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Mathematica
Contre les idées reçues qui en font une discipline élitiste, intimidante et abstraite, David Bessis montre que les mathématiques sont humaines et à la portée de tous ; il présente ici une manière sensible et radicalement nouvelle de les aborder.
Plus qu’un savoir, les mathématiques sont une pratique et même une activité physique. Il n’existe pas de talent inné et il faut croire les plus grands mathématiciens quand ils disent ne posséder aucun don spécial mais une immense capacité à mobiliser leur curiosité, leur imagination et leur intuition.
Par des exemples simples et étonnants, l'auteur relie son expérience mathématique aux grands apprentissages de la vie : observer, parler, marcher ou encore manger avec une cuillère. Comprendre les mathématiques, c’est voir et sentir, c’est parcourir un chemin secret qui ramène à notre plasticité mentale enfantine.
Entre le récit initiatique et l’essai subversif, Mathematica est un livre puissant et accessible à tous, philosophique et imagé, sur notre capacité à construire nous-mêmes notre intelligence.
L’auteur est un mathématicien français né en 1971. Ancien élève de l’Ecole normale supérieure (Ulm), David Bessis a été professeur assistant à Yale puis chercheur au CNRS. Il a créé et dirige aujourd’hui une société spécialisée en intelligence artificielle. Mathematica est son troisième livre.
Plus qu’un savoir, les mathématiques sont une pratique et même une activité physique. Il n’existe pas de talent inné et il faut croire les plus grands mathématiciens quand ils disent ne posséder aucun don spécial mais une immense capacité à mobiliser leur curiosité, leur imagination et leur intuition.
Par des exemples simples et étonnants, l'auteur relie son expérience mathématique aux grands apprentissages de la vie : observer, parler, marcher ou encore manger avec une cuillère. Comprendre les mathématiques, c’est voir et sentir, c’est parcourir un chemin secret qui ramène à notre plasticité mentale enfantine.
Entre le récit initiatique et l’essai subversif, Mathematica est un livre puissant et accessible à tous, philosophique et imagé, sur notre capacité à construire nous-mêmes notre intelligence.
L’auteur est un mathématicien français né en 1971. Ancien élève de l’Ecole normale supérieure (Ulm), David Bessis a été professeur assistant à Yale puis chercheur au CNRS. Il a créé et dirige aujourd’hui une société spécialisée en intelligence artificielle. Mathematica est son troisième livre.
320 pages, Kindle Edition
Published January 21, 2022
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Displaying 1 - 30 of 112 reviews
July 19, 2024
You will get A LOT from this book whether or not you are into math. I'm not mathy. I'm a creative writer, artist, karaoke person & I love learning languages. I picked this up because of a thread I saw online, on a whim, because something in my gut said, "you should take a look at this." And then I listened to the introduction, and I was like, "yeah, this seems good and the world is a dumpster fire so..."
Then, to my delight, I discovered in this one of the best guides to imagination and general creativity I've ever read. I've recommended the book to my friends, especially the artistic types, who were giving me some side-eye until I went into some of the details (which I'll do below).
What you'll get out of this book:
1. Intuition is not just some magical thing that you've either got or your don't. You can train it by thinking through things both creatively and with logic. You can make it BETTER! Because it's your brain! And you'll get some concrete approaches on how to do this.
2. Math people are really good at creative visualization (and other creative sensory imagination -- not just pictures). Not just naturally, but because they've trained this skill doing lots of other seemingly random imaginative exercise. Using imagination, you can calculate stuff without having to know a bunch of formulas. (I could go into more detail, but I'm not giving away the whole book because it's valuable to read/listen to it and try his exercises.)
3. You can then apply these imaginative techniques and to improving your skills in all kinds of seemingly unrelated areas. The way he talks about feeling out formulas, for example, reminds me very much of writing and feeling out the shape of a story. I also directly see how what he's talking in creative visualization will make you better at drawing from the imagination. I can draw from reference, but I have a hard time drawing from imagination. I just assumed I had a bad visual imagination, but never had the (seemingly obvious but I wasn't thinking of it) revelation that I could just focus on improving my visual imagination and memory... duh.
I've actually gained quite a bit more from this book too in regards to thinking about how I think and how I can think better. (Circular much, lol!) But I'm not going to give everything away.
Tl;dr: Whether or not you're interested in improving your math skills (it was fun to improve some of mine but a that's not why I got this book), Mathematica is a winner if you're interested in learning more about how to use and improve your imagination. And it's fun!
NOTES FOR AUDIO: The figures are not included though I hope they are working on a PDF for this audio. Here's what jammed me up and how I fixed it: The icosahedron is basically a 20 sided di (think D&D d20). The super one has a picture on Wikipedia.
For the visualization for adding up to 100 whole numbers, it's helpful to think of it as six sided dice (d6) instead of cubes which felt harder to visualize for me, idk why. Start with doing it to three first, then four, then five so you understand the nature of the question. The you will get the rest of it).
There are some good videos on YouTube which will walk you through the infinity set and different infinity sizing stuff. I recommend Dr. Trefor Bazett's two videos on this from his Cool Math series. They came up when I looked on YouTube about different size infinities, so just look him up with infinity size and you'll probably get them. The videos weren't long and quite a bit of fun.
Still working on the knot thing, but I'll be coming back to it when I find better reference.
Hope this helps! The important thing is to get the process down more than the results, So don't stress it too much.
Then, to my delight, I discovered in this one of the best guides to imagination and general creativity I've ever read. I've recommended the book to my friends, especially the artistic types, who were giving me some side-eye until I went into some of the details (which I'll do below).
What you'll get out of this book:
1. Intuition is not just some magical thing that you've either got or your don't. You can train it by thinking through things both creatively and with logic. You can make it BETTER! Because it's your brain! And you'll get some concrete approaches on how to do this.
2. Math people are really good at creative visualization (and other creative sensory imagination -- not just pictures). Not just naturally, but because they've trained this skill doing lots of other seemingly random imaginative exercise. Using imagination, you can calculate stuff without having to know a bunch of formulas. (I could go into more detail, but I'm not giving away the whole book because it's valuable to read/listen to it and try his exercises.)
3. You can then apply these imaginative techniques and to improving your skills in all kinds of seemingly unrelated areas. The way he talks about feeling out formulas, for example, reminds me very much of writing and feeling out the shape of a story. I also directly see how what he's talking in creative visualization will make you better at drawing from the imagination. I can draw from reference, but I have a hard time drawing from imagination. I just assumed I had a bad visual imagination, but never had the (seemingly obvious but I wasn't thinking of it) revelation that I could just focus on improving my visual imagination and memory... duh.
I've actually gained quite a bit more from this book too in regards to thinking about how I think and how I can think better. (Circular much, lol!) But I'm not going to give everything away.
Tl;dr: Whether or not you're interested in improving your math skills (it was fun to improve some of mine but a that's not why I got this book), Mathematica is a winner if you're interested in learning more about how to use and improve your imagination. And it's fun!
NOTES FOR AUDIO: The figures are not included though I hope they are working on a PDF for this audio. Here's what jammed me up and how I fixed it: The icosahedron is basically a 20 sided di (think D&D d20). The super one has a picture on Wikipedia.
