Mathematica by David Bessis

Embracing the dissonance between intuition and logic

A discovery always begins with the simple and innocent desire to understand. You invent new actions not because you want to do something new and original, but because you can't get where you want to be with the existing techniques. Without any reference point, without someone to guide you, you have to listen to what your body is telling you. You have to get used to feeling your body in a new way.

Commentary

Mathematica by David Bessis is the rare read that bridges complex mathematical thinking with practical wisdom. I found myself highlighting passages with such frequency that it rivaled my experience with Nassim Taleb's Anti-Fragile - a personal benchmark for transformative reading.

This book held me hostage. I literally neglected all non-essential aspects of life to sit and read it. I find this to be fleetingly rare so I tried not to fight it. Instead, allowing my exuberance to pull me through each page.

My excitement for this work stems from my personal journey with mathematics. Like many, I fell into the "I'm just bad at math" trap during my freshman year of high school, resigning myself to the limited world one inhabits with that mindset. Over the past couple of decades, I've made a concerted effort to override this self-imposed limitation.

My first significant breakthrough came by way of Nassim Taleb's Skin in the Game, which showed me how mathematical concepts could be communicated in accessible, non-mathematical language.

What makes Bessis' book extraordinary is how it illuminates Taleb's thinking process in ways that Taleb himself might find too obvious to explain. It's like being given a window into the machinery behind Taleb's popular theories and principles.

I've heard Taleb mention on podcasts that he writes his books purely from intuited conjecture, only later hunting down academic citations to support his ideas.

For anyone who has read Taleb's work, this seems almost impossible - how could someone intuit such sophisticated frameworks for probability, decision-making, risk, and uncertainty?

Bessis' book not only explains how this is possible but offers practical applications to develop our own intuitive thinking skill by borrowing from mathematics greatest thinkers.

In the end, what Bessis has created here isn't just another book about mathematics. It's a manifesto for intellectual growth, a roadmap for developing better intuition, and a kick in the pants to anyone who's ever said "I'm just not good at math."

What I’m stealing

1. Be interested in things with an unwavering commitment, intensity and tenacity

2. Listen to the dissonance between intuition and logic

3. Repeatedly collide your intuition with reality

4. Intuition is your strongest intellectual resource

5. Big ideas are ridiculously simple

6. Words don’t come easily and don’t come right away

7. Listen to what your body is telling you, get used to feeling your body in a new way

8. Books should be at our service, rather than the reverse

9. One person’s clear mental image is another person’s intimidation

10. Listen to the voice of things

11. Progress is always slow because the body needs time to transform itself

12. The essence of mental plasticity is to transform audacity into competence

13. Nothing is more exciting than a big glaring error

14. Understanding is making something intuitive for yourself. Explaining something to others is proposing simple ways of making it intuitive

15. It’s normal not to understand

16. Approach the search for truth like a martial art

17. Doubt is a forcing function for clarification and specificity

18. Rationality is a guide rather than the ultimate judge

19. Candor and sensitivity are powerful intellectual weapons

Supplemental Resources

Eric Nehrlic, Urepentant Generalist

Rene Descartes Discourse on Method and Mediations on First Philosophy

Math is Still Catching up to the Mysterious Genius of Srinivasa Ramanujan

Dog ears, highlights, marginalia

To be passionately curious means to have the ability to be interested in things with an unwavering commitment, with an intensity and tenacity that never fails.

the magic power of mathematicians isn't logic but intuition.


In their minds, the ideas are luminous, simple, and powerful. On paper, they become stunted and sad.


There's a way to become good at math. This method is never taught in school. It doesn't resemble any academic method and goes against the traditional tenets of education. It tries to make things easier rather than more difficult. You can compare it to meditation, yoga, rock climbing, or martial arts. It includes techniques to overcome our fears, conquer our flight reflex in the face of the unknown, and find pleasure in being contradicted. The method's exact scope is actually broader than math. It's a universal method for reprogramming our intuition and, in that sense, it's a method for becoming more intelligent.


They know very well that being bad at math means they won't be able to go into any number of professions, including some of the most prestigious and best paid. Maybe they don't understand why math is so important, but they know that it is. They feel excluded, which gives them an excellent reason for hating it.


I want to convince you that the only possible explanation is that it's all a giant misunderstanding. People aren't good at math because no one has taken the time to give them clear instructions. No one has told them that math is a physical activity. No one has told them that, in math, there aren't things to learn, but things to do.


Studying math the same way that you study history or biology is useless. You might as well take careful notes during a yoga class so that you don't forget anything. If you don't practice any breathing exercises, it's worth nothing at all.

Explaining math is getting others to see things they've never seen before.

