Mathematica: A Secret World of Intuition and Curiosity

by David Bessis, Kevin Frey (Translator)

A fascinating look into how the transformative joys of mathematical experience are available to everyone, not just specialists
 
Math has a reputation for being inaccessible. People think that it requires a special gift or that comprehension is a matter of genes. Yet the greatest mathematicians throughout history, from René Descartes to Alexander Grothendieck, have insisted that this is not the case. Like Albert Einstein, who famously claimed to have “no special talent,” they said that they had accomplished what they did using ordinary human doubts, weaknesses, curiosity, and imagination.
 
David Bessis offers an illuminating guide toward deeper mathematical comprehension and reconnects us with the mental plasticity we experienced as children. With simple, concrete examples, Bessis shows how mathematical comprehension is integral to the great learning milestones of life, such as learning to see, to speak, to walk, and to eat with a spoon.
 
Focusing on the deeply human roots of mathematics, Bessis dispels the myths of mathematical genius and offers an engaging initiation into the experience of math not as a series of discouragingly incomprehensible logic problems but as a physical activity akin to yoga, meditation, or a martial art. He opens the door to changing the way you think not only about math but about intelligence, intuition, and everything that goes on inside your head.

Genres: Mathematics, Nonfiction, Science, France, Philosophy, Audiobook, Popular Science

344 pages, Hardcover

Published May 28, 2024

Reviews

[Haengbok92 · Rating 5 out of 5]

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.

[Rrrrrron · Rating 3 out of 5]

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.

[Ali · Rating 5 out of 5]

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.

[Vicente Mendes · Rating 5 out of 5]

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.”

[Tartiff · Rating 5 out of 5]

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.

[Ana Bernardino · Rating 2 out of 5]

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.

[Roozbeh Daneshvar · Rating 2 out of 5]

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:

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.

[Chris Wang · Rating 4 out of 5]

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.

[Jose Antonio · Rating 4 out of 5]

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

[Samet Öksüz · Rating 5 out of 5]

Okumakla kalmadım, Türkçe tercümesini tamamladım. 2025'in ilk aylarında raflarda olacak,

[Thomas · Rating 4 out of 5]

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.

[Vasil Kolev · Rating 5 out of 5]

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)

[Cody]

Not 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.

[Shahzeb K · Rating 5 out of 5]

Best non-fic read of 2025.

Anyone can do math. Just like anyone can ride a bike.

[Stan · Rating 5 out of 5]

A surprisingly good book. Also, one doesn’t have to be interested in Maths to gain some value out of it.

[Hector · Rating 5 out of 5]

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.

[Julia Green · Rating 5 out of 5]

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!

[Elisabetta · Rating 4 out of 5]

Letto insieme a GDL Planet Earth

Quando scopriamo un nuovo concetto matematico, facciamo fatica a immaginarlo. Ci appare sotto forma di una definizione astratta, di una serie di parole sulla pagina o di parole pronunciate da un professore. Questa sequenza di parole, per noi, non ha alcun significato. Non evoca nulla.

Non lasciatevi ingannare dal titolo, o meglio non lasciatevi spaventare. Sorprendentemente, almeno per me, è che alla fine si parla di pensiero astratto e di come il "trucco" della matematica sia riuscire a costruirsi le immagini mentali di quei concetti che su carta appaiono complessi. Se hai quell'immagine, allora tutto è talmente ovvio da risultare banale. E difficile da spiegare a qualcun altro. Un po' come quando da bambini capiamo che il cerchio è un cerchio. Prima mica era chiaro, poi lo è diventato. Si è creata quell'immagine e dalla nostra mente non si scolla, un cerchio è un cerchio. Punto. Però vai a spiegarlo a qualcuno che un cerchio non sa cosa sia, non è mica così semplice.
Ecco, il problema sta (più o meno) tutto qui.

Per fare una scoperta matematica, bisogna iniziare a inventare nuovi gesti mentali, a creare nuove immagini nella propria testa, senza sapere in anticipo come farlo e senza essere sicuri che funzionerà.

Recensione completa sul blog Bibliofili per natura

[SeaShore · Rating 2 out of 5]

Excellent. Again, the Brain is remarkable. Our thoughts rule.