For the visualization for adding up to 100 whole numbers, it's helpful to think of it as six sided dice (d6) instead of cubes which felt harder to visualize for me, idk why. Start with doing it to three first, then four, then five so you understand the nature of the question. The you will get the rest of it).
There are some good videos on YouTube which will walk you through the infinity set and different infinity sizing stuff. I recommend Dr. Trefor Bazett's two videos on this from his Cool Math series. They came up when I looked on YouTube about different size infinities, so just look him up with infinity size and you'll probably get them. The videos weren't long and quite a bit of fun.
Still working on the knot thing, but I'll be coming back to it when I find better reference.
Hope this helps! The important thing is to get the process down more than the results, So don't stress it too much.
July 15, 2024
Very good opening on the idea that math is for everyone and that the enormous gap between those who can and those who can’t should not be so large. If we translate this math gap into the 100 meter dash, it would be something like the best on the world can run it in 10s but the average person would take a couple of weeks and those who are weak at math will take 500 years, they cant do it in their lifetime
Despite this wonderful opening, none of it is realized. There are absolutely ZERO practical ideas on learning or teaching math in this book. The rest of the book is a couple of stories and philosophical musings, some good and some ok and a bit of lameness.
Despite this wonderful opening, none of it is realized. There are absolutely ZERO practical ideas on learning or teaching math in this book. The rest of the book is a couple of stories and philosophical musings, some good and some ok and a bit of lameness.
June 19, 2025
The book starts off with a promising premise—explaining how mathematics works and showing that it’s for everyone. However, it falls short after the introduction, turning into a collection of anecdotal stories about mathematicians, repeated ideas, and little meaningful content to learn from.
February 3, 2025
I love math and I enjoy it. I am also intimidated by it. I have taken a variety of ways to overcome this fear, somehow find a way to understand it (as much as I can), and eventually see math in the real world and enjoy it. That's the mindset I started reading this book with.
One key takeaway from this book was the following, which I found important:
Maybe a lot of the book is summarized by the above sentence. I believe the book could be significantly shortened, maybe to an article. There were a lot of redundant sections, told and retold again and again. The author took a patronizing tone, maybe in the hope of making the contents more accessible. Add to that a lot of text, swinging between self-promotions and humble-bragging, which made the text harder to read. On top of all, add some philosophizing forced into the contents. I admit that there's a possibility that some of these could be the effects of the translation. Maybe not.
Overall, I had takeaways from this book as well, and reading some parts were enjoyable, although most of them were masked by the poor medium. Below I am bringing some pieces from this book:
On tricks and their impact:
and also your belief in their existence:
On how language can prevent us from forming images
On the impact of naming:
How mathematics is a practice and you are not supposed to know everything:
On the impact of teaching a concept on our own learning of that same concept:
On doubt and its nature:
On paranoia and mathematical reasoning:
On emergent properties:
and some more on various topics:
This quote is from Grothendieck:
One key takeaway from this book was the following, which I found important:
The most simple and fundamental advice you can give to people who want to understand math, which I've repeated throughout this book, is to pretend the things are really there, right in front of you, and that you can reach out and touch them. People who don't understand math are basically stuck in a state of disbelief. They're refusing to imagine things that don't actually exist, because they don't see the point. It just makes no sense to them.
Maybe a lot of the book is summarized by the above sentence. I believe the book could be significantly shortened, maybe to an article. There were a lot of redundant sections, told and retold again and again. The author took a patronizing tone, maybe in the hope of making the contents more accessible. Add to that a lot of text, swinging between self-promotions and humble-bragging, which made the text harder to read. On top of all, add some philosophizing forced into the contents. I admit that there's a possibility that some of these could be the effects of the translation. Maybe not.
Overall, I had takeaways from this book as well, and reading some parts were enjoyable, although most of them were masked by the poor medium. Below I am bringing some pieces from this book:
On tricks and their impact:
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 existence of truths that are counterintuitive by nature.
and also your belief in their existence:
Believing that tricks exist is to accept the idea that there are things you'll never understand and that you have to learn by heart.
On how language can prevent us from forming images
The language trap is the belief that naming things is enough to make them exist, and we can dispense with the effort of really imagining them.
On the impact of naming:
Naming things certainly allows us to evoke them, but not to make them present in our mind with the intensity and clarity that allow for creative thinking.
How mathematics is a practice and you are not supposed to know everything:
Mathematics is a practice rather than knowledge. Mathematicians understand better than anyone the objects they're working on, but their mathematical intuition can never become omnipotent.
On the impact of teaching a concept on our own learning of that same concept:
mathematicians have a saying, that the only thing a math lesson is good for is to allow the professor to understand.
On doubt and its nature:
No one can doubt on a piece of paper. Doubt is a secret motor activity, an unseen action. To doubt something is to be able to imagine a scenario, even seemingly improbable, where the thing could be untrue.
On paranoia and mathematical reasoning:
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.
On emergent properties:
you can spend twenty years of your life reverse-engineering cars, but that won't teach you anything about traffic jams. And yet traffic jams exist and they're entirely made up of cars.
and some more on various topics:
Successful math becomes so intuitive that it no longer looks like math.
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.
discovery always begins with the simple and innocent desire to understand.
Math is mysterious and difficult because you can't see how others are doing it.
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.
In a math book, the most important passages aren't the theorems or the proofs: they are the definitions.
The ability to associate imaginary physical sensations with abstract concepts is called synesthesia. Some people see letters in colors. Others see the days of the week as if they were positioned in the space around them.
This quote is from Grothendieck:
“Finding mistakes is a crucial moment, above all a creative moment, in all work of discovery, whether it's in mathematics or within oneself. It's a moment when our knowledge of the thing being examined is suddenly renewed.”
Logic doesn't help you think. It helps you find out where you're thinking wrong.
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.
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.
As Thurston remarked, “There is sometimes a huge expansion factor in translating from the encoding in my own thinking to something that can be conveyed to someone else.”
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.
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. When you're unable to imagine mathematical objects, you have the sense that you don't really understand them. And you'd be right.
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.
When we believe we're directly seeing the world in three dimensions, we're unconsciously piecing together the two-dimensional images captured by our retinas.
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.
Nothing is counterintuitive by nature: something is only ever counterintuitive temporarily, until you've found means to make it intuitive.
Understanding something is making it intuitive for yourself. Explaining something to others is proposing simple ways of making it intuitive.
In mathematics, the sudden occurrence of a miracle or an idea that seems to come out of nowhere is always the signal that you're missing an image.
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.
Thurston's response offers a radical change of perspective: The product of mathematics is clarity and understanding. Not theorems, by themselves. The world does not suffer from an oversupply of clarity and understanding (to put it mildly). The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification.
Being a paradox is always a temporary status, in wait of a resolution. Presenting a problem as structurally being a paradox is just a pompous way of saying you can't solve it.
When you mathematically model a deep-learning system, you can define a numerical quantity that measures its “perplexity” in a given situation. A system that learns is one that adjusts its weights in order to reduce its perplexity.
lines written by Thurston in 2011: “Mathematics is commonly thought to be the pursuit of universal truths, of patterns that are not anchored to any single fixed context. But on a deeper level the goal of mathematics is to develop enhanced ways for humans to see and think about the world. Mathematics is a transforming journey, and progress in it can be better measured by changes in how we think than by the external truths we discover.”
