Highlights
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no, mathematicians don’t think logically. It is in fact utterly impossible to think logically. Logic doesn’t help at all with thinking.
Sorpresivo!
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The second fallacy is truly toxic. It has the power to make us hopelessly inhibited. It has actually succeeded at convincing most of humanity that math is a strange and dangerous territory. For each of us, including the most “gifted,” it imposes an unsurpassable limit, that of the mathematical intuition everyone is “naturally” endowed with. The third misconception is a simple variation on the same theme: to be like Einstein or Descartes, you have to be born that way; you can’t get there by trying. And when Einstein or Descartes tell us differently, they’re just making fun of us. This vision that we’re incapable of becoming good at math is false, but it derives from an essential truth: the magic power of mathematicians isn’t logic but intuition.
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Secret math, also known as mathematical intuition, can be found in the heads of mathematicians. It consists of mental representations and abstract sensations, often visual, that are for them quite obvious, and that give them a great deal of pleasure. But when it comes to sharing these sensations with the rest of the world, mathematicians are often at a loss. What had seemed so evident to them is suddenly less
Muy cercano a lo propuesto por Andrers Ericsson respecto del desarrollo de habilidades: depende del cultivo de representaciones.
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If you taught children music by giving them the written scores for Mozart or Michael Jackson to decipher without their ever having heard it played, music would be as universally hated as math.
Relación con la cognición corporeizada?
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The mistake is in believing that our mathematical intuition is a static given, an insurmountable limit. The intuition that we have of mathematical objects isn’t innate. It’s not fixed. We can build it up, make it stronger day by day, as long as we follow the right method. Mathematicians are well aware that official math doesn’t tell all the story. They know that the real goal is to understand what’s in the books, to see it, to feel
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There’s a way to become good at math. This method is never taught in school. It doesn’t resemble any academic method and goes against the traditional tenets of education. It tries to make things easier rather than more difficult. You can compare it to meditation, yoga, rock climbing, or martial arts. It includes techniques to overcome our fears, conquer our flight reflex in the face of the unknown, and find pleasure in being contradicted. The method’s exact scope is actually broader than math. It’s a universal method for reprogramming our intuition and, in that sense, it’s a method for becoming more intelligent.
Muy interesante que esté involucrada la variable emocional como incidente en la capacidad metal.
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the competence gap that requires explaining is, in fact, too extreme for genetics. Human beings do exhibit innate biological differences, but overall we are a fairly homogeneous species. People differ in height, muscular strength, cardiac output, and lung capacity, and part of this variability can be traced to genetic factors. Yet those differences never encompass multiple orders of magnitude.
Algo simar se podría decir respecto de la meditación.
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Developing good mental habits, adopting the right psychological attitude, can make you a billion times better at math. But the method for becoming good at math has never been taught in schools. You can reach it only by accident. You’re left to discover, by yourself and by chance, snippets of the method. Most people end up not discovering anything, because certain essential points of the method are surprising and counterintuitive. It’s very easy to overlook them.
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I’ll talk later about the exercises in imagination that I began to do from childhood on. At first, it was nothing more than innocent games. For example, I had fun walking around the room with my eyes closed while trying to remember the layout of the furniture. What did this have to do with what I was learning at school? I wasn’t even particularly good. I often ran into the walls. I never imagined that this game, and other increasingly difficult ones, would allow me to develop, starting out at the same level as everyone else, a particularly powerful geometric intuition.
Muy interesante que el ejercicio es práctico e involucra al sujeto como agente en interacción corporeizada con el entorno.
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Like the spoon or the bicycle, our tools end up becoming extensions of our selves. We use them without thinking. They transform us. They augment us. They make us what we are. Without our tools, we really don’t amount to much. Language is the most difficult thing of all to learn. It’s an incredibly long, frighteningly difficult process. At eighteen months old, hardly anything we babble is intelligible. And yet we keep on trying all day long.
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In ancient Egypt, with the use of hieroglyphics, the art of writing was akin to magic. Scribes were a hereditary caste, passing down their secrets from generation to generation. In medieval Europe, writing was a vocation. Young men became monks, shut themselves off from the world, and devoted their existence to copying manuscripts. What did the peasants think of all that? Did they believe that reading and writing required a special talent, a particular form of intelligence that they didn’t have? Did they find being excluded from written language unfair and frustrating? Or did they simply tell themselves they didn’t have the time, money, or desire, and that in any case there wasn’t anything for them to read?
La intuición matemática como un dispositivo cognitivo.
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I want to convince you that the only possible explanation is that it’s all a giant misunderstanding. People aren’t good at math because no one has taken the time to give them clear instructions. No one has told them that math is a physical activity. No one has told them that, in math, there aren’t things to learn, but things to do.
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Studying math the same way that you study history or biology is useless. You might as well take careful notes during a yoga class so that you don’t forget anything. If you don’t practice any breathing exercises, it’s worth nothing at all.
