2024-12-07 22:47:11
Words don’t come easily, and they don’t come right away.
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Before Descartes, no one knew that you could describe geometric figures using equations. In Geometry, his 1637 treatise, an appendix to Discourse on Method, he established a bridge between algebra and geometry, two branches of mathematics that had previously been thought of as entirely separate.
2024-12-07 22:54:07
When you look at the world, you can’t help but recognize shapes, size, textures, colors. But there are many other things you might see. There are other structures, other types of shapes, other types of relations between objects. Even if you might not be able to see them now, these shapes and structures could eventually become obvious to you.
2024-12-07 22:55:31
You have to keep in mind that most of the things we learn, we learn through imitation. The instinct for imitation is universal. We share it with all other mammals, and not only them.
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But math, because it relies on unseen actions, can’t be learned through imitation.
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discovery always begins with the simple and innocent desire to understand
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Finding the solution means thinking what had been unthinkable. It’s like augmenting the cognitive capacity of human beings.
2024-12-08 07:28:56
Math is mysterious and difficult because you can’t see how others are doing it. You can see what they’re writing on the blackboard or on a sheet of paper, but you can’t see the prior actions they performed in their heads that enabled them to think and write those things.
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But they know that getting lost is a normal stage in the understanding process.
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felt overwhelmed by the existing knowledge.
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The underlying principle is never to force yourself to follow the pages in order, but to follow your own desire and curiosity.
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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.
2024-12-08 16:02:33
In the world of mathematics, toasters arrive disassembled. We all have to put them together in our own heads. “Bad” teachers are the ones who recite the 198 steps to assemble the toaster as if that were the end of the story. “Good” teachers do their best to explain what a toaster is.
2024-12-08 16:03:39
“One person’s clear mental image is another person’s intimidation,” Thurston wrote.
2024-12-08 17:39:27
Laurent Schwartz, who himself was about to receive the Fields Medal, had him read his latest article, which ended with a list of fourteen problems that he had been unable to solve. It’s the kind of list an ambitious student could dig through for a good PhD subject: choose a problem, spend three years thinking about it, get your advisor to help you find an incomplete solution, and everyone’s happy. Grothendieck went off to his room and came back a few months later. He’d solved all fourteen problems.
2024-12-08 17:41:33
The connection of the mathematical experience with madness is a subject we can’t seem to ignore.
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Grothendieck explains his uncommon creativity by the proximity he maintains with his inner child: “In me, and for reasons I have not yet dreamed of exploring, a certain innocence has survived.”
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Grothendieck’s recommendation is to act like the two-year-old. When he wants to understand something, he goes straight at it, without hesitations, as a child would. He doesn’t wait to understand before launching into it. He acts without thinking, a bit haphazardly:
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Mathematical things are the things that nonmathematicians call mathematical concepts or mathematical abstractions. They may consist of numbers, sets, spaces, different kinds of geometric shapes, or other types of abstract structures. Mathematicians prefer to call them mathematical objects, because imagining these things as material objects that one can touch makes it easier to understand them.
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he’s describing actions that we perform in our heads and mental images that we manipulate, but our language is missing the right words. There is no specific vocabulary to talk plainly of these actions and images.
2024-12-08 17:55:16
“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.”
2024-12-08 17:55:54
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. It’s when we fear making a mistake that the error which is inside of us becomes immovable as a rock.
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mathematical understanding is achieved by gradually modifying the way we represent things to ourselves, and making them clearer, more precise, closer to reality.
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All our great learning achievements rely on this mental plasticity. Error plays a fundamental role, as it is the driving force of plasticity. 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.
2024-12-08 18:01:50
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.
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Through the work of writing, intuition becomes less and less vague and less and less wrong. This process is slow and gradual.
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we’ll see how the peculiarities of mathematical language make it an incredible tool of mental clarification.
2024-12-08 18:06:18
“It’s the only way I have of understanding, through sheer persistence, how things work.”
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The definition is bizarrely tortured and complicated, and most of all, it’s circular:
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Writing mathematics, that is, transcribing mental images with enough clarity and precision to allow others to understand and reproduce them, is an art.
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Learning to write math is learning to have clear ideas. Wouldn’t it be a shame to deprive yourself of that?