“Calculation” comes from the Latin calculus, which means “small pebble,” referring to the stones used on an abacus for counting.
April 29, 2025
This is a beautiful book about Mathematics, written by a mathematician.
There are two main points in the book, which could be orthogonal:
- there is an intuitive “secret math” going on inside each person’s head, which is different from the rigid formalism of the “official math”. It is ignored because it is very subjective and hard to explain, yet, it is how people actually understand things and make discoveries. “Official math” is just the only way we have of actually communicating with each other. Official math is to math what musical notation is to music - not the actual thing. Then, secret math is the subjective experience of music inside our heads.
- we vastly overrate the importance of genetics in how good someone is in mathematics: it is more important to be “passionately curious” and to embrace ignorance
The first point is very well explained in the book. I am convinced. An interesting consequence, for readers of “Thinking Fast and Slow”, is that mathematical learning is proof that our “System 1” is reprogrammable. We learn intuitions for circles, numbers and other abstract objects. We shouldn’t necessarily be distrustful of our System 1 mistakes, we should actually look towards them and engage in the learning process to improve our intuitions.
The second point is a minority belief I have and am interested in seeing more discussion around it, however, I thought the book could have gone more in depth, and detailed the idea of the cumulative process that drowns the initial (genetic) signal . The mental Fosbury Flop analogy is a very good one, though.
I also think that a good amount of the discussion is also valid for other subjects other than math, that this is very much about knowledge and how it can be acquired. There is a small portion dedicated to epistemology, but I wanted to read more.
This was a thought-provoking book, very enjoyable to read, that left me wanting more in depth discussion. Bessis writes at the end “I wrote this as a kind of handbook, something that I would have loved to have on my nightstand during my studies to guide me, to encourage me, to help overcome my inhibitions. I believe it would have helped me immensely. I hope it will help you.” This book has been already quite motivating, but for sure if I keep coming back in the next years, I will update this 4 to a 5, for what’s that’s worth!
“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.”
“Mathematics has the reputation of being inaccessible. You have to be one of the elite, to have received a special gift. The greatest mathematicians have written that this isn't so. What they accomplished, as we shall see, they claim to have accomplished through ordinary human means, their curiosity and imagination, their doubts and weaknesses.”
“I wanted to know how to do real math, difficult math.
But all that I was able to learn was the easy math, the math for dummies.
It seems silly to say this, but it really took me years to
realize it was only an optical illusion. The horizon was shifting with me. It was always staying at my level.
Real magic doesn't exist. When you learn a magic trick, it ceases being magical. That may be sad, but you'd better get used to it.
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.”
“Before school came along and got all caught up with it, before our inhibitions and our fear of being judged came
along, we all have experienced great joy in math. Between humans and mathematics, it's been a long and profound love story.”
“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.”
“A particularity about mathematics is that understanding a
discovery is almost as challenging as making the discovery itself. In order to reproduce unseen actions, you can't avoid introspection. You have to listen to yourself, and reinvent the actions within yourself and for yourself.”
There are two main points in the book, which could be orthogonal:
- there is an intuitive “secret math” going on inside each person’s head, which is different from the rigid formalism of the “official math”. It is ignored because it is very subjective and hard to explain, yet, it is how people actually understand things and make discoveries. “Official math” is just the only way we have of actually communicating with each other. Official math is to math what musical notation is to music - not the actual thing. Then, secret math is the subjective experience of music inside our heads.
- we vastly overrate the importance of genetics in how good someone is in mathematics: it is more important to be “passionately curious” and to embrace ignorance
The first point is very well explained in the book. I am convinced. An interesting consequence, for readers of “Thinking Fast and Slow”, is that mathematical learning is proof that our “System 1” is reprogrammable. We learn intuitions for circles, numbers and other abstract objects. We shouldn’t necessarily be distrustful of our System 1 mistakes, we should actually look towards them and engage in the learning process to improve our intuitions.
The second point is a minority belief I have and am interested in seeing more discussion around it, however, I thought the book could have gone more in depth, and detailed the idea of the cumulative process that drowns the initial (genetic) signal . The mental Fosbury Flop analogy is a very good one, though.
I also think that a good amount of the discussion is also valid for other subjects other than math, that this is very much about knowledge and how it can be acquired. There is a small portion dedicated to epistemology, but I wanted to read more.
This was a thought-provoking book, very enjoyable to read, that left me wanting more in depth discussion. Bessis writes at the end “I wrote this as a kind of handbook, something that I would have loved to have on my nightstand during my studies to guide me, to encourage me, to help overcome my inhibitions. I believe it would have helped me immensely. I hope it will help you.” This book has been already quite motivating, but for sure if I keep coming back in the next years, I will update this 4 to a 5, for what’s that’s worth!
“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.”
“Mathematics has the reputation of being inaccessible. You have to be one of the elite, to have received a special gift. The greatest mathematicians have written that this isn't so. What they accomplished, as we shall see, they claim to have accomplished through ordinary human means, their curiosity and imagination, their doubts and weaknesses.”
“I wanted to know how to do real math, difficult math.
But all that I was able to learn was the easy math, the math for dummies.
It seems silly to say this, but it really took me years to
realize it was only an optical illusion. The horizon was shifting with me. It was always staying at my level.
Real magic doesn't exist. When you learn a magic trick, it ceases being magical. That may be sad, but you'd better get used to it.
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.”
“Before school came along and got all caught up with it, before our inhibitions and our fear of being judged came
along, we all have experienced great joy in math. Between humans and mathematics, it's been a long and profound love story.”
“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.”
“A particularity about mathematics is that understanding a
discovery is almost as challenging as making the discovery itself. In order to reproduce unseen actions, you can't avoid introspection. You have to listen to yourself, and reinvent the actions within yourself and for yourself.”
July 26, 2025
Most insightful book I read on mathematical thinking and math education. As opposed to the dry logical flow of standard math texts, Bessis delivers a very readable description of intuitive and creative minds comparing it to slow- fast thinking and exemplifying it with anecdotes from eminent mathematicians and also his personal experiences. A must read.
August 26, 2025
I’d say it lost me in the second half/last third, but frankly the first part contained the best description of what “doing math” feels like as a bodily/sensorial/“human” activity I’ve ever read, so there’s that.
May 30, 2023
Mathematica est le bouquin dont je rêvais depuis des années pour me réellement me réconcilier avec les mathématiques, sauf que l'auteur va plus loin encore en démontant la vision que la société a des mathématiques et du "génie" supposément nécessaire pour devenir bon en maths.
Il y a quelque chose d'à la fois jouissif et contre-intuitif à le voir démontrer à quel point être compétent dans ce domaine demande surtout de la patience et... de l'imagination. Il prouve au passage qu'enseigner froidement les étapes des calculs sans apprendre aux élèves à visualiser le processus sans recourir aux mots est un non-sens qui a faillit couler son propre avenir avec les mathématiques.
Se voir expliquer à quel point les mathématiciens sont des gens normaux qui ont juste compris certaines choses mieux que les autres et cultivé cela est un des grands moments de ce livre.