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Whether you want to or not, you spend much of your time viewing the world abstractly. It’s a physiological characteristic of your body. Your brain is a machine for creating abstractions from your sensory inputs and mentally manipulating them, just as your lungs are machines for extracting oxygen from the air and transferring it to your blood.
Modelos cognitivos que nos ayudan a predecir el futuro.
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You’re the only person capable of seeing what’s in your head. Even if it’s painful, it’s only by making the effort of rigorously translating your vision into words and symbols that you can share it with others. And it’s also the only way to make sure that your intuition is right. Because sometimes your intuition is wrong.
Aplica a cualquier tipo de escritura o representación de cualqier tipo de intuición. La psicoterapia, en gran medida, se basa en el ejercicio de intuir la estructura y dinámica del sujeto, para expresarla de forma que pueda percibirla y aprehenderla.
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The second mistake schools make is to talk at length about the limits of intuition without ever reminding you of its strengths. The message that sticks with you is that intuition is imperfect. And that’s an important message. But schools forget to pass on an even more important message: your intuition is your strongest intellectual resource. In a sense, it’s your only intellectual resource.
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It’s a bit disconcerting, but you have to accept the facts. Einstein was talking of everyday intuition, the kind that we all have, that which is often seen as childish and that school teaches us to distrust. Einstein was simply speaking of our ability to imagine things. It’s a gift that we’re all endowed with. You might think it’s no big deal, but it’s really quite something, and no one gets anything more than that.
El fantaseo, la actividad de la DMN?
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The paradox is that in order to get to the point where something becomes obvious to you, you first have to construct mental representations that allow that to happen. Once constructed, these mental images allow you to see it immediately and without any effort. But it takes a lot of time and effort to construct them.
Parecida a la capacidad de diagnosticar estructuras psíquicas? la diferencia es que las pruebas matemáticas son demonstraciones universales.
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In classical Latin, the word billion didn’t exist (neither did million). To communicate the idea, the easiest thing would have been to call it the product of “a thousand times a thousand times a thousand.” A Roman during the time of Julius Caesar should have been able to understand that, even if it might have given them a bit of a headache. But if you had told them that you were capable of taking this number, subtracting one from it, and picture the answer immediately in your head, they wouldn’t have been able to follow. They would have taken you for some kind of math whiz.
Excelente ejemplo de dispositivo cognitivo.
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In the Amazon, Yanomami languages have an even more restricted numeral system: there’s a word for “one” and another for “two,” but there’s no word for “three,” just a catchall word that basically means “a lot.” For someone who sees the world in this way, discovering that there’s a clear distinction between 25 and 26 that can be perceived in a split second must come as something of a revelation, comparable to what math students experience when they learn that there are many different sizes of infinity that can be precisely described.
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The main difference between a math whiz and you is that their bag of tricks is bigger than yours and they’re more used to playing with them.
Han desarrollado representaciones mentales específicas y potentes.
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Successful math becomes so intuitive that it no longer looks like math. If the example seems stupid to you, it’s precisely because you understand it at the deepest level.
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A great mathematician is, for example, someone born into a culture where people only know how to count to 5, and one day realizes that you can go further than that. No one invented the infinitude of numbers out of the blue. At first, mathematical ideas are shifting and uncertain. You have the feeling that you might be able to go to 6 or 7, but you aren’t able to articulate it because there are no words for “six” or “seven.” You have the impression of being able to go even further, but this impression is fleeting. You don’t completely believe it, you tell yourself something can’t be right. This is what happens when you run up against the limits of language. In order to express what you feel, you have to invent new words, or create a new usage for words that already exist. Fleeting impressions cease being fleeting only after you find a way to pin them down with words. It takes time to get there. Words don’t come easily, and they don’t come right away. The initial phase of a discovery is a spiritual experience. You think outside of language. The world is illuminated. You have epiphanies. You see things that until then were hidden. Things so new they don’t yet have a name.
Pensar esto desde la perspectiva de un descubrimiento afectivo, del tipo de cosas que uno aprende a ver en terapia.
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It’s hard to imagine that before Descartes no one had “seen” cartesian coordinates. It’s almost absurd, like imagining people couldn’t see circles and squares. 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.
Aquí hay una clave para relacionar lo que plantea el autor con el proceso analítico: involucra lograr ser consciente de algo que era invisible aún siendo evidente y estando sobre la mesa, como la carta robada.
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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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It’s all too easy to make fun of math teachers, but try putting yourself in their place. How would you explain to someone how to tie their shoes if that person had never even seen shoes and your only means of communication was by phone? Take a few seconds to imagine the scene and you’ll see how hard it would be. The very idea is so difficult it makes your mind reel. This is the practical reality of teaching math, and we’re all in the same boat. Professional mathematicians have this in common with people who are bad at math: they both know what it feels like to be totally at sea.
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If you really want to, if you have enough time for it and you’ve chosen the right book, it’s well worth the effort. Get ready for a few months of hard work. This initiation rite will transform you. In my lifetime I’ve really succeeded in reading only three or four math books. I don’t regret the time and effort. It gave me unexpected powers, as if I’d drunk a magic potion. This power remains with me today. But the potion was hard to swallow.
