100 books like What Is Mathematics, Really?

I’m interested in how mathematicians create mathematics but this book made me realize that learning mathematics is also a form of creativity. Each of us has created our understanding of mathematics as we were growing up. We are all creative!  

What is amazing about this book is that even children as young as six months possess rudimentary mathematical concepts, in particular, the concept of number. (Actually, Carey shows children have two distinct ways of thinking about numbers). The concept of number is built-in. That’s amazing to me! The mastery of counting numbers, 1,2,3,… is a great creative leap in the development of the child. This leap is followed by a series of further amazing accomplishments, for example, the insight that a fraction like 2/3, is a completely new kind of number (and not just a problem in division). How do kids manage to accomplish such radical changes in their concept… show more.

Lots of people have a priori ideas about what mathematics is all about but Lakatos had the brilliant idea of looking at what actually happened. His book is all about one famous theorem: “for all regular polyhedra, V – E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces.  Think of a cube where V=8, E = 12, F = 6.  

We tend to think that mathematics proceeds from a well-defined hypothesis to conclusion. But that is only the finishing step. Along the way the definitions keep changing as do the hypotheses and even the conclusion. Everything is moving! This is what makes doing mathematics so exciting!

This is another book about the new research into how babies think. I am excited about this research because of its implications for how people learn mathematics and how researchers create math. This book taught me something important about how we all think. Gopnik distinguishes between what she calls flashlight consciousness and lantern consciousness. Flashlight is the way adults think. You focus on one thing at a time and give it your full attention. But babies, she claims, use their minds differently. Their lantern consciousness is unfocused and is aware of the big picture all at once.  

So what happens to lantern consciousness when you grow up? The answer is that creative individuals use it and alternate between lantern and flashlight consciousness. When we are creating or learning something new, we have to drop back to lantern consciousness. Logic comes from flashlight consciousness and, by itself, will never produce anything… show more.

It’s a little weird that this book should find a place on my list. It’s a book about how society has become resistant to anything that is difficult and painful and the kinds of people that we have become as a result. But mathematics is difficult! To understand mathematics you have to think hard, sometimes for a long time. Moreover understanding something hard is discontinuous, it requires a leap to a new way of thinking. You have to start with a problem and this problem might be an ambiguity or a contradiction. A is true and B is true but A and B seem to contradict one another. When you sort out this problem you will have learned something.

The moral here is to embrace things that are difficult if you want to learn significant new things. “No pain, no gain.” You don’t have to worry about some super AI… show more.

David Hilbert was the most important mathematician at the dawn of the 20th century. In 1900, he gave the mathematical community its homework for the next 100 years setting out the list of open problems that had to be solved by 2000. While to the rest of the mathematicians, he may have appeared as their professor, he was also the class clown. As colorful and funny as he was brilliant, you cannot but come away loving this great mathematical genius.

A renowned historian of science, Clagett carries the story of Greek science forward all the way to the sixth century CE—a span of 1200 years. From that point in time, Greek science passed into the hands of Islamic scholars who advanced it further, especially the mathematical sciences.

This book is not, like ours, organized chronologically and developmentally but according to modern scientific domains—biology and medicine, mathematics, physics, and astronomy. And it focuses on specific scientific inquiries, while we focus on more general and fundamental things like ontology (what exists), cosmology (the overall structure of the universe), the laws of nature, and the drivers of change and motion.

This book is thus a complement to ours in its wide historical sweep and in what it highlights.

This remarkable book is authored by Francis Su, the past president of the Mathematical Association of America. The author describes human flourishing as follows: “a wholeness of being and doing, of realizing one’s potential and helping others do the same, of acting with honour and treating others with dignity.” He explains how human beings, of all ages and ability levels, can experience flourishing through the doing of mathematics.

In each of the final twelve chapters, the author explores a trait of mathematics, how it relates to our journey as humans, and how the development of each trait enables us to flourish: Exploration, Meaning, Play, Beauty, Permanence, Truth, Struggle, Power, Justice, Freedom, Community, Love. As a mathematical researcher and educator, I have experienced these traits first-hand, and recommend this book to anyone and everyone.

Discover Condorcet!

Most people, when asked to name a philosopher who wrote about inequality, would think of Rousseau. Condorcet was the last of the Encyclopédistes, young enough to experience the revolution in 1789—sadly, also one of its victims.

Unlike his philosopher colleagues, he participated actively in public policymaking, first in the Ministry of Finance, later as an elected member of the Legislative Assembly after the revolution. He chaired an organization working for the abolition of slavery. He argued for equal rights for women before Olympe de Gouges and Mary Wollstonecraft had published their more well-known pamphlets. He co-authored the Declaration of the Rights of Man and of the Citizen and also wrote a proposal for new constitution for France.

Most importantly, he realized the fundamental role of education as a means to reduce inequality and liberate mankind, and he even developed curricula for the various stages of a general… show more.

From a strictly philosophical perspective, the emergence of category theory as a unifying paradigm rivaling set theory is probably the most important development in mathematics in the last half-century.

But for philosophers without a lot of mathematical background, learning even its rudiments can be daunting. Among many introductory texts (Lawvere and Schanuel, Awodey, Riehl, Spivak), Cheng’s book stands out as perhaps the friendliest and most accessible.

She does not forego rigor, but she isn’t afraid to put aside precise formalism when necessary for intuition and clearer understanding. Her book takes the reader from mathematical beginnings through category theory’s core constructions to glimpses of higher-order categories (one of Cheng’s areas of expertise).

A mathematically novice philosopher who wants to understand the basics of category theory couldn’t do better.

This is a sequel to Alex Bellos's bestseller Alex's Adventures in Numberland, but more focused on applications of mathematics to the real world, especially through physics. Many of these were known to me, particularly when they involved calculus, but I greatly enjoyed Alex's distinctive and novel way of putting across sophisticated ideas, in part by interspersing them with personal interviews with mathematicians of all kinds.