He makes reference to William Thurston, Grothendieck among other Mathematicians who influenced his thinking.
Referencing synesthesia he reminds us of how the brain evolves as children get older and images we take for granted. Consider the image of a spoon with food. Synesthesia is when your brain routes sensory information through multiple unrelated senses, causing you to experience more than one sense simultaneously. Some examples include tasting words or linking colors to numbers and letters. It’s not a medical condition, and many people find it useful to help them learn and remember information. I like the example he uses where we can imagine a large number when given a number like 999,999,999, 999, 000.

[Tara · Rating 4 out of 5]

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.

[Fabrice · Rating 5 out of 5]

game-changing book

Reading this book has completely transformed my point of view on how learning works and the role of intuition in « rational » thinking.

His cheeky but brutal take-down of Kahneman’s systems 1&2 is also very eye-opening.

I highly recommend it to anyone, especially parents, keen to inspire others to be more curious and thirsty of understanding the world.

[Matthieu Miossec · Rating 5 out of 5]

Après la lecture de ce livre, c’est sur, je vais penser les maths très différemment. Ce livre c’est un de ces petits joyaux qu’on trouve des fois par un heureux hasard. C’est pas à quoi je m’attendais, c’est mieux.

[Jacob Vorstrup Goldman · Rating 5 out of 5]

The best, bar none, book on mathematical thinking I’ve read. The perspective on intuition presented through the trio of Descartes, Grothendieck and Thurston is outstanding and worth everybody’s time.

[Arthur Ryman · Rating 3 out of 5]

The main message of this book is that virtually anyone can understand math if they put in the effort, and they would then be able to experience the pleasure of understanding, but few people are motivated to do so because math is not taught correctly and this causes many people to dislike it. Math teaching emphasizes symbol manipulation and fails to teach techniques that improve imagination and intuition.

I applaud the author for writing this book, but I wonder who is the audience? If the audience is people who have been put off math then the book should have included more convincing examples and avoided reference to philosophers. So perhaps the audience is math educators?

The author claims that presenting math as a formal system (official math) hides what mathematicians actually do (secret math). He asserts that mathematics is done by constructing mental images that enable intuitive understanding of mathematical concepts. These images and concepts form over a long period of time and emerge in our minds through the process of brain plasticity. He then claims that we have no way of explaining to other people what goes on in our mind so we have invented the formal language of mathematics. This language is formal and subject to rigorous mechanical checking.

This seems overstated. When mathematicians explain things they are not restricted to formal symbolic language. Rather, they use symbols and formulas but augment them with informal diagrams. The author claims that Euclid resorted to axioms and logical deductions because he couldn’t explain his thought processes. However, plane geometry is very visual and always illustrated with simple line drawings. Wikipedia shows a surviving fragment of an ancient parchment from The Elements and it contains a simple line drawing. One can only assume that virtually every page contained drawings.

Perhaps he is too influenced by the Bourbaki tradition which is extremely formal and nonvisual. Siobhan Roberts in her biography of Donald Coxeter claimed that the only diagram found in Bourbaki’s book is a Coxeter diagram.

Coxeter himself would have felt at home in any lecture by Roger Penrose who makes liberal use of drawings. Penrose received the Nobel Prize for his singularity theorem which he arrived at through a long process of subconscious visualization about so-called trapped surfaces. However, Penrose’s biggest impact on cosmology may well be his invention of Penrose diagrams which provide a compact two-dimensional representation of infinite four-dimensional space-times.

The use of formal methods in geometry is motivated by caution. It is easy to draw misleading diagrams and deduce false statements from them. Kevin Buzzard cites the famous diagrammatic proof that all triangles are isosceles.

The author has been highly influenced by recent advances in AI. The success of neural networks in machine learning provide a compelling model for what is going on in human brains. Concepts emerge in neural networks after repeated exposure to many examples.

Does the author imply that mathematical concept formation does not always result in literal mental images? If so, then I concede that those would be very difficult to share. However, in practice mathematicians make heavy use of visual imagery when sharing ideas.

The author appears to contradict himself slightly. Consider the following paragraph in the Epilogue.
“... We must force ourselves to imagine it and put words to all our impressions, without being distracted by our constant feeling of inferiority. ...”

