Mathematicians can have a childlike sense of wonder. They chart the optimal color of bananas to ripen on schedule, rather than just purchasing the best-looking ones. Where most people see a mess to clean up, they see the partial differential equations written in the red of splattered ketchup. They set up statistical models to predict the right time to commute, instead of trusting Google Maps.
Though it may seem strange, especially to students, this way of viewing reality is common to those who have an organized language with which to interpret the world. Just as a mathematician sees fractals in a snowflake, so a civil engineer sees stress contours in the Brooklyn Bridge and a musician sees the notes of Bach’s "Fugue in G Minor." Their knowledge of the language of their profession gives them an intuition about how the universe unfolds.
The question, as David Bessis reminds us in his book, Mathematica, is how to instill this way of thinking. The first answer is that nothing replaces a good teacher, and unfortunately they’re in short supply. A poll by the National Assessment of Educational Progress found that the biggest difference between high- and low-performing students is whether a teacher is available to help them with schoolwork. Personal interaction with teachers is the best way for gaining an intuition about the mathematical nature of the world, and as online courses become increasingly common, students (many of whom already dislike mathematics) lose personal interaction with the teachers they need.
That’s why Bessis insists that math is as much about the people who guide us and the imaginative process as it is about learning the material. Recently translated from French, Bessis writes of his own mathematical adventures—all in hope of convincing everyone (from those adverse to mathematics to those with doctoral degrees) to discard the commonly held belief that mathematicians are born special.
Subtitled A Secret World of Intuition and Curiosity, the book tries to reintroduce wonder by asking us to peer into the minds of mathematicians. For Bessis, mathematics is not learned by studying a textbook. It is learned through a continuous collaboration between reason and imagination, intuition and curiosity. He thinks that differences in cognitive abilities do not account for most discrepancies in mathematical understanding. Those who succeed in mathematical endeavors, he insists, have learned the hidden world of abstractions—a skill that is difficult to teach because it cannot be demonstrated. Imagination is the key to the success of mathematicians.
The real secret that Bessis hopes his readers will learn is that understanding mathematics requires students to retain some childish curiosity, manifested in the brazenness to ask dumb questions. His theories come primarily from Alexander Grothendieck (1928–2014), an eclectic German-born mathematician. In his mysticism-filled autobiography, Harvests and Sowings: Reflection and testimony on a past as a mathematician (1983), Grothendieck claimed that "discovery is the privilege of the child." He never attributed his mathematical work to his being more gifted than others. Rather, he attributed his deep abstractions to curiosity: The universe is infused with logic, and its beautiful secrets can be uncovered when reason and imagination work in concert. And Grothendieck’s call for a return to childhood is endorsed by Bessis, who demands that we forget "these absurd stories of gifts and talent."
It’s such a charming account that, following along in Mathematica, the reader will want to agree. To step back even a little, however, is to allow doubt to creep in. There is a difference between having childish wonder and pursuing childish things. Without adult reason, childishness becomes unrestrained emotion, unfiltered behavior, and untrained thought. Curiosity must be refined with maturity and directed toward higher ends.
Bessis would answer that reason by itself is even worse. He includes in Mathematica the story of the unabomber, Ted Kaczynski—a mathematician turned terrorist—to show how reason unchecked can become unreasonable. Kaczynski, living in isolation, let his reason convince him that bombings were necessary to fight the oppression of technology. "When applied outside of mathematics and without any safeguards," Bessis says, "mathematical reasoning becomes an actual illness."
With reason and childishness properly balanced, however, we get the likes of Ben Underwood, who became blind by the age of three and learned to navigate by echolocation—clicking his tongue to see the texture of the world around him. Or William Thurston, who despite lacking depth perception learned to see geometric objects in four- and five-dimensional space. Or Maryna Viazovska, a Ukrainian-born mathematician, who solved Kepler’s conjecture on the most compact way to stack spheres in eight dimensions in 2016 (and in 24 dimensions three months later).
Though these stories are inspiring, the hard truth is that this level of understanding is not possible for everyone. In trying to make mathematics seem accessible, Bessis understates the varying degrees of accessibility. Some people simply won’t be able to see the world that Bessis describes.
Still, Bessis has turned what could have been a boring account of his graduate school years into a "quest" (as he puts it) to rediscover the mathematical lens with which to see the world. Students and academics will recognize his experiences of late-night blackboard and whiteboard scribblings, his feelings of imposter syndrome, and his hours laboring over a problem until it just clicks (sometimes while asleep).
Even those who have grown estranged from mathematics may find comfort in the book and thereby renew their interest in the field. Bessis has tried to peel back the complexity of the secret world of mathematics for non-mathematicians. At that task, Mathematica is about as good as we can hope for.
Unfortunately, as classrooms migrate to more online environments, and as the mathematics that get taught in school becomes more about practical matters rather than about discovering a way to grasp truth in the universe, students will stop being curious. And thus stop learning. They will end up seeing math as just an adjunct to a job, not as a language to interpret the world. And that’s a shame. The world is full of wonder, after all.
Mathematica: A Secret World of Intuition and Curiosity
by David Bessis
Yale University Press, 341 pp., $30
Matthew Phillips is an aerospace engineering doctoral candidate at North Carolina State University.