Algo similar se podría decir respecto del esfuerzo de escribir las ideas.
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In fact, you should do whatever you feel like doing. You can leaf through the book for ten seconds, one hour, or three months—whatever. The underlying principle is never to force yourself to follow the pages in order, but to follow your own desire and curiosity.
Aplica al ejercicio de desarrollo de ideas propias en cualquier ámbito.
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the book should never dictate the agenda. We’re the ones asking the questions.
A propósito de la importancia de la consideración del deseo en cualquier proceso de aprendizaje.
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Following your desire is the only way of giving the book a real chance. If you start at the beginning, you run the risk of getting discouraged by page 2.
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At a profound level, math is the only successful attempt by humanity to speak with precision about things that we can’t point to with our fingers. This is one of the central themes of this book and we’ll come back to it a number of times.
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Learning math is learning how to use words that are “empty shells,” defined by logical formalism, as if they were ordinary words. It’s learning to give these words an intuitive and concrete meaning. It’s learning to see the objects they point to as if they were right there in front of our eyes.
Me pasó en mi estudio de la estadística.
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The ability to associate imaginary physical sensations with abstract concepts is called synesthesia. Some people see letters in colors.
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There’s a widespread belief that synesthesia is rare and associated with certain mental conditions. In reality it’s a universal phenomenon and a core building block of human cognition. Here’s a little test to see if 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? What is rare, and what our culture doesn’t push you to do, is to be aware of your capability for synesthesia and to try to develop it systematically. Secret math is a mental yoga whose goal is to retake control over our ability for synesthesia.
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Let me try to explain. 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. They constantly look their students in the eyes, because it’s in their eyes that they will know if they’ve understood. Inflicting the 198 steps of putting together the toaster on someone who doesn’t even know what bread is for is just plain mean. It’s like raising children without telling them stories. You can’t teach humans the same way you would teach robots.
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For Thurston, as for most creative mathematicians, mathematics is a sensual and carnal experience that is located upstream from language. Logical formalism is at the heart of the apparatus that makes this experience possible. Math books may be unreadable, but we nevertheless need them. They’re a device that we rely on in our quest for the true math, the only one that matters: the secret math, the one that lies in our head.
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“The quality of the inventiveness and the imagination of a researcher comes from the quality of his attention, listening to the voice of things.”
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“Discovery is the privilege of the child. I’m talking about young children, children who aren’t yet afraid to make mistakes, to look like fools, not to be serious, not to act like everyone else. They’re also not afraid if the things they’re looking at have the bad taste of being different from what was expected from them, what they were supposed to be.” This quotation from Harvests and Sowings sounds like something we’ve heard thousands of times before, but that is clearly not true. And even if it were true, what good does that do us? We’ll never be young children again. But it’s obviously a metaphor. Grothendieck is alluding to the child who is present “within us” and with whom “we have lost contact.” His book is actually addressed not to us but to the lost child within us, as he makes perfectly clear from the onset: “It’s to the one within you who knows how to be alone, to the child, that I wish to speak, and to no one else.” 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.” He describes this as a “gift of solitude,” the capacity to find himself “alone and listening to things, intensely absorbed in a child’s game.”
Impactante la relación con winnicott.
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Grothendieck was a great yogi who invented his own meditation technique. It’s centered on a radical form of curiosity and indifference to judgment, what we might call the child’s pose.
Relacionable con la idea de que el satori es ujna forma distinta de relacionarse con la más íntima y próxima experiencia fenomenológica, tan pegada a nuestra mirada que es muy difícil tratarla como una variable.
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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.
Me hace sentido que este probing es la actividad natural del sistema SEEKING, y es lo que está a la base de la asimilación y acomodación. En qué medida se puede poner en práctica esta recomendación en el trabajo analítico con pacientes?
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When I’m curious about a thing, mathematical or otherwise, I interrogate it. I interrogate it, without worrying about whether my question is or will seem to be stupid, certainly without it being well thought out. Often the question takes the form of an assertion—an assertion which, in truth, is an exploratory probe. I believe, more or less, in my assertions… . Often, especially at the outset of my research, the assertion is completely false—still, it was necessary to make it to convince myself.
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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.
Notoria la relación con el trabajol analítico y la redescripción representacional.
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Grothendieck did it differently. He knew that it was worthless to gather information about things that you can’t yet see. Instead, he allowed himself to imagine the things right away, without waiting, even when he was well aware that it might not work and his mental images would likely be terribly wrong.
Muy similar a mi approach para buscar señales de complejos en los pacientes.
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Grothendieck actively sought out the error as a young child actively seeks mischief. In his exploration of the world of mathematics, each time he found something bizarre or intriguing, unclear or unsatisfactory, incoherent or disagreeable, that’s where he began digging.