2024-12-09 22:36:50
Mathematicians use logic and language as an apparatus for learning to see.
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On paper, this number is an abstract assemblage, logical and cold. Yet in your head, it’s a simple object, concrete and clear as day
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For example, in a three-dimensional space, you can put together twenty equilateral triangles to make a twenty-sided die that looks like that pictured in the figure. This remarkable object, which has been known for ages, is called a regular icosahedron. Looking at the drawing, you have the impression of seeing an icosahedron floating in space. But that’s not really what’s in front of your eyes. What you’re looking at is a two-dimensional page on which is found the image of an icosahedron. More precisely, this image is what is called a projection: it’s the shadow (in two dimensions) of an imaginary icosahedron (in three dimensions).
2024-12-10 22:11:43
Contrary to common belief, it’s never abstraction that makes math difficult to understand. Abstraction is our universal mode of thinking. The words that we use are all abstractions. Speaking, making sentences, is to manipulate and assemble abstractions.
2024-12-11 22:19:56
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.
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Progress is slow because the body needs time to transform itself. It doesn’t help to force it, which may end up hurting you. You just need to commit to a regular training schedule, keep your cool, keep going even when it seems you’re not making any progress. It’s like going to the speech or physical therapist, except you’re all alone and inside your head.
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A conjecture is a mathematical statement that someone believes is valid but isn’t yet able to prove.
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When we look around us, we have the illusion of a direct relation to the world, as if our eyes were magic windows drilled directly into our consciousness and giving us direct access to reality. If that’s the meaning we want to give to the verb to see, then we have to be ready to face the consequences: in this case, we never really see anything, we just believe that we do.
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The first is this: between raw perception and what we believe we see, there’s a lot of room for maneuver.
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You need a lot of self-control and self-confidence to commit to a process that’s confusing, slow, and uncertain.
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Ten or twenty hours of real exploration, outside of our comfort zone, is enough to discover within ourselves unsuspected powers.
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Each time our System 1 gives us an answer, we’re tempted to use it without calling on System 2, not even to verify that the answer is correct. Because System 2 uses a lot of mental energy and resources, we primarily rely on our instinct. Biologically, we’ve developed a preference for intellectual laziness.
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He has a simple recommendation: learn the list of cognitive biases presented in his book by heart, and each time you recognize one of the typical situations, fight your inclination and use your System 2 while trying to ignore your System 1.
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My protocol suffers from what is called selection bias, in that my friends aren’t necessarily representative of the general population,
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famous tale about Carl Friedrich Gauss (1777–1855), “the prince of mathematics
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If I observed the world with candor and sincerity, if I took the time to soak it all in, it was possible to overcome my limitations and become creative.
2024-12-22 21:44:39
When you become a professional mathematician, your social identity becomes that of someone who is smart.
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You’re at the edge of what’s socially acceptable. Your credibility is at stake. If you acknowledge how lost you are, you’ll look like a fool. The social norm is to let it go.
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The vast majority of math conversations end with this feeling of malaise. They fail for this simple reason: you don’t dare say how lost you are. You’re ashamed, you feel ridiculous, and this idea gnaws at you and makes you incapable of listening. You think only of your own worthlessness. It’s what keeps you from imagining and learning. You come out of these conversations feeling humiliated.
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understood that it was only by explaining it to others that I was able to really understand my own results
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The best way for me to understand my own math is to imagine that I have to explain it to complete beginners. By playing the fool with myself, I end up finding ways to present my results as being obvious.
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was initially afraid that the naïveté of my presentations wasn’t doing me any favors. The risk was that people wouldn’t take me seriously. The opposite happened. The simpler my talks were, the more intelligent people thought I was.
2024-12-22 21:54:39
“You’ll have to explain that to me again, because I didn’t understand anything.”
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’s a bit like with the sum of whole numbers from 1 to 100, where there are two levels of understanding. The first level consists of following the reasoning step by step and accepting that it’s correct. Accepting is not the same as understanding. The second level is real understanding. It requires seeing where the reasoning comes from and why it’s natural
2024-12-22 21:58:26
“Explain it to me again, but very simply, very slowly. I don’t understand anything about your subject. Assume that I have brain damage and can’t focus my attention for more than a few seconds.” That made him smile, and he had the kindness to explain it slowly and calmly, starting from the beginning, with the basics in his field, that I should have known but up till now had never succeeded in understanding.
