Let me preface this review by saying that I’m biased; this is one of my favorite books. Not just favorite math text, but overall favorite. This is one of the few books that I’ve read multiple times. It’s also from 1978, but I’ve seen used copies for sale on many websites. And finally, it clocks at a binary satisfying 256 pages.

This book is a description (with worked examples and discussions!) of various mathematical modeling methods. I can’t stress enough that this is about truly designing intelligent models for problems; you won’t find the term “machine learning” anywhere in the text. After a brief intro to “what is modeling”, it dives right into various techniques with real worked problems. Early sections build a few basic tools, but most chapters are self-contained. It’s easy to jump into a chapter or specific problem and come out with a full understanding. In terms of difficulty, most of the math is actually straightforward. Any technical undergrad or sharp teen can understand. And if you don’t follow, an appendix on probability fills most gaps. By the end, you’ll have explored a swath of applied models.

I want to say again that this text is not about data analysis. Models are fit and backed up with data as appropriate, but the emphasis is on understanding how to construct intelligent models. Your latest neural net might reach the same conclusions, but it will require mountains of data, and you still won’t know why the model is what it is. This really reflects the era of the book. In 1978, computing power was limited, so you invested proportionally more effort into building a great model to minimize required processing. A skill that’s still useful today.

Without giving away spoilers, I want to highlight a few of the worked examples. “The Nuclear Missile Arms Race” in Graphical Methods is one of my favorites. Without needing any numerical data, Bender derives a qualitative effect of missile technology and disarmament. His simple figures make crystal clear his argument, much better than many modern distracting graphics. Also relevant these days is “Are Fair Elections Possible?” in Potpurri. While this is one of the more theoretical math heavy sections, I still found it very accessible. It features a logical proof about trying to avoid a dictator in an election with more than two candidates. And it even gives a hypothetical real world example of deciding between contract choices. This is another situation where we can learn something even in absence of data. Finally, close to my heart is “Dynamics of Car Following” in Local Stability Theory. This is a “typical” problem in the book, with a three part description. After laying out the basic problem, it shows a short table of data that we’ll use to build or verify our model. Then it derives a reasonable model from first principles in (physics, biology, economics, etc). Finally, it applies the data, discusses the results, and acknowledges the limitations. No pretentions, just analysis. If you’re interested in self-driving cars, you also can check out my model (with fun graphics!) on github. This book contains dozens of worked problems; you’ll probably identify with several of them.

While this is a wonderful insightful book, it doesn’t cover everything under the sun. I mentioned it’s from 1978; modeling has advanced since then. Cheap computing opens up new possibilities, and traps. One great thing about easy computing is exploratory data analysis. If you can visualize the data the right way, the model may become obvious. Also, Bender admits that no single model is deeply developed. He even mentions that he had to cut an “in-depth” chapter, I think it’s still worthwhile. My final tally, 97/100 (5 point scales are an archaic blight on the world). A wide variety of math and science topics, all accessible and modular make this a must read!