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What Grothendieck wrote about error is of universal significance, well beyond the field of science. It makes you want to engrave his words on school façades: 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. Not many people realize that the main obstacles in mathematics are psychological, not only at the beginning but all throughout, up to the highest levels. As we leave childhood behind, we learn to fear looking stupid. We learn to be ashamed of our mistakes. We learn to hide, even to ourselves, the fact that we know almost nothing. To get ahead in math, we need to deactivate this reflex for dissimulation. And it’s not easy. At the age when we were still free to ask stupid questions, even to ask the same stupid questions hundreds of times in a row, no one hated math. The great mathematicians invent and put in place special techniques to recover this lost childhood innocence. They all say it’s indispensable.
Redescripción representacional.
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As we’ll see throughout this book, 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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The organ that allows you to see the world in a logical manner doesn’t exist. If you’re counting on that to become good at math, you’ll have a long wait. Our prodigious faculty for learning and invention has its origin in our unconscious ability to constantly reconfigure the fabric of associations of images and sensations that, literally and figuratively, comprise the real structure of our thought.
En qué medida esta descripción no es representativa de lo que sucede en análisis?
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In the world of mental images, the laws of physics don’t apply. You can imagine anything, even inconsistent things, without falling on your face. The error that is inside of us can become immovable as a rock without our even being aware of it.
lo mismo aplica al desarrollo de ideas en general.
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Mathematicians have invented a method that lets them discover the errors inside themselves. This method relies on writing—more precisely, on writing in the official language of mathematics, constructed on logical formalism. Logic doesn’t help you think. It helps you find out where you’re thinking wrong.
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Often, you only have to write it down for you to see it’s incorrect, whereas before writing there was a vagueness, a bad feeling, instead of this evidence. That now allows you to start over without this lack of knowledge, with a question-assertion perhaps a little less off the mark.
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Mathematical writing is the work of transcribing a living (but confused, unstable, nonverbal) intuition into a precise and stable (but as dead as a fossil) text. Or, rather, it would be a simple job of transcription if the intuition was from the outset precise and correct. But intuition is rarely precise and correct from the outset. At first it’s vague and wrong, and it always remains a bit so. Through the work of writing, intuition becomes less and less vague and less and less wrong. This process is slow and gradual. Mathematical creation is a constant back-and-forth between imagination (the art of picturing what you read) and verbalization (the art of putting words to what you see). This simultaneously transforms our intuition and our language. We learn to see and, at the same time, we learn to talk. We learn to picture new things and we invent a language that allows us to name them. The whole process, according to Grothendieck, amounts to “gathering intangible mists from out of an apparent void.”
Esto es trabajo analítico. Esto es función alfa.
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In June 1955, seventeen months earlier, Grothendieck had written to Serre to share with him his first notes. The tone was enthusiastic, as Grothendieck was in the initial phase of discovery. He probed into things, made massive errors, and achieved rapid progress. At the time he still qualified some passages in his notes as “unlaid eggs” that were potentially “screwed up.” In the year that followed, Grothendieck brooded upon his “eggs.” He watched them hatch and patiently fed the bizarre creature that emerged. As the manuscript grew and gained structure, Serre and Grothendieck spoke about it with a growing glibness, going so far as to give it a nickname: the “diplodocus.”
En análisis, el proceso también involucra la incubación de nuevas ideas y perspectivas.
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You don’t explain what a banana is to someone who doesn’t know with a bunch of convoluted phrases. To show what we really think about bananas, the simplest and most honest definition is still the one we give to children: “Try it! It’s good!”
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A mathematical definition is neither a commentary nor an explication: it is the exact assembly guide of a new mental image and the “birth certificate” of the new word chosen to designate it.
A propósito de la agencia ingenua.
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transcribing mental images with enough clarity and precision to allow others to understand and reproduce them, is an art. What makes it so difficult is that your mental images are often a lot less clear than you think. What keeps the knot you make in your shoelaces from coming undone? If you don’t know, you don’t really know what it takes to tie it right.
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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. Understanding a mathematical text that defines a star as a shape whose signature is obtained by repeating n times the pattern “point, pit,” is to get to a point where you forget the formal definition and sense directly what a star is, on command, with the simple mention of the word star.
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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.
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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.
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Contrary to common belief, it’s never abstraction that makes math difficult to understand. Abstraction is our universal mode of thinking. The words that we use are all abstractions. Speaking, making sentences, is to manipulate and assemble abstractions. In that respect, four-dimensional geometry isn’t any more abstract than two-dimensional geometry. The problem with four-dimensional geometry has nothing to do with abstraction. The problem is that it’s hard to visualize and hard to draw.
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Today, when I try to describe this intellectual method, I sum it up like this: I began to listen to the dissonance between my intuition and logic.
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I began to use class as a benchmark for my intuition. I tried to predict what the teacher was going to say. Most of the time I got it wrong, but that let me figure out where my intuition was already correct. The things I understood, I understood so well that I could rely on them and concentrate on the others. I kept going back to what I didn’t understand until I understood why I didn’t understand it. And in the end that’s what allowed me to understand.
Trabajar desde la agencia que te permite operar desde los modelos intuitivos disponibles.
