A Mind-Blowing Way of Looking at Math (with David Bessis)

Today, on September 18th, 2025, mathematician and author David Bessis joins us for a conversation on his book, Mathematica: A Secret World of Intuition and Curiosity. As the host, Russ Roberts, enthusiastically introduces David and his book, it becomes evident that this is not going to be your typical discussion on mathematics.

Russ Roberts kicks off the conversation by highlighting the common misconception that mathematicians are these superhuman beings who effortlessly manipulate complex equations and symbols with ease. However, David challenges this stereotype by emphasizing that mathematicians are just regular people who have honed their intuition and curiosity through their journey with math.

David delves into the inspiration behind his book, recalling his numerous attempts at trying to articulate the essence of math in a way that resonates with the general public. He acknowledges the historical precedence of renowned mathematicians like Descartes and Grothendieck, who also grappled with the challenge of elucidating the inner workings of their minds when it came to mathematics.

Drawing from his own experience as a research mathematician, David emphasizes the importance of meta-cognition in the field of mathematics. He explains that the true art of mathematics lies in how one interacts with abstract symbols and formulas, gradually tuning their intuition to derive meaning from them. This process of internalizing mathematical concepts and building a mental framework is often overlooked in traditional teaching methods.

David asserts that the failure to acknowledge the role of intuition and visualization in mathematics has hindered effective communication and understanding of the subject. He argues that the real magic of math lies not in logic, but in the intuitive leaps and mental processes that mathematicians engage in behind the scenes.

In essence, David’s book serves as a testament to the human side of mathematics, shedding light on the creative, intuitive, and curious nature of mathematicians. By demystifying the perception of math as a daunting and insurmountable subject, David invites readers to explore the secret world of intuition and curiosity that underpins the beauty of mathematics. So, as Russ Roberts aptly concludes, if this conversation piques your interest, be sure to pick up a copy of David Bessis’ Mathematica and embark on a journey of discovery and wonder. Mathematics is often seen as a complex and formal discipline that is meant to be understood through rigorous proofs and logical reasoning. However, in his book, mathematician David Bessis reveals a surprising truth: most mathematicians cannot actually read math books in the traditional sense.

Bessis explains that math books are not like novels that tell a story in a linear and understandable way. Instead, they follow a logical formalism that is meant to build mathematical objects and structures. When you open a math book, the words and equations may seem incomprehensible, making it impossible to read them like a novel.

This revelation may come as a shock to many, as the common perception is that professional mathematicians can effortlessly read and understand any math book they pick up. However, Bessis explains that math books are not meant to be read cover to cover like a novel. Instead, they serve as tools to calibrate and validate a mathematician’s intuition.

So, how are mathematicians supposed to interact with math books if they can’t read them? Bessis suggests starting with a specific problem or question that you want to understand. Instead of reading the book from beginning to end, jump to the section that addresses your question. This way, you can focus on the information that is relevant to your current understanding and avoid getting lost in the intricacies of the entire book.

Bessis also shares a valuable piece of advice that he received from a mentor early in his career: don’t try to read research-level math books from start to finish. This can be overwhelming and lead to frustration. Instead, approach math books with a specific goal in mind and use them as a resource to deepen your understanding of a particular concept or theorem.

In conclusion, the revelation that most mathematicians cannot read math books may challenge the traditional view of how mathematics is learned and understood. By recognizing that math books are not meant to be read in a linear fashion, mathematicians can approach them in a more practical and effective way that aligns with their intuition and problem-solving skills. This insight sheds light on the true nature of mathematical research and the unconventional methods that mathematicians use to navigate the complex world of mathematics. I wonder what the intuition is behind this result? This question is at the heart of a fascinating conversation between David Bessis and Russ Roberts, where they explore the role of intuition in mathematics and economics. Bessis argues that intuition is not just a gut feeling, but a powerful tool that can be trained and refined.

In his book, Bessis challenges the traditional approach to math and economics, which often prioritizes logic over intuition. He suggests that starting with intuition and using logic to validate or correct it can lead to deeper understanding and new insights. This subversive approach turns traditional thinking on its head and opens up new possibilities for learning and growth.