Here he says we must both imagine AND put into words. So we need both the official AND the secret math. They are complementary.

[Matteo · Rating 4 out of 5]

Un libro che probabilmente mancava o di cui, almeno io, sentivo la mancanza, siccome parla soprattutto di matematica e di come imparare la matematica.

Il tutto parte da una battuta di Einstein: "Non preoccuparti delle tue difficoltà in matematica; posso assicurarti che le mie sono ancora maggiori." (dalla lettera alla studentessa liceale Barbara Wilson, 7 gennaio 1943)
L'autore si premura di dire che sembra un po' una top-model che ribadisce l'importanza di essere belli dentro (e non fuori).
Ma cosa succede se diamo veramente ascolto a queste parole? A quali scoperte possono portarci?

Il testo narra del viaggio personale dell'autore, il matematico David Bessis, attraverso la "sua" matematica: cosa intende lui per matematica e, soprattutto, come si fa ad apprendere la matematica. Questa "recherche" è poi arricchita da analisi di scritti di Einstein, Decartes, Grothendieck e William Thurston: tutti eminenti matematici che hanno provato a descrivere i loro processi mentali, le loro tecniche e i loro trucchi nel "fare" matematica.

La cosa più incredibile che viene fuori dalle vite di questi matematici è la matematica vissuta come intuizione costantemente affinata. Tutti gli esempi parlano di come si debba sviluppare un'intuizione matematica e che l'unico modo per farlo siano avere un approccio 'fanciullesco' (da bambini), avere una mente aperta, pronti ad indagare tutto, quindi appasionata curiosità, e molta dedizione.
Lo scopo dell'autore è riuscire a far capire che la matematica sia accessibile, anche se non necessariamente facile!, a tutti. Tutti vi si possono dedicare e, anzi, tutti lo facciamo già anche inconsapevolmente, nella vita quotidiana. Abbiamo sviluppato intuizione per forme e numeri e non ce ne rendiamo più conto. Si può fare lo stesso anche per concetti che, a prima vita, sembrano più complicati.

Anche il linguaggio matematico, che può sembrare così ostico, è in realtà una costruzione logica che ci permetta di scrivere (e rendere condivisibile) i nostri pensieri. Non è un compito esattamente semplice. Proprio per questo sembra così difficile. Se dovessimo insegnare ad una persona che non hai mai visto un paio di scarpe come allacciarle usando solo parole scritte, come faremmo? Probabilmente incontreremmo simili difficoltà.

Personalmente (siccome ho un dottorato in matematica applicata), ho molto apprezzato la schiettezza nel riconoscere la matematica come intuizione che si può "imparare". Anche solo questo semplice messaggio vale la lettura del testo, a mio avviso.

Avrei preferito solo un po' più di analisi scientifiche e un taglio più saggistico in alcuni punti. Se non sbaglio, una parte delle tesi dell’autore rientrano nella branchia del “costruttivismo”: la conoscenza non è trasmessa, ma va elaborata personalmente. Questo ha anche implicazioni filosofiche molto interessanti e, soprattutto, c’è una letteratura enorme sulla materia. Ma niente viene proposto in questo libro. Per questo motivo, sfortunatamente, risulta essere piuttosto limitato e limitante.

[William Easton · Rating 5 out of 5]

9.1/10 At last, a suitable replacement to A Mathematician's Apology and A Mathematician's Lament. In the first couple chapters, I was increasingly excited to read things that I personally strongly believed, but had only rarely heard mention of. This book is targeted mostly at those who don't "get" math, and to those people I could hardly recommend it more highly, but I think the math people of the world also have much to gain from reading it. If nothing else, some validation. I don't agree with everything he says (the revulsion towards math books feels a bit extreme to me), and there are some parts that I wouldn't miss (mainly the thankfully brief discussion of AI), but by and large a very worthy read. And quite well-written too, though I don't know if the author or the translator is more to thank for that. The last gripe I have is that the author sometimes writes as if he alone knows this secret universe, but really many of the ideas are common knowledge in math departments, if somewhat less explicitly stated.