On ressort avec une réelle soif de découvrir la richesse de ce qu'on nous a entre ouvert et d'enfin persuader son propre cerveau de s'élever un peu passage.
Il y a quelque chose d'à la fois jouissif et contre-intuitif à le voir démontrer à quel point être compétent dans ce domaine demande surtout de la patience et... de l'imagination. Il prouve au passage qu'enseigner froidement les étapes des calculs sans apprendre aux élèves à visualiser le processus sans recourir aux mots est un non-sens qui a faillit couler son propre avenir avec les mathématiques.
Se voir expliquer à quel point les mathématiciens sont des gens normaux qui ont juste compris certaines choses mieux que les autres et cultivé cela est un des grands moments de ce livre.
On ressort avec une réelle soif de découvrir la richesse de ce qu'on nous a entre ouvert et d'enfin persuader son propre cerveau de s'élever un peu passage.
December 9, 2024
Es haben nicht alle Kapitel für mich funktioniert, aber trotzdem vier Sterne, weil es ein paar sehr gute Stellen gibt. Z.B. wo der Autor überzeugend zeigt, dass viele auch gute Mathematiker nicht alles verstehen; dass man der Intuition vertrauen soll, sie aber auch korrigieren muss, wenn man draufkommt, dass sie mal nicht stimmt; und generell Mut macht, Mathematik zu betreiben.
December 31, 2025
I’ve heard that in some forms of non-Theravada Buddhism, certain texts are only supposed to be read when your teacher thinks you’re ready to hear them. The rationale behind this seems to be that upon hearing about a concept for the first time, our minds put a certain “novelty” aura on it that allows them to more rapidly rearrange our views. Thus, timing this aura to when your mind will be most amenable to change leads to the best results.
Along a similar line of thinking, maybe it’s best to not listen to podcasts about books that you plan on reading because then you’ll just box your reading experience into the summarized frames you’ve heard on the podcast.
Something like that happens to me with this book. I listened to the Econ talk episode on it, and the concept (that your intuition is your most powerful resource and that you should actively train it through something like Cartesian doubt) fascinated me. But then when I actually read the book a lot of it seemed repetitive and I felt like I had already sussed out a lot of the implications myself. This likely made me gloss over certain parts of it that could have taught me something.
So, I don’t think I had as good as a reading experience than I would have without the “spoilers,” but ultimately this was still a very solid 4 star book. I left it hoping that real life intuition could live up to Bessis’s romantic notion of it.
The book also gave me a bunch of random practical tools, including a way to open myself up to better math conversations (by admitting off the bat that I know nothing). Finally, it gave me a real reason to record my dreams (to help the process of translating intuition to writing).
On the negative side, the chapter on the unibomber seemed whole unnecessary, and the book was a bit repetitive at times.
Finally, at its core this is a self help book, so to really get use out of it you need to find some system to remind yourself of the tools presented and get yourself to actually implement them. For me that’s anki, but other forms of journaling could work.
Along a similar line of thinking, maybe it’s best to not listen to podcasts about books that you plan on reading because then you’ll just box your reading experience into the summarized frames you’ve heard on the podcast.
Something like that happens to me with this book. I listened to the Econ talk episode on it, and the concept (that your intuition is your most powerful resource and that you should actively train it through something like Cartesian doubt) fascinated me. But then when I actually read the book a lot of it seemed repetitive and I felt like I had already sussed out a lot of the implications myself. This likely made me gloss over certain parts of it that could have taught me something.
So, I don’t think I had as good as a reading experience than I would have without the “spoilers,” but ultimately this was still a very solid 4 star book. I left it hoping that real life intuition could live up to Bessis’s romantic notion of it.
The book also gave me a bunch of random practical tools, including a way to open myself up to better math conversations (by admitting off the bat that I know nothing). Finally, it gave me a real reason to record my dreams (to help the process of translating intuition to writing).
On the negative side, the chapter on the unibomber seemed whole unnecessary, and the book was a bit repetitive at times.
Finally, at its core this is a self help book, so to really get use out of it you need to find some system to remind yourself of the tools presented and get yourself to actually implement them. For me that’s anki, but other forms of journaling could work.
December 25, 2025
This is a very cool self-help/inspirational book and has the added bonus for me of sparking good conversations with my pure-math-degree-holding husband :)
December 27, 2025
This is a book about the power of imagination and intuition dressed up as a book about mathematics. It's also a bit of a self-help book, full of convincing arguments that could make even the least mathematically inclined believe they have the capacity to learn. The tone is anti-elitist and pro child-like wonder. It's written in a democratizing spirit.
I took advanced mathematics courses in college but always struggled with formalism. This book helped me understand the mistake in my approach. Math is an act of imagination more than an exercise in rationalism. The seeds planted here could help anyone develop a strong approach for building mathematical intuition and imagination, which are really the same thing.
I would have given the book 5 stars, but I found much of it redundant. That's not necessarily a bad thing - the redundancy managed to get the point across. I also felt it could have used a tiny bit more math, or a few more tangible examples of precisely how intuition can be developed towards solving problems. In any case, it's great and I'd recommend it. Plenty of inspiration here for anyone.
I took advanced mathematics courses in college but always struggled with formalism. This book helped me understand the mistake in my approach. Math is an act of imagination more than an exercise in rationalism. The seeds planted here could help anyone develop a strong approach for building mathematical intuition and imagination, which are really the same thing.
I would have given the book 5 stars, but I found much of it redundant. That's not necessarily a bad thing - the redundancy managed to get the point across. I also felt it could have used a tiny bit more math, or a few more tangible examples of precisely how intuition can be developed towards solving problems. In any case, it's great and I'd recommend it. Plenty of inspiration here for anyone.
September 28, 2025
A mathematician's view of their internal world of maths and thoughts.
I absolutely loved reading this, but I'm incredibly biased. I do maths for neuroscience, so this like really hit the centre of the Venn diagram of my interests. A lot of the descriptions of mathematical research felt incredibly poetic, heart-warming, human, and creative - they are like a longer more elaborated version of William Thurston's famous essay. And the internal mental descriptions are just so intriguing, precisely described anecdotes of some of the mind's vageries.
One of the professors I work with describes his experience doing research as a perpetual struggle, it's never purely fun, it's a constant, addictive nagging. Some sections came the closest I've seen to describing the feeling of being consumed by a problem.
At times a little self-aggrandising and self-helpey, but eh, it's worth it for the other stuff in my opinion.
I absolutely loved reading this, but I'm incredibly biased. I do maths for neuroscience, so this like really hit the centre of the Venn diagram of my interests. A lot of the descriptions of mathematical research felt incredibly poetic, heart-warming, human, and creative - they are like a longer more elaborated version of William Thurston's famous essay. And the internal mental descriptions are just so intriguing, precisely described anecdotes of some of the mind's vageries.
One of the professors I work with describes his experience doing research as a perpetual struggle, it's never purely fun, it's a constant, addictive nagging. Some sections came the closest I've seen to describing the feeling of being consumed by a problem.
At times a little self-aggrandising and self-helpey, but eh, it's worth it for the other stuff in my opinion.
