100 books like The Philosophical Baby

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.

I was fascinated by Erikson’s theory of the eight stages of human life, from infancy to old age. At each stage, he says, we must solve a dilemma, starting with: “trust or distrust?” Our ability to mature properly depends on meeting the challenges of each stage, which then propels us to the next stage.

I was disturbed, however, by the implications of his theory: if we fail to succeed in any given stage, our future development is permanently compromised. In short, we never really fully grow up. 

Reuben Hersh is responsible for a revolution in the way we look at mathematics. His main idea is very simple: mathematics is something that is created by human beings. Isn’t that obvious, you say? Not if you believe that mathematics is there even before life itself, that it is built into the nature of reality in some way. In philosophy, this view is called Platonism. Hersh had the radical but obvious idea that if we want to understand what mathematics is we should look at what mathematicians actually do when they create mathematics. Like all great ideas it can be stated very simply but the implications are enormous.  His ideas are what got me started writing my own books about math and science.

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!

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.

“Never rush childhood but let it ripen in the child”: Jean-Jacques Rousseau was the first philosopher to see children as more than adults-in-training. As someone who loved revisiting childhood as a parent, I simply adored this celebration of the world of a child.

In this philosophical novel, Rousseau is tutor to Emile from infancy to adulthood, and we watch Emile discover the world without any use of schools or books: “I hate books!” says the world-famous author. Purely through experiential learning, Emile matures, falls in love with Sophie, and becomes a model citizen. Rousseau pioneers the idea of learning through natural consequences: if Emile breaks the window in his room, he gets to enjoy a cold room. 

I loved hearing the stories of these men, both Harvard College students (including the young John F. Kennedy) and Boston “townies,” as they mature from ages 20 to 90. The largest long-term study of human development, each of these 600 men was interviewed and studied every two years, creating a vast data set for students of human development.

George Vaillant, a Harvard psychiatrist, decided to test Erik Erikson’s theory of the stages of life using the Harvard Grant data. What he found gives hope to all of us late-bloomers: early deficits could be redeemed by later successes. What matters, he found, is not IQ or perfect health but close relationships with family and friends. “Maturation is the evolution of teenage self-centeredness into the disinterested empathy of a grandparent.” 

David Norton really shakes up our assumptions about human lives. According to him, we develop within the stages of life but not across them. The goal of life, he says, is self-actualization, meaning to become who we really are, but this goal excludes childhood because children don’t have a self to actualize and old age because the elderly cannot actualize their selves.

At each stage, we solve problems unique to that stage: for example, reconciling ourselves to death is a stage that might happen at any age from 18 to 80. Each stage is unique and cannot be compared to other stages. I found Norton’s book to be very insightful and thought-provoking. 

Many undergraduate mathematics books – even those aimed at new students – are dense, technical, and difficult to read at any sort of speed. This is a natural feature of books in a deductive science, but it can be very discouraging, even for dedicated students. Houston’s book covers many ideas useful at the transition to proof-based mathematics, and he has worked extensively and attentively with students at that stage. Consequently, his book maintains high mathematical integrity and has lots of useful exercises while also being an unusually friendly read.

Polya was a great mathematician who knew what counted (after all, he made major contributions to combinatorics, the mathematics of counting). He thought hard about what he was doing when working on problems in mathematics, developing a mental process that lead to creative breakthroughs and solutions. Polya’s problem-solving method is broadly applicable to domains other than mathematics, and this book features many nice puzzles to improve your thinking.

Algorithm design is challenging because it often requires flashes of sudden insight which seem to come out of the blue. But there is a way of thinking about problems that make such flashes more likely to happen. I try to teach this thought process in my books, but Polya got there first.