Infinite Powers by Steven Strogatz
Steven Strogatz is known for his ability to make complex scientific topics more approachable and maybe even fun. In Infinite Powers, he explores the history of calculus.
I’ve always had a soft spot for math, but earlier this year I realized that I’d slowly drifted away from the topic that excited me so much during my high school and university years. Reading Surely You’re Joking, Mr Feynman - a biography of the famous physicist - reminded me how much I enjoy the subject and I’ve been reimmersing myself ever since. Much of the best content today is, somewhat surprisingly, online. YouTube is an incredible tool for finding great educational material that’s suited for exactly your level of familiarity with whatever subject you’re interested in. If you’re like me, you’re probably watching 3Blue1Brown videos as we speak.
Most people are not as excited about math and, for some, calculus is the subject of many a nightmare. Kids are still told, at a very young age, that they are not good at math and I suspect that this frame of mind sets them up for years of struggle in science classes. It’s true, many aspects of math are extremely inscrutable: dynamic optimization, random forest regression models and fourier transformations just to name a few. But those only show up mid-way through a math-heavy university education. Most students will have plenty of time to get to know the basic building blocks of mathematics. They start with sums and multiplications, divisions and exponents and eventually progress to calculus. That’s more than enough to get by in almost any profession.
Infinite Powers is the story of calculus. Strogatz trys to introduce the topic in a way that doesn’t alienate people who normally don’t enjoy science. If you don’t remember what calculus is, and I really don’t blame you if you don’t, it’s the study of functions, derivatives and integrals. You can think of calculus as the beating heart of modern mathematics and physics. It’s the foundation for things like 5G phone networks, heat pumps and launching rockets into space. Even ChatGPT is, under the hood, a complex machine that solves calculus problems in a neural network (a sort of artificial brain that loosely mimics actual neurons). Most laws of nature - gravity, electricity, light, heat flow, and so on - obey the laws of calculus.
The zany scientists themselves are almost enough to make this book worth your time. Pierre de Fermat was a French judge and civil servant who, in the evenings, spent his time tackling unsolved math problems, while his wife tended to his 8 children. He was mostly interested in the intellectual challenge of the problems themselves and rarely published his findings, even when those would end up revolutionizing calculus.
Isaac Newton was a child prodigy who, arguably, is one of the top 3 scientists to ever live. He was a polymath who single-handedly built the future of physics, math, astronomy and many other fields. Like Fermat, he avoided publishing his results for years to avoid potential controversy. In a now famous challenge, Johann Bernoulli, a Swiss mathematician, asked the 17th century scientific community to solve the so-called Brachistochrone problem. Newton responded with the solution overnight. Naturally, he published it anonymously.
Strogatz does his best to appease casual readers, but he doesn‘t always strike the right balance between making calculus approachable and showing its emergent complexity. I would have enjoyed even deeper dives into the actual math that drives calculus. Strogatz alludes to all the ways in which mathematics has evolved over time, but other authors in the space have done a better job of explaining the math without dumbing it down. For example, Derek Muller (from the Veritasium videos) is a great source for understanding some of the concepts - like the Black-Scholes model - that Strogatz only mentions in passing.
Think of it this way. There are two parts to understanding math. First, you need to be able to calculate, count and otherwise process formulas. This mostly involves tedious step by step calculations and memorizing rules like “never divide by zero” and “don‘t take square roots of negative numbers (unless you’re comfortable with imaginary numbers)”. This part is very familiar to all of us from school. You “solve for x” or try to “find the derivative of the function y”.
Math is also a way to understand the world. I’ve always enjoyed this part more. A mathematical education is the foundation for understanding chemistry, biology and industry. It helps ground our opinions in the way the world actually works. Say you need to find out how much firewood you need to buy to heat your house for the winter. No matter how you look at the problem - from the simplest way to the most complex one - math will necessarily need to be involved. The calculation itself will be simple, but coming up with the formula is the real challenge.
I’m planning to read a handful of other books on the history of science. I find it fascinating that there haven’t been major breakthroughs in math or physics in decades. There are more PhDs in the world today than ever before, but we haven’t seen a new Newton, Einstein or von Neumann emerge. Turning that thought on its head, it seems that science only needs one person to completely change its trajectory. What do those people have in common? What does it tell us about progress?
Other things worth reading
Tom Whitwell publishes a yearly list of 52 things that he learned. The stories are often unexpected and fun and this year is no exception.
Solar energy has undergone a quiet revolution. In just 5 years, Jenny Chase expects electricity will be free in much of the world - as long as the sun is shining: How the Sun Won.
Online education has also changed incredibly quickly and for the better. Just watch these university-level maths videos from 3Blue1Brown to see what the future of higher education looks like: Differential Equations.