August 28, 2024
wow. me ha gustado mucho.
el libro está muy bien y lo puedes leer incluso si no estudias mates. me lo topé de casualidad al ver un hilo en twitter del autor y me lo he fundido en 5 dias. habla sobre cómo desarrollar la imaginación y la creatividad y por qué son tan importantes tanto dentro como fuera de las mates.
tmb me han gustado las anécdotas de varios matemáticos que cuenta y sobre el punto de vista de estos a la hora de hacer matemáticas
el libro está muy bien y lo puedes leer incluso si no estudias mates. me lo topé de casualidad al ver un hilo en twitter del autor y me lo he fundido en 5 dias. habla sobre cómo desarrollar la imaginación y la creatividad y por qué son tan importantes tanto dentro como fuera de las mates.
tmb me han gustado las anécdotas de varios matemáticos que cuenta y sobre el punto de vista de estos a la hora de hacer matemáticas
August 2, 2025
DNF - read about 60%. It has some good analogies and antidotes about learning math and the way the brain works that I can share with my students, putting concrete words to ideas I already had. But it started to feel repetitive and drawn out. I was getting bored; which is saying a lot because I’m rarely bored by math.
December 23, 2024
Okumakla kalmadım, Türkçe tercümesini tamamladım. 2025'in ilk aylarında raflarda olacak,
February 26, 2025
I would love to have a hot chocolate with this author, since people like him are the best interlocutors — those who speak in metaphors and cultural references !!, the best, yes.
February 11, 2026
the rare book that makes you want to think harder and feel better doing it.
it also delivers the shock of recognition usually found in great novels (that feeling when you think you were in a certain unique way but then you find that not so unique way has been perfectly articulated on the page by a total stranger). bessis dismantles the math genius myth, revealing math as a playful mental game for any curious mind. challenges the badge of honor worn for math ineptitude. Wonderful and useful reading for everyone, especially young parents
I discovered this book through econtalk: https://www.youtube.com/watch?v=NIpbp... - this is the episode featuring David Bessis where he discusses the core themes of the book with russ roberts
it also delivers the shock of recognition usually found in great novels (that feeling when you think you were in a certain unique way but then you find that not so unique way has been perfectly articulated on the page by a total stranger). bessis dismantles the math genius myth, revealing math as a playful mental game for any curious mind. challenges the badge of honor worn for math ineptitude. Wonderful and useful reading for everyone, especially young parents
I discovered this book through econtalk: https://www.youtube.com/watch?v=NIpbp... - this is the episode featuring David Bessis where he discusses the core themes of the book with russ roberts
August 8, 2025
This was eye-opening.
All really good such books show you something that seems extremely obvious. It's the case here, and helps with one of my main complaints about teaching mathematics, that there's no idea how people get to some result, how the thinking works, or should work.
The style is a bit weird at times, but really worth the read - if you plan on teaching maths (or have kids that are learning it), if you had problems with it at school, or if you just want a good book about thinking.
(and I probably need to reread it)
All really good such books show you something that seems extremely obvious. It's the case here, and helps with one of my main complaints about teaching mathematics, that there's no idea how people get to some result, how the thinking works, or should work.
The style is a bit weird at times, but really worth the read - if you plan on teaching maths (or have kids that are learning it), if you had problems with it at school, or if you just want a good book about thinking.
(and I probably need to reread it)
Read
March 6, 2025Not a math book, but a book about math—or, really, a book about life. A lyrical romp in praise of curiosity and imagination, Bessis continually stresses the role of intuition and creativity in problem solving, and how their interplay with logic is the key to being "good" at math.
November 9, 2025
Lots of interesting ideas, but I don't know what to do with them. Possibly a failure of intuition and curiosity on my part, possibly a function of the amount of time that has passed since the author left academic mathematics and began to explore deep learning AI.
December 6, 2025
This book blew my mind and helped me see math in a way I could have never imagined before. The way math is taught in schools is absolutely criminal. Formal education severs us from our intuition, our childlike wonder and curiosity, and then makes us feel dumb or like we’re bad at math, all of which isn’t true. We’re all natural mathematicians before we’re programmed into “doing” math that is absolutely counterintuitive and dumb.
December 29, 2024
Best non-fic read of 2025.
Anyone can do math. Just like anyone can ride a bike.
Anyone can do math. Just like anyone can ride a bike.
January 28, 2025
A surprisingly good book. Also, one doesn’t have to be interested in Maths to gain some value out of it.
January 29, 2025
Wonderful from start to finish. This felt like the text / perspective on math I’ve been looking for my whole life.
Deeply humanistic view on mathematical formalism and the enterprise of mathematical ‘thinking’. Honestly felt Bessis spoke directly to many of my insecurities and misconceptions wrt this topic. Definitely going to read again.
Deeply humanistic view on mathematical formalism and the enterprise of mathematical ‘thinking’. Honestly felt Bessis spoke directly to many of my insecurities and misconceptions wrt this topic. Definitely going to read again.
December 2, 2025
J'ai beaucoup aimé le concept et l'objectif visé par l'auteur. Certains passages un peu trop philosophiques et longs à mon goût ont fait en sorte que ce n'est pas un 5 étoiles, mais presque.
February 6, 2026
This mathematician told me all the things that I have done.
March 11, 2026
A book about the philosophy of learning mathematics. Gah! And I'm not sure it succeeds even in that.
March 15, 2026
*Introduction: About Popular Math Books*
I fell in love with mathematics a bit more than a decade ago, when I was finally able to overcome my fears by starting from scratch on Khan Academy and completing everything they had there, including Calculus. Since then, I’ve continued reading and studying, with my current goal updated to mastering about everything that would be taught in an undergraduate pure math degree. From previous posts here, you can see that the serious stuff I am plodding through at the moment is Real Analysis, but I still find time to take detours into popular math books, which tend to be more rewarding and faster reads than textbooks.
Pop math has to face a really tough uphill battle, and there is a very narrow Goldilocks zone in which it works. For professional mathematicians and those steeped deeply enough in the field, these books will for the most part be uninteresting and hand-wavy. For the broad public, including the subset of those who read popular science, math tends to be too daunting. And while you can write books about Physics or Biology with very few equations and lots of metaphors and practical examples, this sort of defeats the purpose if the topic of the book is Math itself.
Common strategies are to focus on the biographical stories of great mathematicians, or to cherry-pick a selection of some of its accessible, entertaining, and most popular bits. The usual suspects (infinities, fractals, prime numbers, Fibonacci and the Golden Ratio, counterintuitive probabilities, famous problems, solved or not...) sometimes make it into books of their own.
Another subclass consists of books that delve into the psychology and sociology of mathematical practice and teaching. They try both to describe what doing mathematics is really like, moving away from the stereotypes of the crazy, introverted genius and of Mathematics as drudgery, brain-killing rote computation, and they usually go on a detour to complain that secondary and high school math fail abysmally to show what ‘real mathematics’ is like. Frenkel’s Love and Math has a bit of this, with a very vivid metaphor in which he describes what is done in schools as trying to teach art by getting students to paint fences instead of pictures. Eugenia Cheng’s The Joy of Abstraction goes out of its way to be as accessible as possible in presenting to a broad audience an area as arcane as Category Theory, and in the first parts of the book makes a good case for what the title says (i.e., how abstraction can be both enjoyable and pragmatically useful for whatever interests you have), while lamenting math-phobia and, if memory serves, putting a lot of the blame on how it’s taught. Michael Harris’s Mathematics Without Apologies doesn’t go too much into pedagogical issues, but it does explore in depth what it is like to do mathematics ‘from the inside’ as a professional, and demolishes quite a lot of stereotypes and preconceptions, including among practicing mathematicians.
