Highly recommended, for a wide range of audiences. This book builds up mathematics from the most basic level, namely counting. More than that, though, it presents mathematics in historical, scientific, cultural, and artistic context. It proceeds through the history of mathematics, teaching theorems from geometry, arithmetic, algebra, and probability along the way. I’ve never really liked geometry, but this book made me find it fascinating. And not just the Euclidean geometry that we learn in high school; [book: Mathematics for the Nonmathematician] spends a lot of time explaining how Renaissance painters discovered the laws of perspective and based them on a rigorous geometry that they invented (namely projective geometry). I imagine my artist friends will be able to relate to this book in a way that they’ve never related to a math text before.

Then the sections on physics are astounding, and make me want to go learn the mechanics that I never really grasped in college.

Throughout, Kline sprinkles his historical discussions and his theorems with applications from as many fields as he can find. Without sacrificing much in rigor, Kline calculates the approximate distance to the moon and the Sun, and tells us how we could estimate the distance from Venus to the Sun without having to fly to Venus and set up a telescope there. He discusses the theory of optics that might have allowed the Greeks to design their famous parabolic mirror to light invading ships on fire. The volume of examples here is truly astounding, and make the book just endlessly fun.

Kline wants you to understand why mathematics is *beautiful*. Why, exactly, do people spend their time on this austere, arcane science? Why did the Greeks turn it into the foundation of true knowledge? And for that matter, were the Greeks as amazing as we’ve made them out to be, inventing branches of knowledge and ways of thinking that have persisted for thousands of years? (Short answer: yes.)

The Greeks believed that mathematics taught us the truth. To skip over a lot of careful explanation, Kline traces this belief from the Greeks to its demise in the 1800s. Mathematics teaches us what follows from certain axioms, but we can choose those axioms for the sake of convenience. We can invent different geometries if they’re useful to us; for that matter we can invent entirely new ways of adding numbers together if those are useful. (Kline has a particularly charming example of how you might build a system of arithmetic around baseball batting averages.) The gap between deduction, formerly thought to be the essence of infallible truth, and induction, formerly thought to be messy and error-prone, has narrowed somewhat. The axioms have to come from somewhere, and they don’t come from God. They come from humans, who pick axioms that seem to approximate some portion of the world around them. *Given the axioms*, we proceed step by step to certain conclusions; but the axioms are ours to create.

This is just such a fun book. I recommend it to anyone with the vaguest interest in how mathematics intersects with our world. And I thank my friend Paul for pointing me in this book’s direction.