Sure, your intuition is wrong every now and then, but not always. Often it's right. And you can make it so that it's right more often. You can train it to see more clearly and distinctly. Starting from the same point as you, mathematicians construct a visionary intuition that is powerful and trustworthy. They get there using simple methods, like those taught in this book.

your intuition is your strongest intellectual resource. In a sense, it's your only intellectual resource.

That's why you never have complete confidence in logical arguments and you're much more at ease with what you understand intuitively.


If you've been able to do these things, rest assured: you have the genetic potential and the intellectual faculties to become very good at math. From the biological perspective, that's all that's needed. The other ingredients aren't genetic, and they're also at your disposal. It's simply a matter of sincerity, patience, desire, and courage.

The big ideas are always intuitive and always simple. They're even ridiculously simple.

This is a universal law of human cognition. It states that our science was invented by humans and that humans are, at the deepest level, all made of the same stuff.

The great discoveries are made by people who are simply trying to understand. They just want to make things clear for themselves.


And when you think you see a math whiz, it's never really a math whiz, it's always just someone who has a way of seeing numbers that turns calculations that you find complex and scary into something easy and even obvious.

The main difference between a math whiz and you is that their bag of tricks is bigger than yours and they're more used to playing with them.

If you find that the math you do understand is too easy, it's not because it's easy, it's because you understand it.


This is what happens when you run up against the limits of language.

In order to express what you feel, you have to invent new words, or create a new usage for words that already exist. Fleeting impressions cease being fleeting only after you find a way to pin them down with words. It takes time to get there. Words don't come easily, and they don't come right away.

The initial phase of a discovery is a spiritual experience. You think outside of language. The world is illuminated. You have epiphanies.

You see things that until then were hidden. Things so new they don't yet have a name.


Understanding a mathematical notion is learning to see things that you could not see before. It's learning to find them obvious. It's raising your state of consciousness.


But math, because it relies on unseen actions, can't be learned through imitation.

A discovery always begins with the simple and innocent desire to understand. You invent new actions not because you want to do something new and original, but because you can't get where you want to be with the existing techniques. Without any reference point, without someone to guide you, you have to listen to what your body is telling you. You have to get used to feeling your body in a new way.

Finding the solution means thinking what had been unthinkable. It's like augmenting the cognitive capacity of human beings.

To truly learn a movement, you have to understand it beyond words. You have to feel it within your own body, and find it natural and intuitive.

Math is mysterious and difficult because you can't see how others are doing it. You can see what they're writing on the blackboard or on a sheet of paper, but you can't see the prior actions they performed in their heads that enabled them to think and write those things.

Math itself is simple but the mental actions that allow us to make sense of it are subtle and counterintuitive. These actions are invisible.

We can't simply imitate what others do. We lack the adequate words to explain how to do it, and in any case words will always miss the main point: what we really feel inside ourselves.


But they know that getting lost is a normal stage in the understanding process. They won't get upset. They won't pretend to understand what they can't understand. They won't even try to take notes. They'll simply stop listening.

If they really want to understand, they'll find another way.


If you really want to, if you have enough time for it and you've chosen the right book, it's well worth the effort. Get ready for a few months of hard work. This initiation rite will transform you. In my lifetime I've really succeeded in reading only three or four math books.

I don't regret the time and effort. It gave me unexpected powers, as if I'd drunk a magic potion. This power remains with me today. But the potion was hard to swallow.


In fact, you should do whatever you feel like doing. You can leaf through the book for ten seconds, one hour, or three monthswhatever. The underlying principle is never to force yourself to follow the pages in order, but to follow your own desire and curiosity.

The book should be at our service, rather than the reverse. It will never work if we try to read a math book like a "normal" book, if we let the book dictate the pace, if we wait for it to take us by the hand  and tell us a story. We're not there to listen passively. We don't have the patience and, frankly, we're just not interested.


In the end, the page that interests us the most may end up being the least difficult for us. First of all because we're interested in it: interesting things are a whole lot easier. And also because it's necessarily tied to something we already understand-otherwise it wouldn't interest us.

Following your desire is the only way of giving the book a real chance. If you start at the beginning, you run the risk of getting discouraged by page 2.


At a profound level, math is the only successful attempt by humanity to speak with precision about things that we can't point to with our fingers. This is one of the central themes of this book and we'll come back to it a number of times.

What is rare, and what our culture doesn't push you to do, is to be aware of your capability for synesthesia and to try to develop it systematically. Secret math is a mental yoga whose goal is to retake control over our ability for synesthesia.

He told me how an idea had been invented and how you need to understand it "morally."

"Morally" understanding something means being able to explain it to yourself intuitively and cite the reason why it's true: the moral of the story.


