Mathematica - by David Bessis

ISBN: 9780300270884
Date read: 2025-11-14
How strongly I recommend it: 8/10
(See my list of 430+ books, for more.)

Go to the Amazon page for details and reviews.

Math is imagination, visualization, and intuition. The symbols are just a language to explain the mind’s image. You can train your intuition and develop visualization skills with practice. Math is an inner tool to enhance human cognition, more akin to psychology.

my notes

If we taught children music by giving them the written scores to decipher - without their ever having heard it played - music would be as universally hated as math.

Intuition is the soul of mathematics.

No one really knows how to define mathematics.

By studying math, you can learn how to translate your visual intuition into rigorous proofs.
It will never be a perfect translation.
It takes a lot of words to express a simple intuition.
It all seems so clear in your head.
But once you start to write it down, it seems technical and complicated.

There are many different sizes of infinity that can be precisely described.

Everyone around me seemed to be better at math.
I wanted to know how to do real math, difficult math.
But all that I was able to learn was the easy math.
It was only an optical illusion.
The horizon was shifting with me - always staying at my level.
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.

He doesn’t really read what’s in the book.
He prefers to concentrate on “the thoughts between the lines.”
Once he has a clear idea, the formalism and all the technical details suddenly seem useless and superfluous:
“When the idea is clear, the formal setup is usually unnecessary and redundant. I often feel that I could write it out myself more easily than figuring out what the authors actually wrote.”
“It’s like a new toaster that comes with a 16-page manual: If you already understand toasters and if the toaster looks like previous toasters you’ve encountered, you might just plug it in and see if it works, rather than first reading all the details in the manual.”

You’re capable of synesthesia.
Looking at the word chocolate, are you able to sense a sound, a color, a taste?
Looking at “999,999,999,” do you get the feeling of something large?
Be aware of your capability for synesthesia and try to develop it systematically.
Secret math is a mental yoga whose goal is to retake control over our ability for synesthesia.

Mathematics is a sensual and carnal experience that is located upstream from language.

The quality of your inventiveness and your imagination comes from the quality of your attention, listening to the voice of things.

You find yourself in the pilot’s seat of a commercial jet or the command post of a nuclear generator:
There are a lot of buttons and screens, but you have no clue how they work and an intense desire not to make a mistake.
You would love to know how it all works, but you don’t.
The normal reaction is to stay seated and not touch anything.
Before making any move you need to study and think about it.
But if you put any two-year-old in the pilot’s seat, they’ll act differently.
They’ll push all the buttons, starting with ones that are red or blinking.
Act like the two-year-old.
When you want to understand something, go straight at it, without hesitations, as a child would.
Don’t wait to understand before launching into it.
Act without thinking, a bit haphazardly.

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

It’s worthless to gather information about things that you can’t yet see.
Instead, allow yourself to imagine the things right away, without waiting.
Even when you’re well aware that it might not work and your mental images will likely be terribly wrong.
Don’t be afraid of failure.
Even be certain that you’ll be wrong, and that’s exactly what you’re looking for.
Actively seek out the error as a young child actively seeks mischief.

Each time you find something bizarre or intriguing, unclear or unsatisfactory, incoherent or disagreeable, that’s where you began digging.
Finding mistakes is a crucial moment, a creative moment, in all work of discovery.
It’s a moment when our knowledge of the thing being examined is suddenly renewed.

Fear of mistakes and fear of the truth is one and the same thing.
The person who fears being wrong is powerless to discover anything new.

Learning to see, to walk, use a spoon, tie your shoelaces, talk, read and write, is always about reconfiguring your brain.
And it’s never done in one shot.
Children don’t learn how to walk until they’ve tried and failed.
They need to fall in order to learn how to stand up.
It’s the accumulation of errors that allow them to develop their intuitive sense of balance.

Logic doesn’t help you think.
It helps you find out where you’re thinking wrong.

The written transcription of what seems obvious to you might be 10, 100, 1000 times as long as the summary you make for yourself in your head.
Even then, you’ll have to leave aside a bunch of details you won’t have the heart to write down.
Visualize what you do when you tie your shoes.
Now take pen and paper and try to describe each movement exactly, so that an absolute beginner could follow your instructions and get the same result.

Mathematical comprehension is 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.

Mathematicians use logic and language as an apparatus for learning to see.

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

When you look at photos, you have the sense of actually seeing scenes that occur in three dimensions.
It doesn’t require any particular effort.
It doesn’t tire you out.

Transforming mathematical definitions into mental images is so important.
When you’re unable to imagine mathematical objects, you don’t really understand them.