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The shy little voice that’s telling you that you don’t understand, that’s your mathematical intuition. Don’t confuse it with the loud noisy voice that’s telling you that you’re not smart enough. The little voice will guide you. That’s the one you need to lend an ear to. That’s the one you need to take care of, and protect throughout your entire life.
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Mathematical work isn’t a series of lightning insights and strokes of genius. It’s first of all a work of reeducation based on the repetition of the same exercises of imagination. Progress is slow because the body needs time to transform itself. It doesn’t help to force it, which may end up hurting you. You just need to commit to a regular training schedule, keep your cool, keep going even when it seems you’re not making any progress. It’s like going to the speech or physical therapist, except you’re all alone and inside your head.
En qué medida una versión de esto aplica al trabajo terapéutico?
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“People don’t understand how I can visualize in four or five dimensions. 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.”
Esto tiene profundas resonancias difíciles de captar.
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No mathematician solves problems the way you’re taught in school. It’s biologically impossible to create truly innovative mathematics by following this method, just as it’s biologically impossible to learn how to walk by solving Newton’s equations.
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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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Reconnecting with your early childhood capacity for learning means to stop believing in these absurd stories of gifts and talent. It means to become once again capable of devoting ten or twenty hours to something that may or may not be impossible, without being distracted by the feeling of your own uselessness. It means to rediscover the world with an open mind, trying something just to see what happens, for fun, because you want to.
Begginer’s mind. La importancia de evitar el miedo para lograr pensar en el mejor nivel posible.
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Mathematical creativity has the reputation of being a great mystery that science can’t explain. In my experience, however, it emerged as a natural phenomenon once I had adopted the correct psychological attitude. But the greatest effect of this new approach was in my personal life. If I was able to hack my visual cortex and modify my way of perceiving space, if I was capable of changing even my way of understanding the notion of truth, what about all the rest? What about, for example, all that I’d believed were givens in my life, these “strengths” and these “weaknesses” that people spoke about and made up my so-called “personality”? What about my shyness, my mental blocks, my insecurities, and everything that was supposed to be holding me back? What about my social identity? How could these things be any less adaptable, less malleable, less freely reprogrammable than my perception of space and truth? I remember with delight this beautiful day when, the very moment I stepped out into the street, I convinced myself that these things couldn’t be fixed and determined, that they were necessarily open to reconfiguration, and that it was up to me to try. Believing that you have a fixed personality, I thought to myself, is nothing but a superstition.
La experiencia que describe es una especie de satori.
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the most troubling aspect is that Kahneman reasons as if our intuition were hardwired, with no possibility for us to reconfigure or reprogram it. Had he lived in the ancient Roman era, he would almost certainly have said that it was impossible to represent mentally the result of the operation “1,000,000,000–1,” because the number greatly exceeded the capacities of human intuition.
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System 3 is an assortment of introspection and meditation techniques aimed at establishing a dialogue between intuition and rationality. You use it each time you 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.
Función alfa?
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It’s obviously stupid to call this approach System 3. It should simply be called thinking or reflecting. But the meaning of these words has been hijacked by a tradition that wants to make us believe that we should think contrary to our intuition. We’re told that our intuition is the mortal enemy of reason, that any dialogue between the two is impossible, and thinking means you have to submit blindly to System 2. I’m personally incapable of thinking against my intuition and I have serious doubts as to the sincerity of people who claim they can.
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My intuition isn’t any less fallible than yours. It’s always getting things wrong. I have, however, learned never to be ashamed of it. I don’t disdain my mistakes, I don’t push them aside, because I don’t think that they betray my intellectual inferiority or some cognitive biases hardwired in my brain. On the contrary. Nothing’s more exciting than a big glaring error: it’s always a sign that I’m not looking at things in the right way, and that it’s possible to see them more clearly. When I’m able to put my finger on an error in my intuition, I know it’s good news, because it means that my mental representations are already in the process of reconfiguring themselves. My intuition has the mental age of a two-year-old—it has no inhibitions and always wants to learn. If you stop mistreating your own, you’ll see that it’s exactly like mine, only asking to be allowed to grow.
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This is where System 1 has an edge: it isn’t bound by the constraints of language and writing.
Mapea bien con las características del proceso primario.
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Understanding something is making it intuitive for yourself. Explaining something to others is proposing simple ways of making it intuitive.
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System 3 is so entirely ignored by our culture that I can’t find the right word to characterize it. As I said above, I would like to say that System 3 simply corresponds to our capacity for thinking. But the verb to think doesn’t mean much since it’s been used as an injunction to submit to System 2.
Función alfa? actualización del paisaje adaptativo?
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System 3 specifically aims at establishing a dialogue between Systems 1 and 2, in order to understand their misalignments and resolve them. Rather than a free meditation, it’s one constrained by the principle of noncontradiction. Its ultimate goal is to revise and update System 1 while taking into account the results of System 2.
id988933271
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. Your way of looking at things isn’t the right one. Something is missing. There exists a better way, simpler, clearer, deeper, that you don’t know yet and that, perhaps, no one yet knows. Looking for and finding the right way of seeing things is the driving force of mathematics. It’s the main source of pleasure you can take from it.