The trips mathematicians take allow the diffusion of new ideas with an efficiency otherwise impossible. We love long-term stays. We need time to talk, take coffee breaks, scribble on the blackboard, and pick up the discussion the next day with a question that came to us when we woke up. A Japanese mathematician, Kyoji Saito, wanted to understand the "thoughts between the lines" of one of my articles, so he often invited me to Kyoto. For my part, it allowed me to better understand the "thoughts between the lines" of his articles. These types of trips are part of a mathematician's life.


"Bad" teachers are the ones who recite the 198 steps to assemble the toaster as if that were the end of the story. "Good" teachers do their best to explain what a toaster is. They constantly look their students in the eyes, because it's in their eyes that they will know if they've understood.

"One person's clear mental image is another person's intimidation, Thurston wrote.

It's hard to share mental images, as they are evanescent and profoundly subjective. Our common language is incapable of transcribing them with precision. It's precisely because our intuition is so secretive and so unstable that logical formalism was invented.


for the most creative mathematicians, mathematics is a sensual and carnal experience that is located upstream from language.


Grothendieck had a different explanation: "The quality of the inventiveness and the imagination of a researcher comes from the quality of his attention, listening to the voice of things."

Interrogating things, listening to the voice of things, means trying to imagine them, examining the mental images that form within you, seeking to solidify these images and make them clearer, working at unveiling more and more details, as when you try to recall a dream.

mathematical understanding  is achieved by gradually modifying the way we represent things to ourselves, and making them clearer, more precise, closer to reality.


Our prodigious faculty for learning and invention has its origih in our unconscious ability to constantly reconfigure the fabric of associations of images and sensations that, literally and figuratively, comprise the real structure of our thought.


It is in this continuous effort to articulate the inarticulable, to define what is as yet unclear, that the particular dynamic of mathematical work (and perhaps as well all creative intellectual work) is perhaps found.

Mathematical writing is the work of transcribing a living (but confused, unstable, nonverbal) intuition into a precise and stable (but as dead as a fossil) text.

Or, rather, it would be a simple job of transcription if the intuition was from the outset precise and correct. But intuition is rarely precise and correct from the outset. At first it's vague and wrong, and it always remains a bit so. Through the work of writing, intuition becomes less and less vague and less and less wrong. This process is slow and gradual.

In that sense, mathematical definitions have the power of creation: they bring things into existence. It may seem silly to speak so pompously, but that's what really happens.

When you see things that others don't yet perceive, sharing your vision requires finding a way to get others to re-create those things in their own heads. A mathematical definition serves this purpose. It provides detailed instructions allowing others, starting with things they are already able to see, to mentally construct those new things.


But before getting to this simple understanding, while you still don't have the right mental images, you'll have to go through a lot of trial and error.


Writing mathematics, that is, transcribing mental images with enough clarity and precision to allow others to understand and reproduce them, is an art.

 What makes it so difficult is that your mental images are often a lot less clear than you think.


the art of mathematical writing is really a dual task of the clarification of ideas and the refinement of language.


Learning to write math is learning to have clear ideas. Wouldn't it be a shame to deprive yourself of that?


The others aren't blind. They are biologically capable of seeing the same shapes as you. But they haven't yet learned how. Their brain receives the same raw visual information but doesn't structure it in the same way.


Mathematical comprehension is precisely this: finding the means of creating within yourself the right mental images in place of a formal definition, to turn this definition into something intuitive, to "feel" what it is really talking about.


The real pleasure of mathematics is waking up one day and realizing that you can see stars in your head, which you'd never been able to do before.

The secret techniques of mathematicians aim to facilitate and accelerate this intuitive understanding. Mathematicians use logic and language as an apparatus for learning to see.


If you've never learned to think in multiple dimensions, you've missed out on one of the great joys of life. It's like you've never seen the ocean, or never eaten chocolate.


Visual intuition makes certain mathematical properties clear, that without the mental image wouldn't be clear at all. This is why transforming mathematical definitions into mental images is so important.


Contrary to common belief, it's never abstraction that makes math difficult to understand. Abstraction is our universal mode of thinking. The words that we use are all abstractions. Speaking, making sentences, is to manipulate and assemble abstractions. In that respect, four-dimensional geometry isn't any more abstract than two dimensional geometry. The problem with four-dimensional geometry has nothing to do with abstraction. The problem is that it's hard to visualize and hard to draw.


Vector spaces are usually represented by letters and linear maps by arrows connecting these letters. But when I chose to picture vector spaces as bigger or smaller barrels (according to their dimension) and linear maps as bigger or smaller pipes (according to their rank) connecting those barrels, then all the exercises on these concepts suddenly became obvious.

It wasn't much. That was a tiny subject and there were plenty of other exercises that I was unable to solve. But in this subject, not only was I able to solve the exercises, they became as obvious to me as 1,000,000,000 I = 999,999,999. They became so obvious that it seemed absurd that you'd even have to ask, and even more absurd that there were people who didn't know how to solve them.

