David Basis: “A society that is truly capable of teaching mathematics is more democratic”

Mathematics is everywhere and yet it is widely regarded as inaccessible. David Basis, an alumnus of Normal Supery in Ecole, was an assistant professor at Yale University and a researcher in mathematics at CNRS.

Founder of a company specializing in artificial intelligence, he called for a profound change in attitude towards the Queen of Science. Through the practice of thought and the analysis of the stories of mathematicians, he taught us that our mathematical insights and understandings can be worked on and shared.

Your book seeks to return to the notion that mathematics is reserved for the elite. You compete to reduce math to a subject of logic.

There are two stubborn superstitions about mathematics: the idea that it would be a gift that very few people have access to, and that is a matter of logic. There are two different ways to miss this issue. If it were a matter of logic, everyone would have easy access to it. In fact, mathematical understanding is based on insight, not logic. We never teach this important truth: Insight is not static, we can develop it. So the superstitions on this gift of mathematics come from the sky. If you think that insight is innate, you can’t really teach math.

After all, it would be a question of rediscovering our “child-like mental plasticity” path. What is the “little baby pose” you are talking about?

When you learn a new mathematical concept, it is always a psychomotor learning. To play with ideas, you need to train yourself to communicate with your imagination. You have to have the courage to be inexperienced, as a small child learns to walk, to play.

All psychomotor education begins with adventure. We start as a cheater.

Young children are not afraid to make mistakes. They broke their mouths, they tried again. They ask questions, ask again, endlessly. This ability to ask “dumb” questions is lost in later life.

All psychomotor education begins with adventure. We go there though we don’t know how. We start as a cheater. Learning mathematics is as difficult as learning math from a polytechnic program when you are preparing. The second characteristic of this posture of a small child is the relation of imaginary things to play. The child imagines what he can do with an object, he plays with mental images to bring them together. You just have to be more discriminating with the help you render toward other people.

How to practice math “a physical and physical activity”? Is it a question of working on our insights, looking better “in our heads”?

Contrary to popular belief, manipulating abstract ideas is a physical activity. When you imagine a circle, you see it in your head and it corresponds to a cerebral activity that we can measure. Every mathematician develops his own insights, special ideas of movement, excitement. Argument itself is not the end, it is a tool for working and exchanging insights. Mathematical insights are created and refined through practice.

Considering introspection as a continual restructuring is a fairly recent concept: it is the concept of mental plasticity. It assumes that abandoning spiritualist beliefs, still so pervasive, imagines the presence of a small monster inside us, provoking our thinking.

In fact, you describe mathematics as a set of mental gestures. This imagination is the central activity of our brain, much more than reason, in your opinion. Have you ever engaged in fantasy exercises?

Every mathematician has his own exercise. Lying in the dark of the evening, I often imagined a room where I slept and I worked in the past to move myself somewhere else. I also practiced imagining the place where I was as I could see from another advantage point. These exercises are difficult at first, but you can make progress.

The same is true of mathematics. Because the progress of insight is slow and invisible, education never mentions it. It is a property of the human body, a long-standing misunderstanding, ignored by the general public, but experienced daily by mathematicians.

You compare the toaster manual with the math book. Why?

This comparison is the work of the mathematician Thurston. No one can read a mathematical text in a linear and complete way. We only read Toaster’s instructions if it doesn’t work, only if we have questions. Same in math.

You oppose the widespread and successful spread of literacy for the sake of math disaster.

Once understood, mathematics becomes obvious and empowers us in very deep ways like reading and writing. Other than that, with mathematics, we are still in the Middle Ages: very few people have access to the comprehension allowed by mathematics. It’s like most people can’t read. This is a terrible obstacle in a digitalized world where mathematics plays a central role.

The mythology of a transcendental language reserved for the aristocracy prevented them from being taught. We need to stop this epistemological idiosyncrasy.

My guess is that in fifty years, we will solve this problem and find out how to teach math to everyone. We have long been mistaken about the nature of mathematics, which we see as magical or mysterious. The mythology of a transcendental language reserved for the aristocracy prevented them from being taught. We must stop the epistemological idiosyncrasies of mathematics and the effectiveness of our thinking.