One key point that Bessis makes is that intuition is malleable and can be trained. Just like a deep learning network adapts to new data points, our intuition can evolve and improve over time. Mathematicians, in particular, rely heavily on their intuition and are not afraid to challenge their own assumptions. By confronting their intuition with logical formalism, they can refine their thinking and overcome the fear of being wrong.

Bessis uses the example of children learning numbers to illustrate the process of developing intuition. Just like kids start with a basic understanding of numbers and gradually build up to more complex concepts, adults can also refine their intuition through practice and exploration. Intuition, in this sense, becomes a treadmill for the mind, a tool for growth and discovery.

The conversation between Bessis and Roberts sheds light on the power of intuition in mathematics and economics. By embracing intuition and using logic as a tool for validation, we can deepen our understanding and uncover new insights. Intuition, far from being a mere gut feeling, is a superpower that can be trained and refined to unlock new possibilities and drive innovation. Intuition is a powerful tool that we all possess, whether we realize it or not. It is not just a gut feeling or a hunch; it is a complex process that involves the interconnection of neurons in our brain. Intuition is the result of our brain processing all the data we have absorbed throughout our lives, looking for patterns and connections. It is a different kind of thinking, one that is not always logical or rational, but can sometimes save our lives.

Descartes, a central figure in modern science, introduced the concept of intuition as the clear idea of something. He believed that truth was based on clarity, and that intuition was a mystical connection to God. However, we now understand intuition as a physiological process that is much more complex and rich than we can articulate with language. Intuition is not about language; it is about the meaning and connections that live in our minds.

In our daily lives, we rely on intuition to understand basic instructions and have conversations with others. When we read a banana cake recipe, for example, we effortlessly imagine the bananas, the process of mashing them, and all the different images associated with the word ‘banana.’ Our brains are constantly switching between different images and meanings based on context, allowing us to navigate the world around us.

Intuition is essential for our survival and success, even though it is not always rational or logical. It is a tool that allows us to make quick decisions based on our subconscious processing of information. While intuition can sometimes lead us astray, it can also be the difference between life and death in certain situations. Trusting our intuition and understanding its power can help us navigate the complexities of life with confidence and clarity. Mathematics is a subject that can be both fascinating and intimidating. Many people have a love-hate relationship with math, finding it either incredibly challenging or deeply satisfying. The journey of becoming better at mathematics is a journey of attaching richer, deeper, and more diversified semantics to mathematical abstractions.

One way to understand this journey is through the example of a circle and a straight line. How many points on a circle does a straight line touch? The answer, as Russ Roberts explains, is one or two. The line can be perfectly tangent to the circle, touching it at one point, or it can pierce the circle and touch it at two points. It cannot touch the circle at three points. This might seem obvious to many people, as we can easily visualize a circle and a line in our heads.

The ability to visualize mathematical concepts is a crucial aspect of understanding and working with mathematics. As David Bessis points out, most people can effortlessly imagine a circle and a line intersecting it. This visualization helps us form an intuition about the problem and reach a conclusion. However, not everyone can visualize things in their mind, a condition known as aphantasia. Even for those individuals, the intuition that a line cannot intersect a circle at three points is still present, even if they cannot explain it visually.

Intuition plays a significant role in mathematical reasoning. It allows us to reach conclusions even when we cannot articulate the reasoning behind them. In the case of the circle and the line, our intuition guides us to the correct answer, even if we might struggle to prove it formally. This intuition can be visual, as in the case of imagining geometric shapes, or it can be non-visual, as in the case of aphantasic individuals.

The journey of improving at mathematics involves developing and honing this intuition. It is about forming a deeper understanding of mathematical concepts and being able to apply them in various contexts. Whether through visual or non-visual reasoning, intuition helps us navigate the complex world of mathematics and make sense of its intricacies. By attaching richer, deeper, and more diversified semantics to mathematical abstractions, we can truly enhance our mathematical abilities and become better mathematicians. David Bessis introduces the concept of System 3 thinking as an augmentation to Daniel Kahneman’s System 1 and System 2 thinking in his book. He explains that System 1 is our instinctive, intuitive response to simple questions or situations that require minimal cognitive effort. On the other hand, System 2 is our more deliberate, analytical thinking process that involves conscious reasoning and computation.