[Nicola · Rating 3 out of 5]

VOTO: 3.5

PRO: lettura stimolante ed accessibile, sfata il mito del genio matematico

CONTRO: qualche divagazione di troppo, frequenti rimandi ai capitoli successivi, trattazione astratta

Un saggio sull'apprendimento della matematica, che sfata alcuni miti legati all'insegnamento tradizionale e rivela la relazione profonda tra intuizione e formalismo ed il ruolo dell'immaginazione, della creatività e della curiosità nel pensiero matematico.
Una lettura che, nelle intenzioni dell'autore, dovrebbe aiutare a migliorare la comprensione della matematica e, attraverso di essa, della realtà che ci circonda, ma che deluderà chi è alla ricerca di qualche consiglio pratico.
Quello che ci si ritrova tra le mani sembra assomigliare ad un trattato di filosofia (niente di così strano considerato che nell'antichità i matematici erano prima di tutto filosofi e pensatori), ma alla lunga la scelta di focalizzarsi sulla propria esperienza personale, comune ad altri grandi matematici del passato, rende la lettura abbastanza ripetitiva e offuscare il messaggio centrale, ossia l'importanza da un lato di usare l'immaginazione per costruire immagini e modelli mentali dei concetti matematici per renderli comprensibili e manipolabili, dall'altra di saper affinare la nostra intuizione tramite un processo di formalizzazione, che ne evidenzi difetti e limiti.

[Fabio · Rating 4 out of 5]

The book narrates a professional mathematician's experience in explaining his creative process. The first third, which focuses on his personal experience, is the most interesting part. The author argues strongly that he is not the genius he thought mathematicians were and initially believed his results were somewhat fraudulent. He notes this is a common view among people in the field and those who try to dominate conversations often lack expertise. To make the topic more relatable, he mentions that he owns almost no math books and that they are rarely read because most are too dense and difficult to get through.

The bulk of the book highlights how becoming accustomed to mathematical thinking takes time and is not about following formal proof logic, which comes last. It involves visualizing concepts and using these mental images to build an instinctual understanding. The author extends the idea of fast and slow thinking by suggesting a third category: reshaping intuition when it is incorrect, which he believes is where the best mathematical thinking lies.

In the second third of the book, the author gets "lost" trying to explain the processes of other mathematicians like Descartes.

[Jonathan Neumann · Rating 5 out of 5]

A must read for children and for adults interested in grasping how to approach mathematics. The main message is this: Maths is a way to train your imagination and intuition and ultimately, your grasp of life and of the world, AND THAT'S ALL THAT MATTERS.

Isn't it mind-blowing? How much does it go against everything we've been taught in school which has led to this fear of maths we all have?

I really like the emphasis of the author on the subjective experience, on the role of intuition and on debunking maths as an unfathomable topic. He makes maths feel a lot more... familiar. A lot more approachable. A lot more achievable, for all of us!

The one little piece of criticism I would have, is that the author's approach could have more of a step by step guide at the end of the book for those of us interested in revisiting and really grasping mathematics. Yes, I know it's a subjective experience and you gave a few tips and tricks that worked for you, but I feel like there is more to it. Something that could apply to anyone. Like some sort of guidance to teachers and school children that could be followed. Some kind of success criteria. There is definitely more that can be developed, and I hope you will seriously consider this question.

[Eric Nehrlich · Rating 5 out of 5]

This is one of the few books I've read that changed my perspective. Bessis describes the feeling of grappling with concepts that are mentally just out of reach, as well as the critical step of checking those concepts against reality. I loved his description of "System 3", extending Kahneman's systems 1 and 2, which always felt limited to me, so Bessis's conception of guided intuition as a third system aligns with my experience.

Beyond that, he has an inspirational message - all of us humans are more capable than we think we are, if we can stay in the discomfort of not knowing. He shared his breakthrough moment, when a famous mathematician asked him to explain something after a talk he gave because "I didn't understand any of that". That fearless willingness to embrace what one doesn't understand unlocked Bessis's own capabilities as a mathematician - he stopped trying to impress others with how much he already understood, and instead focused on learning (to use Carol Dweck's terminology, he let go of a fixed mindset to embrace a growth mindset).

Inspiring, hopeful and insightful. Highly recommend.