These barrels and pipes made my life simpler, but where did they come from? Was that how you're supposed to do it? What was going on in other people's heads? How were they imagining mathematical concepts?


The second approach consists of refusing to learn. It takes on math as a sensory experience. The sole function of mathematical statements is to help you generate mental images, and only these images will lead to comprehension. Once you have the correct mental images, everything else becomes clear.


At the time, thankfully, this didn't last long. After a few weeks, without any conscious effort, I ended up accepting the idea that you could reason with letters, that is, reason with numbers without knowing the actual number. I understood that not knowing the actual number was in fact the whole point. Reasoning with letters was a way of reasoning with all numbers at once. It was doing an infinite number of computations with a finite number of words.


I knew perfectly well that they were false. It was obvious. But since it was so obvious, was I able to say exactly in which way they were false?

Today, when I try to describe this intellectual method, I sum it up like this: I began to listen to the dissonance between my intuition  and logic,


I failed to express myself clearly.

I didn't have the right words to talk about these issues. It took me decades to find a proper way to talk about them.

**Note:** Taleb mentions a similar sentiment. Thinking for 20 years before writing his books.


In the space of a few weeks, my way of studying was transformed.

I began to use class as a benchmark for my intuition. I tried to predict what the teacher was going to say. Most of the time I got it wrong, but that let me figure out where my intuition was already correct.

The things I understood, I understood so well that I could rely on them and concentrate on the others.

I kept going back to what I didn't understand until I understood why I didn't understand it


Behind our false beliefs about mathematical intelligence, behind our superstitions and inhibitions, there lies our ignorance about menral plasticity and the laws that govern it.


It's important to be able to guess what is true, what is false.

Then I don't remember statements which are proved. I try to have a collection of pictures in my mind. More than one picture, all false but in different ways, and I know in which way they are false.


Mathematical work isn't a series of lightning insights and strokes of genius. It's first of all a work of reeducation based on the repetition of the same exercises of imagination.

 Progress is slow because the body needs time to transform itself.

It doesn't help to force it, which may end up hurting you. You just need to commit to a regular training schedule, keep your cool, keep going even when it seems you're not making any progress. It's like going to the speech or physical therapist, except you're all alone and inside your head.


A conjecture is a mathematical statement that someone believes is valid but isn't yet able to prove. Making a conjecture is feeling something is right without being able to say why. It is by nature a visionary and intuitive act.


I don't know what Thurston really saw, but when I look at his mathematical work, I haven't the slightest doubt that he saw many things I don't. His style of writing gives the impression that he's just trying to share them with us. He'd love to show the things directly, in real life, but he knows that's impossible. So he writes math papers.


And maybe you've already had the fascinating experience of waking up one morning and knowing, without even having opened your eyes, that it snowed during the night, because the texture of the silence has changed.


No human is capable of solving these equations in the way that school teaches us to solve equations, by applying a conscious and mechanical method. But the specificity of our mental plasticity is to:

give us an unconscious means of solving problems without ever stat ing them, by training our minds to recognize a multitude of subtle patterns that evade our consciousness.


You have the same magic abilities as anyone else. It's just a question of will, patience, and openness to the world.


As for echolocation, it seems that you can get significant results by working an hour per day for two to three weeks. In the end, it's a bit like learning how to drive.

When you want to learn a new sport, a new language, or a new job, you go through a similar process. You have to throw yourself into it and accept that you'll be feeling about for a bit, thinking that you'll never be any good, until the moment you find, as if by magic, that you're getting the hang of it.

The essence of mental plasticity is to transform audacity into competence.

The process is slow and invisible, and at first success seems unachievable: that's the biological reality of our learning mechanisms.


That's why we so often limit ourselves to learning only what's of ficially possible to learn (things that have introductory or professional development courses), what you can learn by imitating others, or what comes naturally.

The rest, the secret and invisible apprenticeships, are said to be "gifts," "talents," "supernatural powers." No one tells us that we can learn to see in five dimensions, get our bearings through echolocation, or tell the sex of dogs and cats by looking at their heads, so we never even try.


Reconnecting with your early childhood capacity for learning means to stop believing in these absurd stories of gifts and talent. It means to become once again capable of devoting ten or twenty hours  to something that may or may not be impossible, without being distracted by the feeling of your own uselessness. It means to rediscover the world with an open mind, trying something just to see what happens, for fun, because you want to.

My basic technique hadn't changed: lending an ear to the dissonance between my intuition and logic. This technique remained my instrument for exploring the world,

I stopped believing that our way of seeing and thinking about the world was a given fact, and that we each had a predefined amount of intelligence that we had to make do with.

