Book Review: Mathematica

A mathematical mind isn’t a gift. It’s something cultivated from within. Mathematicians aren’t born, they are made.

Mathematics is the by-product of our own power of imagination and intuition, a power we can learn and grow. David Bessis’s book is an excellent debunk of the notion that mathematical skills are innate abilities. According to him, mathematics is about intuition, not about logic. To really understand a mathematical idea, one must be able to intuitively “get it”, one must be able to play with it mentally and effortlessly. Logic is here merely to verify the intuitive insights.

Numerous mathematicians have tried to show the world that their mental faculties aren’t a blessing from God or a genetic gift. René Descartes wrote the “Discours de la Méthode” to explain how he got to discover and demonstrate so many important truths. Unfortunately, few seem to have believed him, and most people still believe that mathematicians are born, not made. David Bessis demonstrates that we can all develop our inner mathematician.

In his book Thinking, Fast and Slow, Daniel Kahneman talked about System 1 (fast, intuitive) and System 2 (slow, careful, logical). Bessis affirms that there’s a third system where System 1 and 2 collide. The purpose of System 3 is to sharpen our intuitive System 1 by tuning it with the help of our logical System 2. Intuitive errors and confusion aren’t shameful aberrations to be discarded; they are clues that can lead us to an improved System 1.

To develop our intuition we have to imagine various situations in our head, work through them, and then check if what we imagined holds up, and if it doesn’t update our mental image and try to really get it intuitively. We should be skeptical about our own knowledge and be curious when something feels confusing or unclear. When we don’t understand something, it’s time to dig deeper. Mathematics is the ultimate mind expander. It changes and expands our understanding and perception. It rewires cognition; it’s a transformative experience. It sharpens our imagination like no other discipline.

Imagination is a carnal and sensual skill, like juggling, biking, skating, but it’s more difficult to cultivate because everything happens inside our head. We can’t see others’ imagination. Unfortunately our language is largely inadequate at describing what goes on in our brain. Visualization and imagination, and therefore mathematics, can’t easily be verbalized.

Our language is “squishy” and “fuzzy”; mathematical language is hard and sharp. Our words are unclear; their semantics is fuzzy and situational.

For example, a sphere is a precise mathematical construct, yet when we use the word “sphere” it’s an ill-defined concept. The earth can be called a sphere, but because the earth is slightly flattened at the poles, we should call it an ellipsoid, yet the word sphere is fine in most contexts.

Our language’s meaning is heavily dependent on the context; it’s imprecise. The language of math books, by contrast, is extremely precise and detailed, so much so that it’s alien to us and difficult to process.

One way David Bessis made the transition from mind to the written word was by writing down his dreams. He kept a notebook beside his bed and wrote whatever he remembered when waking up from a dream. At first it was difficult, but the more he did it the more details he remembered. I liked this idea, and I may try it. It seems like a great way to exercise verbalizing our imagination.

Humility and confidence are essential to learning. To become good at math we have to get past our feeling of inadequacy and be willing to look stupid, while at the same time have the confidence to ask these stupid questions. We must understand that even if we don’t grasp things now we will eventually build a mental model to intuitively play with the principle under investigation.

To illustrate his thesis David Bessis writes about folks who have developed amazing mental abilities like Srinivasa Ramanujan or Ben Underwood. Underwood was a blind teenager who learned to navigate the world using echolocation. He would make a click with his tongue and listen to its echo. This kid was the living proof that we have incredible mental plasticity, that our brain is capable of developing seemingly inhuman abilities.

I loved this book. It really spoke to me; I felt like David Bessis wrote this book for me in particular. The stories of his journey as a mathematician and the description of the mental games he played as a child brought me back to my own childhood. Mathematica: A Secret World of Intuition and Curiosity is a great read for those who want to reconnect with their inner child and learn more about the incredible mental abilities we can all develop.