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If the person you’re talking with doesn’t place themself at your level, and refuses to start with the basics and lead you by the hand, there’s no use getting distraught. You’ve probably stumbled across an actual fraud, someone who pretends to explain math that is beyond their own comprehension. The real imposters are the ones without the syndrome.
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This requires a great command of your body and emotions, because we have the instinct to hide our ignorance.
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In his social media profile on MathOverflow, Thurston wrote something similar: Mathematics is a process of staring hard enough with enough perseverance at the fog of muddle and confusion to eventually break through to improved clarity. I’m happy when I can admit, at least to myself, that my thinking is muddled, and I try to overcome the embarrassment that I might reveal ignorance or confusion. Over the years, this has helped me develop clarity in some things, but I remain muddled in many others.
2024-12-23 23:06:48
As proof of what he’s saying, he accompanies his text with three scientific texts, including Geometry, a mathematical work so revolutionary that it remade our language and imagination (for the first time in history, Descartes used the letter x to designate an unknown).
2024-12-25 22:38:45
For Descartes, evidence is the core criterion of truth. Not superficial evidence, the initial intuition that is often false, but evidence constructed through a deliberate and systematic effort of clarification, verbalization, and explanation,
2024-12-25 22:38:59
For Descartes, evidence is the core criterion of truth. Not superficial evidence, the initial intuition that is often false, but evidence constructed through a deliberate and systematic effort of clarification, verbalization, and explanation, which aims to make things perfectly intelligible until they become fully transparent: “Things that we conceive of very clearly and distinctly are always true.”
2024-12-26 22:43:28
’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.
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But once you’re able to imagine a scenario where the thing can be untrue, doubt immediately starts the process of reconfiguring your mental representations.
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Cartesian doubt is a universal technique for reprogramming your intuition.
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Descartes discovered that when we make a sincere attempt at introspection, when we’re attentive to our cognitive dissonance, when we force ourselves to grasp our most fleeting mental images and put words to them, when we have the courage to face the internal contradictions of our imagination, when we have enough calm and self-control to look beyond our prejudices and see things as they really are, it has the result of modifying our mental representations, of making them more powerful, solid, coherent, and effective.
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Kepler’s conjecture is a perfect example. It provides a tentative answer to the following question: What is the best way to stack oranges? The great astronomer and mathematician Johannes Kepler (1571–1630) had an intuition for the solution in 1611 without, however, being able to prove that it was correct.
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we’ve seen, a statement that you think is true but for which you can’t give a rigorous proof is what mathematicians call a conjecture.
2025-01-01 23:57:21
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, how people and objects are positioned in space, their actions and movements. Describing what you see and feel as simply and faithfully as possible. Capturing the moods, the music, the smells, the textures. If you can do that, you can do anything.
2025-01-01 23:59:02
As an adult, I’ve developed my own way of making use of the special state of mind just before falling asleep. Rather than focusing on subjects that preoccupy me, I’ve learned to simply let myself be filled with them. The nuance is subtle but fundamental. Focusing is thinking intensely, in search of solutions. It never works and it keeps you from sleeping. Being filled with something means contemplating it without a goal, in a decentered and disinterested manner. It’s almost like dreaming.
2025-01-02 23:15:01
you want to practice switching viewpoints, here is a good exercise: 1. Choose a random reference point around you, for example, the corner opposite from you in a room, or the window of a house when you’re walking in the street. 2. Try to imagine what you’d see if you were looking in your direction from this reference point
2025-01-05 19:54:45
In mathematics, as in many other fields, creativity is simply the ultimate form of understanding, which itself is but a natural product of our mental activity. It emerges when we force ourselves to continue looking at things that intimidate us until they finally become familiar and obvious.
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For example, I never really was interested in numbers, and never spent enough time thinking about them. That being said, through repetition and habit, I did manage to develop a degree of familiarity with them.
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The product of mathematics is clarity and understanding. Not theorems, by themselves.
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In mathematics, words are defined “axiomatically”: via formal definitions that characterize them entirely.