In place of that, I began to believe that we had the freedom to ceaselessly refashion our way of seeing and thinking, and to construct our own intelligence day after day.

This change of approach had one initial practical consequence: I became a creative mathematician. I began to have ideas no one had had before, to see things no one had seen before, to prove theorems no one had yet proven at first easy theorems, and later in my career theorems that had till then seemed far beyond my capabilities. Mathematical creativity has the reputation of being a great mystery that science can't explain. In my experience, however, it emerged as a natural phenomenon once I had adopted the correct psychological attitude.

System 3 - When I need to make an important decision in my life, if my intuition tells me to choose option A and my reason tells me to choose option B, I tell myself there's something going on and I'm not ready to make the decision.

That's the moment to resort to what I call System 3.

lending an ear to the dissonance between my intuition and logic.

I force myself to translate my intuition into words, to tell it like a simple and intelligible story. Vice versa, I try to picture what logical reasoning is actually expressing, to experience it in my body, to hear what it's trying to say. I ask myself if I really believe it. I fumble about. It takes time but it's not a real effort. It's more like a meditation on running water, something going on in the background that might stop and start, then all of a sudden become clear days, months, or even years later.

I'm personally incapable of thinking against my intuition and I have serious doubts as to the sincerity of people who claim they can.

**Note:** The IYI

Your intuition will always be more powerful and better informed than the most sophisticated of language-based reasonings.

My intuition isn't any less fallible than yours. It's always getting things wrong. I have, however, learned never to be ashamed of it. I don't disdain my mistakes, I don't push them aside, because I don't think that they betray my intellectual inferiority or some cognitive biases hardwired in my brain. On the contrary. Nothing's more exciting than a big glaring error: it's always a sign that I'm not looking at things in the right way, and that it's possible to see them more clearly.

When I'm able to put my finger on an error in my intuition, I know it's good news, because it means that my mental representations are already in the process of reconfiguring themselves.

My intuition has the mental age of a twoyear-old-it has no inhibitions and always wants to learn. If you stop mistreating your own, you'll see that it's exactly like mine, only asking to be allowed to grow.


System I has an edge: it isn't bound by the constraints of language and writing.

The most important messages, the ones you should always bear in mind, are these:

1. You can reprogram your intuition.

2. Any misalignment between your intuition and reason is an opportunity to create within yourself a new way of seeing things.

3. Don't expect it all to come at once, in real time. Developing mental images means reorganizing the connections between your neurons. This process is organic and has its own pace.

4. Don't force it. Simply start from what you already understand, what you can already see, what you find easy, and just play with it. Try to intuitively interpret the calculations you would have written down. If it helps, scribble on a piece of paper.

5. With time and practice, this activity will strengthen your intuitive capacities. It may not seem like you're making progress, until the day the right answer suddenly seems obvious.

You'll need a number of training sessions. Exactly how many, I don't know. It's not worth tiring yourself out-better to split it up into short five-minute sessions, and think about it in the shower or while on a walk. Above all, take your time. It's good to think about it only once a week or once a month. Most important, keep at it and don't let it drop. It will come eventually.

Understanding something is making it intuitive for yourself. Explaining something to others is proposing simple ways of making it intuitive.

But a few people choose to rely on their System 3. They're not aware that they're doing anything special. Math just feels easy to them. It doesn't even feel like work. They're just seeing pictures in their heads, and spending a couple of minutes a day looking at these pictures and asking themselves naïve questions.


There are no tricks. There never were any and there never will be.

Believing in the existence of tricks is as toxic as believing in the exisof truths that are counterintuitive by nature.


Look ing for and finding the right way of seeing things is the driving force of mathematics. It's the main source of pleasure you can take from it.


When you give free rein to your imagination, it is nearly without limits.

These domains each have their own vocabulary and intuitions. It's like they correspond to different ways of using our bodies, different  regions in our brains, different ways of focusing our attention

**Note:** Referring to the different domains of mathematics, but applies more broadly than that.

the probabilistic approach is a sort of kung fu that focuses your attention on a single number while your subconscious does all the heavy lifting.


If you want to validate your intuition and transform it into rigorous thinking, if you really want to understand why the average is 50.5 rather than 50, you need to listen to yourself, to your unconscious processes and their mechanisms.


To get to really know a mathematical object, you have to observe in for a long time, with intensity and detachment, with curiosity and open-mindedness. You need to take the time to play with it and create an intimate relationship, a relationship that takes place outside of language.


mathematical intelligence was something you constructed for yourself. It's the natural byproduct of a physical activity that everyone is free to practice: mathematical imagination.

Mathematics is the science of imagination. Between those who allow themselves to imagine, observe, and manipulate mathematical objects and those who don't, there's an enormous divide.


Contrary to popular belief, logic isn't the enemy of imagination.