Mathematica falls in line with those latter three books. We’ll make a summary of its contents in a minute, but before that, I will abuse your patience a bit longer with a section about my expectations for this book.
*What did I expect?*
I am a very picky reader, so I wouldn’t have picked up this book if I hadn’t expected to enjoy it and/or learn something from it. I follow David Bessis on X, and am superficially acquainted with who he is and with some of the arguments he makes. He is (was?) a professional research mathematician, and given that I am not much of a contrarian, this gives him a lot of auctoritas in my eyes, i.e., I assume he knows what he’s talking about when he writes about Mathematics and that he knows it better than I do.
Two things that I strongly disagree with him about do tend to appear frequently in his tweets. I am a full-blooded mathematical Platonist: the core value I see in Math is that it is the discipline that most closely approximates my yearning for absolute Truth with a capital T, one completely independent of human social construction; and, given that I assume a correspondence theory of truth, this forces me to acquiesce in something like the existence of a Platonic realm of mathematical objects and/or structures. Bessis, on the other hand, is more of an intuitionist and has words of praise for Reuben Hersh, who tends to reduce the discipline to a human activity rooted in our culture and history, similar in this regard to laws, institutions, or money. This is anathema to me.
The second bone of contention I have with him is his general views on math education and the intelligence required for doing professional mathematics. Nobody is very happy with the results of math education -and least of all the teachers- but I don’t feel it is fair to lambast it from the point of view of what a pure mathematician would expect it to do, as, in fact, high school math does a lot of things that are necessarily misaligned with those goals by default (like acting as a gatekeeper to more demanding university studies, or helping students internalize some core mathematical subjects that are instrumentally useful for science and economics). As for intelligence, genius, and IQ in math, this is an ideological minefield into which I loathe to step, but readings in recent years have strongly stirred me away from blank-slatism, which is the de facto dogma of our societies. Math, in fact, is one of the few areas in which it is generally rejected, and before reading the book, Bessis did give me the impression of being anti-IQ for what were likely ideological reasons. Making mathematics ‘accessible to all’ and ‘conscious of gender and race oppressions’ are empty, sectarian mantras to which I am no less averse than I am to viewing Mathematics as a human fiction like magical spells and Harry Potter books.
I was therefore expecting the author to challenge my views with more elaborate intellectual arguments than what can be gleaned from social networks alone. He didn’t disappoint in this regard, although the book only marginally touches on the philosophy of mathematics.
*The book’s core thesis*
Bessis believes that what mathematics is and how people learn to do it are grossly misrepresented and misunderstood. Tweaking the traditional two-systems classification by Kahneman where System 1 is a set of fast, mostly effective but also mostly unreliable and incoherent intuitions and System 2 is the slow, procedural, rational (and inhuman) type of reasoning we associate with logic, mathematics and proofs, he postulates that (doing) mathematics is really working in System 3, which is a balancing of both, and one in which we clarify, sharpen, naturalize and internalize our intuitions from System 1 by forcing them to interact and take into account the straightjacket of System 2. Doing and learning mathematics, therefore, happens through intuition, mental imagery, and repeated internal practice, through imagining, manipulating, testing, and refining abstract objects until they’ve become familiar and obvious. In a way, this is no different from how we push ourselves to learn new, unexpected stuff which after learning becomes trivial and self-evident, like riding bikes, using spoons or putting correctly shaped blocks into holes. Perhaps the greatest difference with math though is that you have to become savvy at manipulating imaginary objects purely in the mind.
*Now, in more detail*
The book begins by rejecting three common myths: that doing mathematics reduces to thinking logically, that some people are born with special mathematical intuitions, and that great mathematicians have brains that are substantially different from normal people’s. Against this, Bessis proposes that math is fundamentally a corrected intuition that can be trained and strengthened. This intuition is seldom talked about and even more seldom taught, even if it constitutes an indispensable tool for working mathematicians, and constitutes a type of ‘secret math’ that has to be discovered by them, as opposed to the ‘official math’ of the formal language of proofs and mathematical textbooks, which (apparently) nobody reads from cover to cover. The book’s project, as stated, is to describe this hidden practice and make it intelligible to non-mathematicians.
In the first part of the book we get examples by analogy, in which mathematical learning is compared to learning how to use tools, language, or bodily techniques. In that sense, mathematics is more like yoga or a martial art: progress depends on practice, posture, rhythm, patience, and the ability to remain with confusion without fleeing it. It starts from normal human powers. It helps to have a ‘child’s pose’ in which we do not worry about making mistakes; in fact, it is key to value these mistakes as the way of honing in on the correct intuitions that we should be pursuing. Progress depends on not fleeing confusion, on resisting discouragement, and on being willing to have one’s intuition corrected rather than collapsing into shame or defensiveness.
As the book progresses, these initial arguments get intertwined with personal and historical examples of the thesis to which the author refers (René Descartes, Alexander Grothendieck, Jean-Pierre Serre, William Thurston), set within colorful anecdotes. While doing mathematics implies this jump into intuitions, which are then strengthened and transferred to the more opaque and rigorous clothing of mathematical writing, and while it forms a kind of self-reinforcing loop, it is also open to dangers of paranoia and obsession, exemplified by the figure of Ted Kaczynski. Toward the end, Bessis delves into an exploration of the philosophy of mind and language, in a transition that is paralleled by his own abandonment of academic math education and move toward AI research. Bessis explores how abstraction actually works in perception, language, and the brain, and argues that concepts are not sharply defined entities but layered, neural, perceptual constructions. In this regard, mathematics differs from ordinary language, which is vague and referential by definition, not because it escapes abstraction, but rather because it creates unusually stable and coherent conceptual networks. Therefore, in the last pages, the author presents what he sees as the true function of mathematics: to transform human cognition. Mathematics works like a truth-constrained fiction: we imagine abstract objects as real, and this disciplined imagination has the power to change the structure of our understanding itself.
The epilogue uses Hardy and Ramanujan as emblematic figures of the two poles of mathematics: formal rigor and powerful intuition. Hardy represents the formalist side associated with proof, axiomatization, and the modern ideal of rigor; Ramanujan, the extreme intuitive side, produced astonishing true results without ordinary proofs. Their relationship serves as a final case study of the book’s central claim that mathematics lives in the tension and dialogue between intuition and formalism, not in either pole alone.
*My impressions*
Overall, I both liked the book more than I expected and have updated on some views after reading it. It is a very readable volume, with very little math, so it should be accessible to a very wide audience.
