to be unique to this work).

Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter
[Finished 7 August 2012] This is one of those books that it seemed that everyone but me had read (or at least had on their desk) when I entered nerd school. Coming to it now a quarter century later, I can see the appeal of it, and I think it would be fun to teach a course using this book as the central text. I was most interested in the Gödel part of the book, which was a central concern of the first two-thirds and was really well-done, providing a good in-depth look at the meaning and consequences of the incompleteness theorem. As Hofstadter moved into the domain of artificial intelligence, the book began to really show its age. Aside from the obvious case of Hofstadter believing that a computer would never beat the best human chess player (which, in fact, happened in 1997, 18 years after the book was published), the field of artificial intelligence has moved in directions rather different from what Hofstadter imagined at the time of this writing. Still, despite its age and its flaws, it’s a book well worth reading.

My Best Mathematical and Logic Puzzles by Martin Gardner
[Finished 22 December 2010] A fun collection of puzzles. They seem to be arranged, more or less, in order of increasing difficulty. Some of the earliest puzzles in this volume I was able to do in my head, while as things moved on, I needed paper or a peek in the back of the book.

Number Theory and its History by Oystein Ore
[Finished 30 October 2009] Written well before the modern computing age, this book has a tendency to focus a bit on computational exercises, although I have to admit that having worked through the book that the heavy computation does a lot to bring a deeper understanding of the mathematical concepts underlying the discussions. It would be a nice introduction to number theory for advanced high school students or non-specialist college students.

How to Lie With Statistics by Darrell Huff
[Finished 7 October 2005] Even though it’s half a century old, this is still a great overview of understanding what statistics mean. I’ve assigned this as reading to my math for liberal arts majors students after I became frustrated with the standard approach of teaching the material which only focuses on how to calculate the statistics. Frankly. there’s no reason to memorize formulas for finding a correlation coefficient, for example, but students had really better understand what margin of error and confidence level mean if they want to be able to function in today’s society. This book is a pretty good start along that road.

A History of Pi by Petr Beckmann
[Finished 11 February 2005] A curious book. The mathematical history is generall pretty good, but as he himself notes, Pi tends to attract crackpots and he himself tends to slip into that category quite frequently. The book is frequently marred by somewhat bizarre diatribes against religion (especially Christianity in general and Catholicism in particular), Latin culture, communism, liberalism and nearly anything else that strikes Beckmann’s fancy. I’m guessing that a big part of this is his own personal history: He came to the US from Soviet-dominated Czechoslovakia in the 60s and apparently decided that Goldwater conservatism was for him (although his strong anti-religion stance would make him somewhat alienated in the contemporary Republican party).

Some of the mathematical presentations are needlessly difficult to follow, and he has a tendency to skim over material that would bear more careful explication, but the mathematical history outside of the diatribes is worth reading. That is, if you can find your way past the diatribes.

What Is Mathematics?: An Elementary Approach to Ideas and Methods by Richard Courant and Herbert Robbins, Revised by Ian Stewart
[Finished 4 February 2005] Sort of the rigorous version of The Art of the Infinite, this book covers, at a fairly quick rate, a wide smattering of mathematical thought including a lot of byways that aren’t taught a whole lot anymore (projective geometry, anyone?). Perhaps some of the more interesting parts come down to asides about axiomatics towards the ends of the book.

It’s hard to tell who the book is aimed at. It purports to be something that a college freshman could understand, but at the same time it also assumes some familiarity with most if not all of its topics.

At times, the coverage is remarkably thin, giving only the vaguest sense of what a topic is about (for example, while knot theory is mentioned along with the Alexander, Jones and HOMFLY polynomials, no indication of how those polynomials are derived is given).

That said, it was stil an enjoyable read. I’m thinking that this would be best read as a culmination of an undergraduate math major or perhaps as a prolegemenon to graduate studies in mathematics.

How to Solve It: A New Aspect of Mathematical Method by G. Polya
[Finished 25 November 2004] At times, this book seems a bit quaint, clearly written in a different age. And the examples make it unclear exactly who the book is written for, high school teachers, presumably. It was a bit repetitive at times, but ultimately rewarding despite its flaws.

The Nothing That Is: A Natural History of Zero by Robert Kaplan
[Finished 10 November 2004] You’d think that there’s not enough material about zero to make a book, and you’d be right. This was nowhere near as successful a book as Kaplan’s later The Art of the Infinite. The first half, detailing the spread of zero, was frequently boring and the mathematical section was painfully short (although the history of limits and Napier’s contribution was quite interesting). Better to skip this one and stick with The Art of the Infinite.

Q. E. D. Beauty in Mathematical Proof by Burkard Polster
[Finished 26 October 2004] A fairly short (58pp) book, which seeks to give a quick overview of some attractive proofs in elementary mathematics (primarily Euclidean and solid geometry, but with some excursions into algebra and elementary number theory). The main focus is on showing the beauty of the proofs, and by including one short page of text facing one page of woodcut-style illustrations the book does a pretty good job of accomplishing its modest goal.

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire
[Finished 17 May 2004] One of the more compelling books I’ve read about mathematics. There were times when Derbyshire’s attempts to make the math easier to digest frustrated me (I kept thinking, “just give me the formula already!”), but for the most part it was pretty easy to follow. This would be a good introduction to analytic number theory for just about anyone and were I teaching an advanced number theory course at the university level, I think that I would assign this as reading over the break to all my students.

The Art of the Infinite: The Pleasures of Mathematics by Robert Kaplan and Ellen Kaplan
[Finished 18 December 2003] A wonderful wonderful book, covering a wide spectrum of mathematics. The heavy lifting is largely relegated to an appendix leaving the body of the textbook accessible to mathematically inclined high school students as well as those with more mathematical experience. At times, the ventures into poetry seem a bit forced, but the biographical details add some color for the casual reader and the ability to connect widely varying areas of mathematics is just plain delightful.All high school math teachers should read this book. All prospective math teachers should read this book. This should be required reading for all math majors. Can you tell that I love this book?

In Code: A Young Woman's Mathematical Journey by Sarah Flannery with David Flannery
[Finished 7 September 2002] An interesting exploration of basic number theory and encryption concepts written by a talented high school student whose father is a professor of mathematics. It provides an easy way to approach the topic and the discussion is more than a little accessible.

The really hard mathematics is stashed into an appendix so that the casual reader can follow the story of how the author came up with an encryption scheme (which turned out, later, to be rather insecure) as a high school student.

Flatland: A Romance of Many Dimensions by Edwin A. Abbott
[Finished 28 November 1998] Classic nerd fiction. Some of the allegory is a bit forced and frankly it gets a bit dull at times. Back when I thought four-dimensional geometry was the coolest thing on earth though, this would have been my favorite book.

Fundamental Statistics for the Behavioral Sciences by Robert B. McCall
[Finished 5 May 1998] Generally fairly lucid, although the ANOVA section lost me pretty badly as did the beginning section on inferential statistics.