It can even be a close ally. The real enemy of imagination, that which blocks understanding and makes us feel like fools, is fear.

Fear is our real limitation.

paying attention to the dissonance between my intuition and logic

Mathematics is a practice rather than knowledge.


You're at the edge of what's socially acceptable. Your credibility is at stake. If you acknowledge how lost you are, you'll look like a fool.

The social norm is to let it go.

This situation is typical of all failed math conversations, those we learn nothing from, those that serve only to reinforce our certainty that we're the worst of the worst.


I have fashioned a method by which, it seems to me, I have a way of adding progressively to my knowledge and raising it by degrees to the highest point that the limitations of my mind and the short span of life allotted to me will permit it to reach.


there are two levels of understanding. The first level consists of following the reasoning step by step and accepting that it's correct.

Accepting is not the same as understanding. The second level is real understanding. It requires seeing where the reasoning comes from and why it's natural.

It's normal not to understand. It's normal to be afraid. It's normal to have to struggle to contain your fear. It is, in fact, precisely what's at stake.

Descartes. In order not to leave any doubt, he opens his book with a sentence in the form of a slogan: "Good sense is the most evenly distributed thing in the world!"


we all know that rationality serves a purpose, just as we all know that math is extraordinarily powerful. Even if we have a hard time explaining why.

Discourse on Method is a self-help book whose message is simple: we have the ability to construct our own intelligence and selfconfidence.

If you open Discourse on Method looking for a glorification of System 2, you'll be sadly disappointed. Descartes's great innovation was to put intuition and subjectivity at the heart of his approach to knowledge. He was distrustful of established knowledge and what was written in books. He placed little credit in authorities. He preferred to reconstruct everything by himself, in his head. His method closely resembles that of Einstein, Thurston, and Grothendieck. It is, of course, System 3, the slow and careful dialogue between intuition and logic, with the aim of developing your intuition


. However, this is precisely where Descartes has the most to teach us.

As long as we conflate rationality and logic, as long as we reduce truth to its social and linguistic dimensions, as long as we see it only as a matter of consensus or authority, we completely miss the point of the Cartesian approach.

For Descartes, truth was a matter of life and death. He perfectly embodies this singular and powerful aspect of mathematical psychology his relationship to the truth is physical, almost carnal: “I constantly felt a burning desire to learn to distinguish the true from the false, to see my actions for what they were, and to proceed with confidence through life.”

Descartes cared less for easy truths, those that are supposed to be true because tradition or such-and-such a person says so, or simply because they seem true. What interested him were solid truths, those that weren't going to change overnight, the ones you can rely on to become stronger and more confident, to make the right choices in life.

He approached truth as a martial art, an instinct you develop and that becomes embodied in action.

Everything else‚ the philosophical arguments, the "opinions" of intellectuals with no skin in the game was all just talk and of no interest to him: "For it seemed to me that I could discover much more truth from the reasoning that we all make about things that affect us and that will soon cause us harm if we misjudge them, than from the speculations in which a scholar engages in the privacy of his study, that have no consequence for him."

He had forced the Christian world to consider this existential question: is the truth necessarily what is written in books, or do we, as human beings, have the ability to discover it ourselves?

A bit later, Descartes realized that the dictionary was damaged.

Then the man and the books disappeared. Without waking up, Descartes interpreted the dictionary as a symbol of science and the etry collection as a symbol of philosophy and wisdom. The substance of the dream was precisely that: it is necessary to reconstruct science while being inspired by the techniques of poets who, through "the divinity of enthusiasm" and "the force of imagination," are able to uncover "the seeds of wisdom (which are found in the spirit of all men, like the sparks of fire in stone)."

Descartes thus invented rationality. Upon awakening, he was convinced that the Spirit of Truth had descended upon him to "open the treasures of all the sciences" by revealing that truth is not to be found in books, but in our heads. We have the ability to discover it ourselves, by the power of our thought.

For Descartes, the experience of understanding mathematics is the sole means of understanding what "understanding" really means.


This lead him to adopt a dualist stance: he imagined a separation between mind and body. Our mind is of an immaterial nature, created by God in his image, and thus capable of attaining Truth as if by magic.

You can't doubt with words, you can doubt only silently, in your head. Doubt is personal and intimate. If you only pretend to doubt, if you don't go all the way, if you don't take the plunge, it's worth nothing.

It's only through a relentless confrontation with doubt that forces you to clarify and specify each detail until it all becomes transparent that you're finally able to create obviousness, doubt is a technique of mental clarification. It serves to construct rather than destroy.

His "burning desire," as we've seen, was the exact opposite: "to proceed with confidence through life."