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Unfortunately, you’ve since lost this intimate relationship to numbers. Following your early childhood, you fell into what I call the language trap, which is what stops you from “seeing” the sum of whole numbers from 1 to 100 like Gauss or Thurston. 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.
id988933273
“Don’t think of a pink elephant.” This is considered a linguistical paradox, since the sentence itself forces us to think of a pink elephant. Except that this passive, reluctant way of thinking about pink elephants isn’t one that will allow you to get to know and really understand them. Try to imagine a life-sized pink elephant standing before you. Take the time to look at it and study it closely. This intentional image will be incredibly more profound, more absorbing, more precise than the fuzzy image formed in your mind at the beginning of this paragraph. When you give free rein to your imagination, it is nearly without limits.
Quizás en psicoterapia, la indicación de entrar en un estado de producción fantasiosa, intencional pero no dirigida, es condición necesaria para el desarrollo de nuevas soluciones e ir más allá de la “trampa del lenguaje” que describe aquí Bessis.
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When you read “the sum of whole numbers from 1 to 100,” if you content yourself with the fuzzy image that forms in your head, you won’t really see anything. Instead of letting yourself be lulled by words, force yourself to think that the sum is physically present in front of you. Force yourself to imagine the whole numbers from 1 to 100 in physical form, made manifest in the real world, carefully lined up in front of you. If you manage to see them and you take the time to carefully examine the scene, you’ll find a way to calculate their sum.
El poder de la visualización y su relación con el pensamiento.
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The language trap is the extreme version of the phenomenon described by Thurston. An expression like “the sum of whole numbers from 1 to 100” is a convenient way to designate a very precise mathematical object. It allows you to speak of it, but it’s also a way of getting rid of it, to put it at some distance, so that it doesn’t bother you any longer.
Probablemente el mismo fenómeno de ser capaz de nombrar un complejo sin tener vínculo afectivo. La diferencia entre declarar y abreaccionar.
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This childlike relationship with numbers, this need for a bodily interaction with abstract things, is the right state of mind to do mathematics. By seeing three oranges in place of the number 3, you begin freeing yourself from the language trap. You stop confusing the writing of a number with its value.
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To get to really know a mathematical object, you have to observe it for a long time, with intensity and detachment, with curiosity and open-mindedness. You need to take the time to play with it and create an intimate relationship, a relationship that takes place outside of language.
Y crear un objeto como una relación transferencial requeriría explorarla mientras sucede, en un frame que no te demanda posicionarte, sino que te permite experimentarla y hablr de ella en tiempo real.
id988933278
Contrary to popular belief, logic isn’t the enemy of imagination. It can even be a close ally. The real enemy of imagination, that which blocks understanding and makes us feel like fools, is fear.
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The third and most unexpected of these breakthroughs happened later, when I was in my thirties. I learned to chase away my fear of being perceived as not smart enough. Up until then, despite an honorable beginning to my career and some initial success, I remained convinced that I wasn’t a real mathematician. I attributed my success to luck. I told myself that I was an imposter, and that I would end up being found out. When I was teaching at Yale, I was having actual nightmares. Our deepest fears are often social. For mathematicians, we’re often afraid we’re not as smart as the others, and that they’ll see it.
Aplica perfectamente a mi experiencia de no ser un clínico suficientemente bueno.
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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. Objects they aren’t familiar with still raise difficulties. You can be an exceptional athlete, Olympic champion with the javelin, in peak physical condition, but that won’t stop you from being crushed at tennis by a decent junior player.
Aplica al trabajo con distintas estructuras psíquicas.
id988933281
Whatever your level of math, you know what I’m talking about. 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.
Una mala sesión.
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This talk was a revelation for me. I understood that it was only by explaining it to others that I was able to really understand my own results. This is a well-known phenomenon, and mathematicians have a saying, that the only thing a math lesson is good for is to allow the professor to understand.
Una de las razones por las cuales es buena idea escribir mis ideas cuando están en un nivel de intuición.
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If you start up a math conversation, it’s to learn something, not to be humiliated. Sometimes you spend half the time reviewing the basics that you’d misunderstood, and sometimes that’s all you do. At any rate, that’s better than to talk about things you can’t make any sense of. 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.
Pienso que el encuentro analítico debe plantearse de esta forma: como una práctica que busca crear nuevas imágenes mentales. En ese encuadre, la libertad de no saber permite despejar el miedo y poder pensar.
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This requires a great command of your body and emotions, because we have the instinct to hide our ignorance.
Bullshit test para psicoanalistas mulas.
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You might as well make fun of revealing what makes you ashamed and what you’d like to hide. Humor is the best weapon I know against fear. By pushing your own intellectual limitations to absurd levels, you can create a temporary zone of childlike freedom where any and all questions are allowed.
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He was extremely kind, one could ask apparently completely stupid questions. Being with him, I wasn’t shy at all asking questions which would be completely stupid, and I’ve kept this habit until now.
Cuál será el correlato de esta posición en el analista? En qué conductas y disposiciones se traduce?