As to its key argument, I feel a complaint can be made that while the author presents ‘what math truly is’, he gives very little in the way of practical advice and of methods to train it. It should come as no surprise that there’s no Royal Road to Geometry: what Bessis proposes is likely as difficult as any of the traditional study paths, but with the redeeming quality that it becomes more of a matter of grit and persistence than of innate talent. Personally, I would like to believe this to be true, and I have some sympathy for the argument: as an ESL teacher, I am aware how frustrating many aspects of language learning are, and how they work through unpredictable phase shifts, i.e., you spend some time x making really slow advances and then suddenly, quite unexpectedly, you discover that you can actually understand conversations and recordings much better than you thought possible, and a lot of the words come to you. It brings to mind the famous “And then a miracle happens” cartoon.
The argument that math is hard to teach not just because it is abstract, but because the crucial operations happen in the head and cannot be directly imitated the way one imitates spoon use, music, or sport feels very convenient for the author’s argument, and I’d consider it an extraordinary claim that warrants some extraordinary evidence. As far as I’m aware of, though, no educational institutions are trying to follow anything like the program and prescriptions that Bessis is making in this book. Is the argument still too new? Is there a lack of experimental evidence that it is indeed correct? These are questions which a pop math book probably can’t try to answer, but I am left rather cluelessly pondering them.
After finishing the book, I still think the author is underplaying the importance of mathematical talent. Even if one accepts most of this argument, it still seems obvious to me that it doesn’t explain actually existing mathematical genius, which I don’t feel can be reduced to just having had the luck of developing ‘the right way of looking at things’. People are notoriously bad at arriving at average standards and capacities when those differ greatly from oneself, and I still think that even relatively small differences in IQ can, potentially, have rather massive effects (as evidenced by distribution tails or nonlinear dynamical systems). How much effect? I really don’t know, but I am open to mathematical ability being a combination of the kind of mental intuition and training that Bessis proposes and some degree of innate skill, in whatever ratio.
Even so, I think the book remains valuable as an attempt to describe what mathematical thinking feels like from the inside, and as a corrective to crude views of mathematics as mere formal rule-following. And this is not only a pet peeve of David Bessis. In the aforementioned Mathematics Without Apologies, Michael Harris makes a case for two paths that modern Mathematics is exploring, and which somewhat map to early 20th century philosophical disquisitions: the more formalist, Type 2 view approximates Hilbert’s dreams of a mechanical evolution of the field, now instantiated by machines that can work in this way better than any humans can hope for; and a more humanistic one, focused on making proofs and mathematical endeavors significant to society and to its practitioners: not so much an issue of finding empty and impossible to understand proofs (like the infamous one for the 4 Color Theorem) but of understanding, of beauty and of elegance. And now, through Bessis, we get a further item to hang from the mathematical clothes-peg: the hope of a better training of one’s mind and an expansion of its capabilities.
I fell in love with mathematics a bit more than a decade ago, when I was finally able to overcome my fears by starting from scratch on Khan Academy and completing everything they had there, including Calculus. Since then, I’ve continued reading and studying, with my current goal updated to mastering about everything that would be taught in an undergraduate pure math degree. From previous posts here, you can see that the serious stuff I am plodding through at the moment is Real Analysis, but I still find time to take detours into popular math books, which tend to be more rewarding and faster reads than textbooks.
Pop math has to face a really tough uphill battle, and there is a very narrow Goldilocks zone in which it works. For professional mathematicians and those steeped deeply enough in the field, these books will for the most part be uninteresting and hand-wavy. For the broad public, including the subset of those who read popular science, math tends to be too daunting. And while you can write books about Physics or Biology with very few equations and lots of metaphors and practical examples, this sort of defeats the purpose if the topic of the book is Math itself.
Common strategies are to focus on the biographical stories of great mathematicians, or to cherry-pick a selection of some of its accessible, entertaining, and most popular bits. The usual suspects (infinities, fractals, prime numbers, Fibonacci and the Golden Ratio, counterintuitive probabilities, famous problems, solved or not...) sometimes make it into books of their own.
Another subclass consists of books that delve into the psychology and sociology of mathematical practice and teaching. They try both to describe what doing mathematics is really like, moving away from the stereotypes of the crazy, introverted genius and of Mathematics as drudgery, brain-killing rote computation, and they usually go on a detour to complain that secondary and high school math fail abysmally to show what ‘real mathematics’ is like. Frenkel’s Love and Math has a bit of this, with a very vivid metaphor in which he describes what is done in schools as trying to teach art by getting students to paint fences instead of pictures. Eugenia Cheng’s The Joy of Abstraction goes out of its way to be as accessible as possible in presenting to a broad audience an area as arcane as Category Theory, and in the first parts of the book makes a good case for what the title says (i.e., how abstraction can be both enjoyable and pragmatically useful for whatever interests you have), while lamenting math-phobia and, if memory serves, putting a lot of the blame on how it’s taught. Michael Harris’s Mathematics Without Apologies doesn’t go too much into pedagogical issues, but it does explore in depth what it is like to do mathematics ‘from the inside’ as a professional, and demolishes quite a lot of stereotypes and preconceptions, including among practicing mathematicians.
Mathematica falls in line with those latter three books. We’ll make a summary of its contents in a minute, but before that, I will abuse your patience a bit longer with a section about my expectations for this book.
*What did I expect?*
I am a very picky reader, so I wouldn’t have picked up this book if I hadn’t expected to enjoy it and/or learn something from it. I follow David Bessis on X, and am superficially acquainted with who he is and with some of the arguments he makes. He is (was?) a professional research mathematician, and given that I am not much of a contrarian, this gives him a lot of auctoritas in my eyes, i.e., I assume he knows what he’s talking about when he writes about Mathematics and that he knows it better than I do.
Two things that I strongly disagree with him about do tend to appear frequently in his tweets. I am a full-blooded mathematical Platonist: the core value I see in Math is that it is the discipline that most closely approximates my yearning for absolute Truth with a capital T, one completely independent of human social construction; and, given that I assume a correspondence theory of truth, this forces me to acquiesce in something like the existence of a Platonic realm of mathematical objects and/or structures. Bessis, on the other hand, is more of an intuitionist and has words of praise for Reuben Hersh, who tends to reduce the discipline to a human activity rooted in our culture and history, similar in this regard to laws, institutions, or money. This is anathema to me.
The second bone of contention I have with him is his general views on math education and the intelligence required for doing professional mathematics. Nobody is very happy with the results of math education -and least of all the teachers- but I don’t feel it is fair to lambast it from the point of view of what a pure mathematician would expect it to do, as, in fact, high school math does a lot of things that are necessarily misaligned with those goals by default (like acting as a gatekeeper to more demanding university studies, or helping students internalize some core mathematical subjects that are instrumentally useful for science and economics). As for intelligence, genius, and IQ in math, this is an ideological minefield into which I loathe to step, but readings in recent years have strongly stirred me away from blank-slatism, which is the de facto dogma of our societies. Math, in fact, is one of the few areas in which it is generally rejected, and before reading the book, Bessis did give me the impression of being anti-IQ for what were likely ideological reasons. Making mathematics ‘accessible to all’ and ‘conscious of gender and race oppressions’ are empty, sectarian mantras to which I am no less averse than I am to viewing Mathematics as a human fiction like magical spells and Harry Potter books.