Descartes discovered that when we make a sincere attempt at introspection, when we're attentive to our cognitive dissonance, when we force ourselves to grasp our most fleeting mental images and put words to them, when we have the courage to face the internal contradictions of our imagination, when we have enough calm and selfcontrol to look beyond our prejudices and see things as they really are, it has the result of modifying our mental representations, of making them more powerful, solid, coherent, and effective.


Practiced right, doubt can induce a state of profound compre hension that amazed Descartes, as it amazes all those who experience it. It's an experience that leaves you transformed, and that in itself is well worth the effort.

Doubt is not only the secret behind Descartes's achievements, it's also the secret of his incredible chutzpah.

Seen in this way, Discourse on Method is a master class in self-confidence. His version of rationality is concrete, personal, rooted in our deepest aspirations. The whole point is to make us stronger:
"This assurance is one side of a mindset, whose other side is an openness to doubt: an attitude of curiosity that excludes all fear as regards one's own mistakes, that allows us to detect and constantly correct them."

This last quote, taken from Grothendieck's Harvests and Sowings, perfectly summarizes this fundamental lesson from Descartes, and captures a unique aspect of the mathematical ethos.

“Arrogant people who love being contradicted, show-offs who smile when you prove them wrong, dogmatists ready to change their mind in a heartbeat: I've encountered this singular attitude only among very good mathematicians.”

In losing the possibility of imitation, we lose much more than our main learning method. We also lose our main driver of desire.


If I could show it, the math that goes on in my head would intrigue a lot of people. But it serves no purpose to say so, since I don't have any means of showing it.


As an adult, I've developed my own way of making use of the special state of mind just before falling asleep. Rather than focusing on subjects that preoccupy me, I've learned to simply let myself be filled with them. The nuance is subtle but fundamental. Focusing is thinking intensely, in search of solutions. It never works and it keeps you from sleeping. Being filled with something means contemplating it without a goal, in a decentered and disinterested manner. It's almost like dreaming.

I might be wrong, but it seems to me that this technique of falling asleep increases my chances of waking up the next morning with interesting ideas.


If you want to practice switching viewpoints, here is a good exercise:

1. Choose a random reference point around you, for example, the corner opposite from you in a room, or the window of a house when you're walking in the street.

2. Try to imagine what you'd see if you were looking in your direction from this reference point.

It's not a binary exercise in which there are those who can and those who can't do it. The exercise is hard for everyone.


creativity is simply the ultimate form of understanding, which itself is but a natural product of our mental activity. It emerges when we force ourselves to continue looking at things that intimidate us until they finally become familiar and obvious.


Here is an exercise of the imagination that helped me a lor when I was trying to improve in these subject areas. I looked at an object, for example, a bottle of shampoo sitting in the bathroom, and asked myself the following question: if my body were shaped like the bottle, how would it feel physically

When you're sincerely preoccupied with something, when you're in trouble, when you have problems at work or problems at home, you instinctively call on the method used by mathematicians.

At night, in your bed, you try to understand the issue. You mull it over. You replay in your head mental images that you dig up from the depths of your memory and imagination. You play Lego with these images. You try to organize them, fit them together and assemble something meaningful, something that makes sense

Descartes's method has the effect of modifying your mental representations and your intuitions and gradually reinforcing their internal consistency.

It's in fact the whole point of the approach. By anchoring our convictions in indisputable evidence and rigorous deduction, we can turn them into certitudes that, over time, become as strong as reinforced concrete.

Except that sometimes these certitudes are false.


Here is the reason why there was a riddle: human language is structurally incompatible with logical reasoning, and we can never have 100 percent certainty in truths expressed in human language and arrived at through deductive logic.

That goes for all kinds of "truths," the ones coming from official science and the ones coming from our small everyday reasoning that we employ all the time.

Outside of mathematics, rationality remains under constant threat from the fragility of our language and our way of perceiving the world.

without this rigidity, no language can be compatible with logical reasoning. It may be annoying, but that's how it is.

In the end, rationality should be used as a guide rather than an ultimate judge. The reality that's before our eyes always merits more attention than the certitudes in our heads. Rationality is great, but empiricists do have a point.

Trusting reason too much, using human language as if it had all the attributes of mathematical language, as if words had a precise meaning, as if each detail merited being interpreted and the logical validity of an argument sufficed to guarantee the validity of its conclusions, is a characteristic symptom of paranoia. When applied outside of mathematics and without any safeguards, mathematical reasoning becomes an actual illness.


To find out, to understand how math really works and what it can really do for us, we cannot continue to overlook its most direct practical aspect: math works on our brain and modifies how we see the world.


Is the elephant young or old? Dangerous or harmless? Is it angry? Does it have a lot of self-confidence? Do you feel sympathy or distrust?

You've never taken a class on interpreting drawings and yet you know how to answer these questions, immediately and without any effort. All that from a few lines drawn on a page.