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math is first and foremost an attitude,
Puede aplicar al análisis? cuál sería esa actitud? cómo se asemeja y distingue de la de la matemática?
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Discourse on Method is a self-help book whose message is simple: we have the ability to construct our own intelligence and self-confidence.
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If you open Discourse on Method looking for a glorification of System 2, you’ll be sadly disappointed. Descartes’s great innovation was to put intuition and subjectivity at the heart of his approach to knowledge. He was distrustful of established knowledge and what was written in books. He placed little credit in authorities. He preferred to reconstruct everything by himself, in his head. His method closely resembles that of Einstein, Thurston, and Grothendieck. It is, of course, System 3, the slow and careful dialogue between intuition and logic, with the aim of developing your intuition.
Muy en la línea de mi eclecticismo.
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He approached truth as a martial art, an instinct you develop and that becomes embodied in action. Everything else—the philosophical arguments, the “opinions” of intellectuals with no skin in the game—was all just talk and of no interest to him: “For it seemed to me that I could discover much more truth from the reasoning that we all make about things that affect us and that will soon cause us harm if we misjudge them, than from the speculations in which a scholar engages in the privacy of his study, that have no consequence for him.”
La imagen del arte marcial es útil para describir la práctica del cultivo del conocimiento: debe ser algo quje fluye natural y espontáneamente luego del entrenamiento de los encadenamientos neuromusculares.
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If in the series of things to be examined we come across something which our intellect is unable to intuit sufficiently well, we must stop at that point.
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Generations of students have been beaten over the head and forced to regard the Discourse as a philosophical treatise. And almost no one has taken the trouble to try it for real.
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Teaching Cartesian doubt is difficult because it’s neither a knowledge nor a mode of argumentation, and is therefore impossible to evaluate. 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.
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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.
Cómo aplica a la clínica?
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Doubt thus corresponds to the shadow zones of intuition. To really doubt something, you can’t just claim that you doubt, you have to sincerely believe that this thing might not be true. In order to do that, you have to construct an image in your head that shows there’s a place for doubt. If you can’t do that, you can’t doubt—you’re certain, like you can be with 2 + 2 = 4. 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.
El verdadero método cartesiano.
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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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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.
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The transition between waking and sleep, as well as sleepiness upon awakening, have since that time played a central role in my intellectual development. In every project I begin, once it gets serious, once I really get interested, once I’m confronted with a real challenge, it starts to occupy that liminal space. Transcribing my dreams was my first real attempt at writing. I was about seventeen when I started to get interested in this. At first, I found it too hard to directly write down my dreams, so I tried to record them while saying them aloud. My project was to collect them, like you would keep a journal or a photo album. But something unexpected happened that forced me to give up the project. Night after night, as an effect of my trying to memorize them and put them into words, my dreams grew in richness and precision. I was dreaming better and better. I was dreaming so well that it started to become annoying. At first, I could remember only small fragments, bits and pieces of a single dream. But after two or three weeks, I was recounting five or six different dreams every day, each with a complete story and enough details to fill up a lot of pages with writing or long minutes of recordings. It became overwhelming. The memories of my dreams were taking up too much space in my head and in my days. I felt like this exercise in introspection would end up consuming me.
Muy interesante el cruce con la productividad de la escritura del sueño para el psicoanálisis.
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Remembering your dreams isn’t something you’re born with. 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. For a long time I kept a notebook on my bedside table, with a pen in the middle as a bookmark, to note down all my dreams and all the ideas that came to me during the night. I even taught myself how to write in complete darkness. When I stop writing down my dreams I quickly lose my ability to remember them. When I force myself to write them down again, even if it’s just a word or two, the ability gradually returns. Sometimes you have to keep it up for several weeks. The most difficult part is trying to recapture the first bit of a dream after a long period of not being able to do it.
Puede ser una recomendación explícita a mis pacientes.
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I’m placing quotation marks around “see” because I know that I can’t really see it with my eyes. It’s visual, and yet at the same time it’s not really visual. I wouldn’t be able to draw it. It’s just a weird sensation that is located within my visual field, like if something had been highlighted. In a sense, you could say that it’s a hallucination. But it’s an educated hallucination, one that’s constructed and controlled. When I look at a bridge, I see stress lines. I see which parts of the bridge are subject to compression, and which ones are subject to tension. I can call up these perceptions on command, with a bit of concentration. They help me experience and understand the world. You have similar sensations. You can “see” that a rope is stretched too tightly and is about to break, or that a balloon is overinflated and is about to burst. You’ve learned to “see” the tension of objects as if it was another layer of augmented reality, supplementary information embedded in your visual field, exactly like a color but located in another layer of reality.
Notable lo claro de los ejemplos para hacer referencia a esta capa informativa entrenable que se superpone sobre la percepción. Me parece que es exactamente lo mismo a lo que significa Ericsson con el concepto de representación mental.
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If I don’t see why something should be true, I’m wary of it. This kind of information can stay in this intermediate status for a long time. Maybe for a few hours, days, weeks, years, or even decades.