I was therefore expecting the author to challenge my views with more elaborate intellectual arguments than what can be gleaned from social networks alone. He didn’t disappoint in this regard, although the book only marginally touches on the philosophy of mathematics.
*The book’s core thesis*
Bessis believes that what mathematics is and how people learn to do it are grossly misrepresented and misunderstood. Tweaking the traditional two-systems classification by Kahneman where System 1 is a set of fast, mostly effective but also mostly unreliable and incoherent intuitions and System 2 is the slow, procedural, rational (and inhuman) type of reasoning we associate with logic, mathematics and proofs, he postulates that (doing) mathematics is really working in System 3, which is a balancing of both, and one in which we clarify, sharpen, naturalize and internalize our intuitions from System 1 by forcing them to interact and take into account the straightjacket of System 2. Doing and learning mathematics, therefore, happens through intuition, mental imagery, and repeated internal practice, through imagining, manipulating, testing, and refining abstract objects until they’ve become familiar and obvious. In a way, this is no different from how we push ourselves to learn new, unexpected stuff which after learning becomes trivial and self-evident, like riding bikes, using spoons or putting correctly shaped blocks into holes. Perhaps the greatest difference with math though is that you have to become savvy at manipulating imaginary objects purely in the mind.
*Now, in more detail*
The book begins by rejecting three common myths: that doing mathematics reduces to thinking logically, that some people are born with special mathematical intuitions, and that great mathematicians have brains that are substantially different from normal people’s. Against this, Bessis proposes that math is fundamentally a corrected intuition that can be trained and strengthened. This intuition is seldom talked about and even more seldom taught, even if it constitutes an indispensable tool for working mathematicians, and constitutes a type of ‘secret math’ that has to be discovered by them, as opposed to the ‘official math’ of the formal language of proofs and mathematical textbooks, which (apparently) nobody reads from cover to cover. The book’s project, as stated, is to describe this hidden practice and make it intelligible to non-mathematicians.
In the first part of the book we get examples by analogy, in which mathematical learning is compared to learning how to use tools, language, or bodily techniques. In that sense, mathematics is more like yoga or a martial art: progress depends on practice, posture, rhythm, patience, and the ability to remain with confusion without fleeing it. It starts from normal human powers. It helps to have a ‘child’s pose’ in which we do not worry about making mistakes; in fact, it is key to value these mistakes as the way of honing in on the correct intuitions that we should be pursuing. Progress depends on not fleeing confusion, on resisting discouragement, and on being willing to have one’s intuition corrected rather than collapsing into shame or defensiveness.
As the book progresses, these initial arguments get intertwined with personal and historical examples of the thesis to which the author refers (René Descartes, Alexander Grothendieck, Jean-Pierre Serre, William Thurston), set within colorful anecdotes. While doing mathematics implies this jump into intuitions, which are then strengthened and transferred to the more opaque and rigorous clothing of mathematical writing, and while it forms a kind of self-reinforcing loop, it is also open to dangers of paranoia and obsession, exemplified by the figure of Ted Kaczynski. Toward the end, Bessis delves into an exploration of the philosophy of mind and language, in a transition that is paralleled by his own abandonment of academic math education and move toward AI research. Bessis explores how abstraction actually works in perception, language, and the brain, and argues that concepts are not sharply defined entities but layered, neural, perceptual constructions. In this regard, mathematics differs from ordinary language, which is vague and referential by definition, not because it escapes abstraction, but rather because it creates unusually stable and coherent conceptual networks. Therefore, in the last pages, the author presents what he sees as the true function of mathematics: to transform human cognition. Mathematics works like a truth-constrained fiction: we imagine abstract objects as real, and this disciplined imagination has the power to change the structure of our understanding itself.
The epilogue uses Hardy and Ramanujan as emblematic figures of the two poles of mathematics: formal rigor and powerful intuition. Hardy represents the formalist side associated with proof, axiomatization, and the modern ideal of rigor; Ramanujan, the extreme intuitive side, produced astonishing true results without ordinary proofs. Their relationship serves as a final case study of the book’s central claim that mathematics lives in the tension and dialogue between intuition and formalism, not in either pole alone.
*My impressions*
Overall, I both liked the book more than I expected and have updated on some views after reading it. It is a very readable volume, with very little math, so it should be accessible to a very wide audience.
As to its key argument, I feel a complaint can be made that while the author presents ‘what math truly is’, he gives very little in the way of practical advice and of methods to train it. It should come as no surprise that there’s no Royal Road to Geometry: what Bessis proposes is likely as difficult as any of the traditional study paths, but with the redeeming quality that it becomes more of a matter of grit and persistence than of innate talent. Personally, I would like to believe this to be true, and I have some sympathy for the argument: as an ESL teacher, I am aware how frustrating many aspects of language learning are, and how they work through unpredictable phase shifts, i.e., you spend some time x making really slow advances and then suddenly, quite unexpectedly, you discover that you can actually understand conversations and recordings much better than you thought possible, and a lot of the words come to you. It brings to mind the famous “And then a miracle happens” cartoon.
The argument that math is hard to teach not just because it is abstract, but because the crucial operations happen in the head and cannot be directly imitated the way one imitates spoon use, music, or sport feels very convenient for the author’s argument, and I’d consider it an extraordinary claim that warrants some extraordinary evidence. As far as I’m aware of, though, no educational institutions are trying to follow anything like the program and prescriptions that Bessis is making in this book. Is the argument still too new? Is there a lack of experimental evidence that it is indeed correct? These are questions which a pop math book probably can’t try to answer, but I am left rather cluelessly pondering them.
After finishing the book, I still think the author is underplaying the importance of mathematical talent. Even if one accepts most of this argument, it still seems obvious to me that it doesn’t explain actually existing mathematical genius, which I don’t feel can be reduced to just having had the luck of developing ‘the right way of looking at things’. People are notoriously bad at arriving at average standards and capacities when those differ greatly from oneself, and I still think that even relatively small differences in IQ can, potentially, have rather massive effects (as evidenced by distribution tails or nonlinear dynamical systems). How much effect? I really don’t know, but I am open to mathematical ability being a combination of the kind of mental intuition and training that Bessis proposes and some degree of innate skill, in whatever ratio.
Even so, I think the book remains valuable as an attempt to describe what mathematical thinking feels like from the inside, and as a corrective to crude views of mathematics as mere formal rule-following. And this is not only a pet peeve of David Bessis. In the aforementioned Mathematics Without Apologies, Michael Harris makes a case for two paths that modern Mathematics is exploring, and which somewhat map to early 20th century philosophical disquisitions: the more formalist, Type 2 view approximates Hilbert’s dreams of a mechanical evolution of the field, now instantiated by machines that can work in this way better than any humans can hope for; and a more humanistic one, focused on making proofs and mathematical endeavors significant to society and to its practitioners: not so much an issue of finding empty and impossible to understand proofs (like the infamous one for the 4 Color Theorem) but of understanding, of beauty and of elegance. And now, through Bessis, we get a further item to hang from the mathematical clothes-peg: the hope of a better training of one’s mind and an expansion of its capabilities.
February 17, 2025
I genuinely this this book was so interesting and explained the feeling of math so well. I loved all the stories incorporated into the message, I would recommend this book to anyone!
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