How is such a miracle even biologically possible?


Intelligence is what is called an emergent property: individually our neurons are primitive and limited, but vast assemblies of neurons make incredibly sophisticated behaviors "emerge" that can't be attributed to any neuron by itself these large-scale behaviors are what we call intelligence.


From the outset in this book, I've discussed a number of power ful and mysterious phenomena that have perplexed me throughout my mathematical career: mental plasticity, the inevitable ambiguity of human language, the role of time and trial and error in trying to understand things, the necessity to ask stupid questions, the feeling of obviousness that comes after the fact.

another phenomenon takes place in the background. It happens at a much slower pace, and is so discrete that we can't perceive it. The correct metaphor isn't lightning, but organic growth. It's the process through which we learn. It's the basis of what we have called System 3, our ability to gradually modify the way we represent the world to ourselves.

Math is first and foremost an inner tool. Its main purpose is to enhance human cognition, With the correct exercises of imagination, we have the ability to develop an intuitive and familiar understanding of mathematical notions. We can appropriate them and make them an extension of our bodies.


They can understand mathematical objects only in a perceptual manner, via false human interpretations, approximations, translations from mathematical vocabu lary into human language.

In fact, this is precisely why math is so beneficial for us: it forces us to enrich our human vocabulary and our human perception.

from a purely practical standpoint, math is indistinguishable from fiction.

Learning math is an activity of pure imagination.


Yes, ideas come unexpectedly, "as if summoned from the void," but that's normal. Yes, plasticity is a slow and silent mechanism that occurs without any real effort on our part, provided we're exposed to the right mental images. Yes, we learn precisely when we force ourselves to imagine things that we don't yet understand, which unfortunately is the same exact thing that most people run away from. Yes, paying attention to the small details that trouble us is of the utmost importance, and the fastest way to learn is to follow the path of maximum perplexity. Cartesian doubt, within this framework, can be interpreted as an "adversarial" hack to accelerate our learning.


How do you teach math to someone who believes that their intuition and perception of reality are given and impossible to reprogram?


Yet using our imagination isn't navigating an ethereal layer of the cosmos. Nor is it a parasitical activity that we should seek to suppress. It is, instead, a genuine physical activity that is central to human cognition.


If we day dream, it's because this allows us to fabricate understanding. What we imagine modifies the actual wiring of our brain and literally changes the way we see the world.

There are a thousand and one ways to imagine. We haven't yet learned to recognize them all, and still less to name them. Think, meditate, reflect, visualize, analyze, fantasize, reason, dream: we use these words haphazardly, without really knowing what they mean, and without realizing how much they have in common.

It's through this vagueness that all the misunderstandings slip in.

No one cared to even tell us that there are right and wrong ways of using our imagination. Some make us stupid. Some make us crazy.

And some have the power of making us incredibly smart.

34 Now that we're teaching machines the secrets of intelligence, it's about time we start teaching humans.


The secret math, the one that deals with human understanding, will never possess the rigor and objectivity of official math. Because of this, it will never be considered as a "serious" topic.

This "nonserious" topic, however, is arguably much more important than most properly mathematical questions.


Failing to place human understanding at the center of mathematics is failing to acknowledge the very nature of mathematics.


I have tried to approach it in my own way, starting from a simple premise: talking about math as I have experienced it, in the simplest way possible, examining what it really consists of, the things you do inside your head, and how to approach it concretely.


This is precisely why math is difficult: it requires looking straight at what is beyond our comprehension. We must become genuinely interested in it. We must force ourselves to imagine it and put words to all our impressions, without being distracted by our constant feeling of inferiority. And we must do that precisely when our instinct tells us to run away as quickly as we can.

I know now that my candor and my sensitivity are my most powerful intellectual weapons. The mathematical approach is one of integrity and being in tune with oneself.

My advice is to have no shame in starting at the beginning, with the most classical and elementary proofs. Since it won't be easy for you to know if you really understand them, try explaining them to someone else, perhaps a child.

When we try explaining things to others, we often realize that our own ideas aren't as clear as we thought. It's a painful and humiliating experience, but one that you can get over, and it's precisely by getting over it that you get ahead in math.

The later works of Wittgenstein are an excellent complement to this chapter as well as to chapter 19. The most accessible is perhaps On Certainty, a short text assembled posthumously from his notes (Oxford: Wiley-Blackwell, 1975).


Proofs without Words: The Example of Ramanujan Continued Fractions notes: http://www.xavierviennot.org/coursIMSc2017/lectures_files/Ramanujan  Inst_2017.pdf; video recording

Other works cited in this chapter:

Alfred North Whitehead and Bertrand Russell, Principia Mathematica, vol. 1 (Cambridge: Cambridge University Press, 1910).