The fourth dimension is whatever you want it to be.
When you do geometry in two dimensions, on a plane, a point is determined by two coordinates, generally called x and y, that represent exactly what you want them to represent.
In a space with ten dimensions, a point is determined by ten coordinates that are generally called x1...x10.
If you want these coordinates to represent something, it can be whatever you want.
If you wanted to describe the geographic expansion of an invasive population of rabbits, you would need to think in four dimensions, since you’d need four coordinates: longitude, latitude, time, and population density.

The sole function of mathematical statements is to help you generate mental images, and only these images will lead to comprehension.
Once you have the correct mental images, everything else becomes clear.

Mathematical intuition is so banal, simple, and stupid, that you need a lot of self-confidence not to throw it in the trash.

The work of reeducation is based on the repetition of the same exercises of imagination.
Progress is slow because the body needs time to transform itself.
You just need to commit to a regular training schedule.

Five-dimensional shapes are hard to visualize - but it doesn’t mean you can’t think about them.
Thinking is really the same as seeing.
For a mathematician, “seeing” signifies thinking in a rapid and intuitive manner, directly, without need for reflection, as if the object really existed, as if it were right there in front of you.

The essence of mental plasticity is to transform audacity into competence.
The process is slow and invisible, and at first success seems unachievable.
That’s the biological reality of our learning mechanisms.
By an unfortunate coincidence, that’s also the perfect recipe for discouragement.
You need a lot of self-control and self-confidence to commit to a process that’s confusing, slow, and uncertain.

Try to recall your dreams, to put words to the fleeting impression that left a strange taste in your mouth, to sort out your most confused and contradictory ideas.

The stupid images in my head had a tendency to correct themselves once I made the effort to describe and name them, when I got into the habit of lending an ear to the dissonance between my intuition and logic.

If my intuition tells me to choose option A and my reason tells me to choose option B?
I tell myself there’s something going on and I’m not ready to make the decision.
That’s the moment to resort to an assortment of introspection and meditation techniques aimed at establishing a dialogue between intuition and rationality.
In practical terms, here’s what that means.
When my intuition tells me A and rationality tells me B, I put myself in the position of a referee.
I force myself to translate my intuition into words, to tell it like a simple and intelligible story.
Vice versa, I try to picture what logical reasoning is actually expressing, to experience it in my body, to hear what it’s trying to say.
I ask myself if I really believe it.
The goal is to understand where things are going wrong.
In the end, it’s almost always my intuition that wins.
When I force it to listen to what logic is saying, it takes that into account and adjusts its position.
Logic is something inert, like a pebble.
My intuition is organic, it is living and growing.
I’m personally incapable of thinking against my intuition and I have serious doubts as to the sincerity of people who claim they can.

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

You can reprogram your intuition.
Any misalignment between your intuition and reason is an opportunity to create within yourself a new way of seeing things.
Don’t expect it all to come at once, in real time.
Developing mental images means reorganizing the connections between your neurons.
This process is organic and has its own pace.
Don’t force it.
Simply start from what you already understand, what you can already see, what you find easy, and just play with it.
Try to intuitively interpret the calculations you would have written down.
If it helps, scribble on a piece of paper.
With time and practice, this activity will strengthen your intuitive capacities.
It may not seem like you’re making progress, until the day the right answer suddenly seems obvious.

Nothing is counterintuitive by nature.
Something is only ever counterintuitive temporarily, until you’ve found means to make it intuitive.
Explaining something to others is proposing simple ways of making it intuitive.

Every time we practice a given activity we habituate our System 1 to the specifics of that activity.

By telling us there was a “trick,” he sent us the wrong message.
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.

Superstition: This belief that our intuition isn’t worth a dime and that we have to mechanically apply methods that we don’t fully understand.

It can happen that things work without our understanding.
But it’s always a temporary situation that’s just waiting for an explanation.
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.

Each time someone talks to you about “tricks,” they’re telling you to stop thinking at precisely the moment when it starts to get interesting.

Mathematics is the science of imagination.
The real enemy of imagination, which blocks understanding and makes us feel like fools, is fear.

Descartes positioned himself in opposition to official knowledge.
He lived in a world where truth was still conflated with authority: truth was tradition, what was written in books.

Take doubt seriously.
Everyone can get personal benefit from Cartesian doubt.

You can only doubt with your gut.
All Cartesian doubt is visceral.
Doubt is personal and intimate.

