## Aaron Kaufman reviews Luke Heaton’s “A Brief History of Mathematical Thought”

I got this book in the mail. It looked cool but I didn’t feel I had time to read it. A few decades ago I read this wonderful book by Morris Kline, “Mathematics: The Loss of Certainty,” so I figured I’d have a sense of what most of Heaton’s new book would cover. I would’ve just read the last chapter to see what’s been happening in math during the past 30 years, but then Aaron happened to come by my ofc and it seemed to make more sense for him to review the whole thing.

So here’s Aaron’s review:

Luke Heaton’s A Brief History of Mathematical Thought is a delightful, accessible, and (mostly) well-written jaunt through foundational mathematics. Its greatest strength lies in its sense of narrative: Heaton draws a curvilinear path, starting with the Babylonians and Egyptians, passing through the Greeks, Romans, and the mathematicians of the Arab world, and encompassing everything from mechanical calculus to non-Euclidean geometry and computation.

Despite his tendency to wax confusingly poetic, his conceptual explanations are brilliantly lucid and accompanied by concrete examples. In Chapter 2 Heaton describes Euclid’s Algorithm, a method for determining the ratio of lengths, and its fundamental contributions to early art and architecture. To find the ratio of length a to length b, construct a rectangle with sides of those lengths. Then, Heaton charmingly describes the following steps:

1. Draw the largest square that fits inside your rectangle.

2. Squeeze in as many of those squares as you possibly can. If you can fill the entire rectangle with your squares, you are finished. Otherwise, there will be a rectangular space that has not been covered by squares. We now take the remaining rectangle and apply step 1.

Having subdivided the rectangle completely, the size of the smallest square is the greatest common divisor of the lengths of the two sides. In such parsimonious language, Heaton discusses methods for calculating square roots, functions of π, and the Golden Ratio, as well as elucidating infinite series, the fundamental theorem of calculus, P = NP, and many other concepts.

In his concluding chapter titled “Lived Experience and the Nature of Fact”, which everything from language, truth, and society to the nature of purpose and meaning, Heaton goes a little far afield. While he is uniquely both well-read in his history and adept at providing intuition, he is clearly possessed of an awe for the beauty in the mathematical world which he struggles to impart. But none of that is to detract from the huge amount of value there is in understanding a subject in its historical context. No matter how much my dad likes the Grateful Dead, it’s impossible for me to appreciate them as much as he does, absent that particular zeitgeist. In the same way, much of the beauty of non-Euclidean space and partial derivatives can only be fully grasped relative to what came before. This is where Heaton’s work shines.

They’re not kidding when they say that math is “unreasonably effective.” I talk a lot about the substitution of programming for math—my slogan is, “In the 20th century if you wanted to do statistics you had to be a bit of a mathematician, whether you wanted to or not; in the 21st century if you want to do statistics you have to be a bit of a programmer, whether you want to or not—but math is needed to make a lot of the programs work. I’m not talking about the silly math with “guarantees” and all that; I’m talking about the real stuff that drives NUTS and VB and EP and all the rest. Also the simpler math that I throw up on the board in my survey sampling class, the sqrt(1/n)’s and the 12^2 + 5^2 = 13^2 and all the rest.

As a frequent user of math, I have a much better sense of its role in life than I used to, back when I was a student. I guess it could be worth rereading Kline’s book, seeing if it all makes sense, and thinking about what its hypothetical updated final chapter should say.