Let us begin by clarifying the difference between the formal face of mathematics and its intimate face, which is the experience of human understanding. If we tell kids that math is a way to develop their intuitive understanding of the world, we will be on the right track.

What is the significance of Descartes in relation to mathematical adventure and insight?

Descartes was a great mathematician, inventor of algebraic geometry. The biggest misinterpretation of this is its dualism, the separation between the spirit of divine origin and the thoughtless body. Yet he understood the central role of insight. The “discourse on methodology” should be read as a testimony and not as a philosophically modern text.

You have dedicated interesting passages to Alexander Grothendik and his recently published autobiography. What role does he play in your own adventures?

He is certainly the greatest mathematician of XXe Century I was very impressed by reading “Récoltes et semailles” by Alexander Grothendik (1928-2014). I have found myself in different places as a mathematician. But, like Descartes, he makes the mistake of having a mysterious relationship with the world and with his insights. He even imagines that God is whispering in his ear. This transcendental relationship with knowledge is the end result of education. So I wanted to write my book, to take some distance. Transcendental relations are opposed to the ambition to democratize access to mathematics.

You criticize the logical formalism of mathematics, this conventional and off-putting method of invisible writing.

I’m not criticizing it, I’m just saying it’s disgusting and inhumane. But it is necessary. Logic is like a tutorial that enhances knowledge. It’s not pretty, but we need it. Insights are necessarily intimate, personal, subjective. Formality is the only strict way to talk about things that cannot be specified, the emotional representation that is in everyone’s head.

Mathematician’s accounts speak of all the secret systematic epiphanies that have not been taught.

Another superstition, the brain of mathematicians. Gauss’s brain, like Einstein’s, with so much analysis, is nothing short of amazing in your opinion. In your opinion, there is no strategy, just a way of looking.

I would like to believe that we are not equal before anything, but genetic differences cannot explain the difference in levels of mathematics, where some people are literally a billion times better than others. In addition, mathematicians’ accounts speak of all the secret systematic epiphanies that have not been taught. They say they have learned to use their imagination and insights in different ways, but are sent back to their talents without hearing what they have to say. For many, it is already inhumane to claim that we have all the means to understand mathematics.

What is the answer to this other hypothesis regarding the relative usefulness of mathematics?

Everyone knows their technical utility: without math, any computing, telecommunications, encryption technology, etc. But many feel it is not for them. Probably because of love, but also because of misunderstandings. Mathematics is not just a technical knowledge, it is a way to become stronger to understand the world using imagination, insight. Mathematics should be part of the general culture, yet it is stuck in a cultural quagmire.

What I have learned in artificial intelligence ultimately allows me to relate the rationale to bizarre and confidentiality in mathematician accounts.

After becoming a mathematics researcher at CNRS, you created a company that specializes in artificial intelligence. What is a connection?

The link is not clear. I showed an old guess like mine and I wanted to explore new things. My encounter with deep learning algorithms was a revelation. Until then, I’ve always wondered what it means to “understand” things. Mathematicians talk clumsily about their insights, as if it were magic. What I have learned in artificial intelligence has finally allowed me to associate this experience with a credible biological explanation, thus rationally linking what was bizarre and secret to the mathematician’s account.

I think this approach will make it possible to form a new epistemological and educational consensus. A society in which one is truly able to teach mathematics is more democratic. We will get there because we need it, and I trust both primary and secondary teachers to support this development. There is no sustainable future without the basis of conversations on scientific topics.


After studying at the Norcole Normale Supérieure in Paris and Paris-VII, David Basis earned a doctorate in mathematics. Assistant Professor at Yale (USA), he joined CNRS. Author of scientific articles and several books, he has taught in Russia, China and Japan, and runs an artificial intelligence company. Inside Mathematics is an adventure in our own heartsIn his latest book, he deals with the stubborn superstitions surrounding mathematics

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