2025-01-06 10:19:01
It’s surprising that Wittgenstein’s philosophy isn’t more well known outside of specialized circles, as it holds a very practical life lesson: we should accept going step by step, clarifying our language as we move forward, and being regularly surprised. It’s an effective antidote for paranoia.
2025-01-06 10:19:21
“The product of mathematics is clarity and understanding. Not theorems, by themselves.”
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ambiguity of human language
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When you let a deep-learning algorithm run for a long time, for example, by making it “learn” from millions and millions of photos taken at random from the internet, you notice that each neuron gradually comes to specialize in the detection of a certain “concept.” The concepts of the first layers are very primitive, while those of the deeper layers are much more sophisticated.
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(It’s always an abuse of language to speak of “real time,” because no system actually functions in real time. It takes a neuron around half a millisecond to fire up.)
2025-01-06 14:05:12
The correct metaphor isn’t lightning, but organic growth. It’s the process through which we learn. It’s the basis of what we have called System 3, our ability to gradually modify the way we represent the world to ourselves.
2025-01-06 14:05:44
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.
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Physiologically, that corresponds to the ability of synaptic connections to strengthen or weaken.
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Mental plasticity is nothing more than this: the decentralized action of your neurons that, individually, seek to reinforce the consistency of their score.
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In
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your head, the elephant was at first a composite object, mobilizing a large number among the 100 billion neurons in your brain. One of the neurons that fired up during this first elephantine encounter had a special destiny. Little by little, by gradually adjusting its weights, it became more and more specialized. Over time, it became your elephant neuron.
2025-01-06 14:09:51
Our brain, like any animal brain, is a perceptual machine that constantly fabricates abstractions. We construct and we maintain a representation of the material world through the tangled network of our neural connections. This representation of the world is a piling up of layers upon layers of abstractions. Down to its very core, it’s conceptual in nature.
2025-01-06 14:10:33
The flaws of our language are but a reflection of its neurological underpinnings. The meanings that we assign to words are perceptual: we know how to recognize an elephant but we can never really define what it is.
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Every definition is an approximation. The meaning of words is always fluid, ambiguous, changing. Nothing is ever clear-cut. Inside our head, the world is abstract and vague.
2025-01-06 14:43:58
All the technological objects that you use in your day-to-day life are designed and built using advanced mathematics. Every piece of information that is recorded or transmitted over a distance can only be so thanks to sophisticated mathematical processing. Every time you interact with your smartphone, you interact with interwoven stacks of mathematical abstractions.
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For centuries, math has played a prominent role in science and technology. The digitization of our world and our lives has amplified this phenomenon by orders of magnitudes.
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Math is first and foremost an inner tool. Its main purpose is to enhance human cognition.
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In your head, mathematical concepts behave differently than other concepts. They’re much more difficult to learn. But once in place, they provide you with mental images of an incomparable clarity and stability.
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Even if you think you’re terrible at math, the conceptual matrix formed by the math that already lives inside your head is the most solid anchor point of your relationship to the world. Without
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With time, this math has become so concrete and obvious to you, so “real,” that it no longer feels like math. By comparison, the math that you don’t yet understand will always seem abstract, absurd, “imaginary.”
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is beautifully expressed in these striking 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.”
2025-01-06 14:58:51
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.
2025-01-06 15:06:08
What people really reproached him for was having made tangible what should have stayed evanescent.
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Euclid’s Elements is the most influential mathematics treatise in history. It dates back twenty-three hundred years and, throughout the centuries, it has come to serve as a blueprint for mathematical reasoning itself.
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“Calculation” comes from the Latin calculus, which means “small pebble,” referring to the stones used on an abacus for counting.
2025-01-06 15:22:19
Yes, plasticity is a slow and silent mechanism that occurs without any real effort on our part, provided we’re exposed to the right mental images.
2025-01-06 15:32:26
We haven’t yet learned to recognize them all, and still less to name them. Think, meditate, reflect, visualize, analyze, fantasize, reason, dream: we use these words haphazardly, without really knowing what they mean, and without realizing how much they have in common.
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’s through this vagueness that all the misunderstandings slip in. No one cared to even tell us that there are right and wrong ways of using our imagination. Some make us stupid. Some make us crazy. And some have the power of making us incredibly smart.
2025-01-06 15:33:39
Now that we’re teaching machines the secrets of intelligence, it’s about time we start teaching humans.