To doubt is to give an argument the sniff test and sense that there’s something off.
It’s allowing yourself to ask, “What? Really?”
To doubt something is to be able to imagine a scenario, even seemingly improbable, where the thing could be untrue.
Doubt not only what others say, but also, and above all, your own certitudes.
Cartesian doubt is a universal technique for reprogramming your intuition.
Doubt is a technique of mental clarification.
It serves to construct rather than destroy.

Arrogant people who love being contradicted?
Show-offs who smile when you prove them wrong?
Dogmatists ready to change their mind in a heartbeat?
I’ve encountered this singular attitude only among very good mathematicians.

If only the mental actions of mathematicians were visible.
In losing the possibility of imitation, we also lose our main driver of desire.
When you were a child, no one needed to make you want to ride a bike.
No one had to convince you that it would serve you well later on in life, or that it would look good on your CV.
You saw other kids riding bikes.
You liked it and you wanted to do the same.

Transcribing my dreams:
As an effect of my trying to memorize them and put them into words, my dreams grew in richness and precision.
Each with a complete story and enough details to fill up a lot of pages.
I wasn’t looking for meaning in them.
I just wanted to master the art of writing them down.
To me, this is the essence of writing.
Starting from images and sensations and seeking a way to render them in words, to make them clear and solid.
Transcribing the situations, what’s at stake - capturing the moods, the music, the smells, the textures.
If you can do that, you can do anything.
It’s an ability you develop through practice.
There are techniques to begin and techniques to get better.
The more faithfully you learn to transcribe what you see, the more you see.

The special state of mind just before falling asleep:
On subjects that preoccupy me, I let myself be filled with them.
Contemplating it without a goal, almost like dreaming.
I try to remember all the rooms I’ve slept in.

To practice switching viewpoints, a good exercise:
1.
Choose a random reference point around you, for example, the corner opposite from you in a room, or the window of a house when you’re walking in the street.
2.
Try to imagine what you’d see if you were looking in your direction from this reference point.

Creativity emerges when we force ourselves to continue looking at things that intimidate us until they finally become familiar and obvious.

Our well-being depends on many things we do that are hard to explain in an intellectual way.
Bare reason is likely to lead you astray.
None of us are smart and wise enough to figure it out intellectually.

When someone comes up with a reasoning where everything all fits together too neatly, suspect that something isn’t right.
In some cases, rationality leads you astray from the truth.
You can apply the methods used by mathematicians outside of math.
But when we use it outside of mathematics, we need to be careful: it’s only within mathematics that this method is able to produce unshakeable truths.
By anchoring our convictions in indisputable evidence and rigorous deduction, we can turn them into certitudes that, over time, become as strong as reinforced concrete.
Except that sometimes these certitudes are false.

The riddle of the chicken and the egg is supposed to present us with a “paradox.”
A sort of unbreachable wall to human comprehension, before which we have no other choice but to bow down.
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.
This way of solving the riddle, however, leaves aside its most troubling aspect:
Why was there a riddle in the first place?
How can it be that, starting from a hypothesis that is indisputably true, following a reasoning that is indisputably correct, we arrive at a conclusion that is indisputably false?
Language is structurally incompatible with logical reasoning, and we can never have 100 percent certainty in truths expressed in human language and arrived at through deductive logic.

Rationality should be used as a guide rather than an ultimate judge.

When we say that something is “true,” we never mean it literally.
We only ever use the word as a shortcut for all these other things, because otherwise we’d have no occasion to use it.

In acknowledging the intrinsic limitations of our language, Wittgenstein made one of the great philosophical breakthroughs:
He broke with a multi-millennial tradition dominated by metaphysics, in which philosophers believed that it was possible to attack, using rationality, problems

Math is an inner tool.
Its main purpose is to enhance human cognition.
To develop an intuitive and familiar understanding of mathematical notions extends our intuitive understanding of the world.
The math that you understand augments reality and adds layer of intelligibility.

Without numbers,
without your perception of points and trajectories in a three-dimensional space,
without x and y,
without the concepts of distance, speed, and acceleration,
without probabilities,
... the whole world around you would suddenly become so blurred and unsteady that you’d feel like you’d been lobotomized.

In the 1800s, there were still serious mathematicians who claimed that negative numbers were nothing but a fairy tale.
In the 1500s, even their advocates labeled them absurd numbers.
Since then, it’s as if reality itself had changed.

In practice, mathematics doesn’t have much to do with the hard sciences.
It’s rather more related to psychology, of which it’s a kind of esoteric and applied sub-branch.

To understand math is to reprogram your intuition.
It’s a matter of neuroplasticity.

We learn precisely when we force ourselves to imagine things that we don’t yet understand.