Bessis uses Kahneman’s classic example of the ball and the bat to illustrate the differences between System 1 and System 2. In this scenario, most people instinctively respond that the ball costs 10 cents, but upon closer examination, the correct answer is 5 cents. Bessis recounts how he was able to quickly provide the correct answer without conscious computation, much to the surprise of his friend who was studying cognitive science.

As a mathematician, Bessis explains that his training has actually made him rely less on his System 2 thinking for computations. Instead, he has developed a “Super Slow” mode of thinking, which he calls System 3. This involves not only recognizing when intuition is wrong but also delving deeper into why it led to an incorrect conclusion.

By continuously exploring and challenging his intuition with formal logic, Bessis believes that one can self-correct and improve their thinking processes over time. This mindset of never giving up on intuition and constantly seeking to reconcile it with formal logic is the key to not just being good at math, but excelling in various fields.

Bessis emphasizes that this approach to thinking extends beyond mathematics and can be applied to social sciences and how we perceive the world around us. By engaging in this rigorous self-reflection and constant questioning of our intuition, we can enhance our critical thinking skills and make more informed decisions in all aspects of life. Even though we might not have a degree in economics, sociology, or psychology, we are constantly taking in the data of the world around us and trying to make sense of it. This process of understanding the complexities of the world can be challenging, but it is essential for growth and improvement.

One key aspect of this process is acknowledging that things are complicated. There are often factors at play that we may not have considered, and it is important to approach situations with an open mind and a willingness to learn. This means being able to say ‘I don’t know’ when we don’t have all the answers. By admitting our limitations and being open to new information, we create opportunities for growth and development.

A powerful example of this mindset is illustrated in the story of mathematician Jean-Pierre Serre attending a seminar given by David Bessis. Serre, one of the greatest mathematicians of the last century, walked into the seminar and listened attentively to Bessis’ talk. Despite his reputation and expertise, Serre was not afraid to admit that he did not understand a word of the presentation. This level of humility and openness to new knowledge is truly remarkable.

Bessis, inspired by Serre’s willingness to admit his lack of understanding, realized the importance of approaching learning with humility and an open mind. He understood that true understanding goes beyond simply grasping the words and logical flow of information. It requires a deeper comprehension of the underlying principles and reasons behind a concept.

By embracing this attitude of humility and openness, Bessis learned that it is okay to not have all the answers. In fact, admitting what we don’t know can be a powerful catalyst for growth and discovery. It allows us to approach challenges with a beginner’s mind, ready to learn and adapt to new information.

In a world full of complexities and uncertainties, the ability to say ‘I don’t know’ is a valuable skill. It opens the door to new possibilities and allows us to expand our knowledge and understanding of the world around us. By following the example set by Jean-Pierre Serre and David Bessis, we can cultivate a mindset of curiosity, humility, and continuous learning that will serve us well in our personal and professional lives. Mathematicians are often seen as hermits, isolated from the rest of the world, delving into complex and specialized topics. They gather in remote locations for week-long retreats, where they immerse themselves in discussions about their latest research projects. At meal times, they sit together, discussing their work in intricate detail.

During these conversations, mathematicians often talk about topics that are so specialized and abstract that it can be difficult for others to follow along. It’s not uncommon for someone to feel embarrassed for not understanding what their colleagues are discussing. In these situations, many mathematicians pretend to understand, nodding along without really grasping the concepts being discussed.

For years, I followed this pattern, nodding along to conversations I didn’t fully understand. It wasn’t until later in my career that I decided to take a different approach. I decided to ask for clarification when I didn’t understand something, using what I call the “Serre trick.” This simple act of humility changed the way I engaged with my colleagues and deepened my understanding of their work.

I remember one particular instance where a younger mathematician was explaining his research topic to me. He was trying to impress me with complex and abstract concepts, presenting his work as if it were a high-end restaurant menu filled with fancy dishes that no one really wants to eat. Instead of pretending to understand, I asked him to explain it to me in simpler terms. This simple act of vulnerability opened up a deeper and more meaningful conversation, allowing me to truly engage with his work.

By being willing to admit when I didn’t understand something, I was able to learn and grow as a mathematician. I realized that true understanding comes from asking questions and seeking clarification, not from pretending to know everything. This shift in mindset has helped me to forge deeper connections with my colleagues and approach mathematics with a greater sense of curiosity and openness.