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This explained why the soil wasn’t sliding down along the hillside. It also explained another phenomenon I’d seen before and that had struck me: of how unbelievably difficult it was to remove a tree stump.
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It’s only been a few years since I’ve completely accepted the idea that airplanes can fly. I learned how to feel it physically. In order to do so I had to learn how to feel the density of the air and the phenomenon of lift. I had to find out that airplanes are much lighter than they appear to be. I had to learn to feel the wings, the way that they lift up and bend, their internal structure, how they are attached to the fuselage and why they don’t break. My intuition ended up finding all that normal. Planes became like an extension of my body.
Siempre he tenido la sensación de que las experiencias visionarias con psiquedélicos me abrieron un campo mental pobremente redescrito, pero intuitivamente disponible. Probablemente sea el mismo mecanismo. Indagar en la relación con la función de consolidación de aprendizajes del sueño.
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What mathematics taught me was that it was really possible to get ahead and move forward in life without giving up the childlike desire to remain down to earth, to accept only what is concrete and evident.
Nuevamente una referencia al cuerpo como ancla fundamental para una verdadera comprensión.
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In certain subjects that at first I wasn’t very good at, I’ve managed to get better later on. To understand the abstract structures of algebra that caused me so many difficulties when I was twenty, I’ve developed a particular form of sensory intuition. These are powerful sensations but impossible to put into words. I grasp certain mathematical concepts through nonvisual motor sensations, tensions and force fields within my own body, as if I could transport myself and experience these objects from the inside. I feel this math in my neck and in my spine.
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Here is an exercise of the imagination that helped me a lot when I was trying to improve in these subject areas. I looked at an object, for example, a bottle of shampoo sitting in the bathroom, and asked myself the following question: if my body were shaped like the bottle, how would it feel physically?
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It all started with a casual observation. One day, I noticed that a question I was asking myself about the geometry of certain braids in dimension 8 could easily be translated into the language of category theory. The unexpected link between two very different intuitions offered a novel way of looking at problems I’d been struggling with for years. It was as if I’d built a bridge between two regions of my brain that up until then hadn’t communicated with one another. There was a big jolt of comprehension, then a series of smaller aftershocks. But it was just the beginning. My mathematical imagination was about to undergo a massive reconfiguration. Each day I woke up with new ideas. Some of them were tied to the problem I wanted to solve (a conjecture dating from the 1970s) but others took me in completely different directions. I thought they were beautiful, but I had to quit following them up because it was impossible to explore so many leads at the same time. It was just too much. I tried to take notes but my understanding progressed faster than my ability to write it down. This state of hyperlucidity lasted six weeks, the time that I finished proving the conjecture. I wasn’t able to sleep. I was exhausted. Once I woke up at 4 in the morning with the urge to look at a book I’d bought ten years earlier (the second volume of Representations and Cohomology by Dave Benson), which, at the time, I had barely opened. I found the book on a shelf, grabbed it, sat on the floor, and read a hundred pages in one go, as if it were a comic book. I’d never been able to read a math book like that before. If I was able to read it so fast, it was because I already knew what was written down: it was as if I had just seen it in my dream. Throughout these six weeks, I had the feeling of understanding more new mathematics than everything I’d understood in the previous twelve years, since I had begun as a PhD student. It was going so fast that I was seasick. It was physically overwhelming and I could no longer cope. I was hurting. I wished that it would stop, so that I could get some rest. But it wouldn’t stop. It was like mathematics had taken control of my brain and was thinking from inside my head, against my will. For the first time in my life, I realized that extreme math was a dangerous sport.
Un acto analítico? Un cambio de posición subjetiva? Muy útil la imagen de efecto no linear de esta combinación de representaciones mentales.
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Paranoid delirium is a close relation to mathematical reasoning. In a way it’s the evil twin. Some people even have difficulty telling them apart.
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Thurston defines mathematics as a collaborative human project oriented toward sharing and understanding, not a search for eternal truths. Without human understanding, theorems have no value. Who cares who proved this or that result first? What counts is the meaning that we give to those results. Real math is the one that lives in each of us. Thurston’s response might seem innocuous, but it’s a profound questioning of the way that math has been presented for over two millennia.
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When you’re sincerely preoccupied with something, when you’re in trouble, when you have problems at work or problems at home, you instinctively call on the method used by mathematicians. At night, in your bed, you try to understand the issue. You mull it over. You replay in your head mental images that you dig up from the depths of your memory and imagination. You play Lego with these images. You try to organize them, fit them together and assemble something meaningful, something that makes sense. Sometimes you have the feeling you understand it all. Your mental images come together. You reinterpret a past event in a new way. You pick up on a detail, a new element, something that had been right under your nose that you hadn’t yet perceived. Now that you see it, everything makes sense. It’s a revelation, a discovery that gets you excited and makes you want to share it with others.
Nuevamente: ejercicio de mentalización y fantaseo generando efectos reales en nuestra capacidad de comprensión, la cual tiene sus propios efectos e implicancias.
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At a distance, dictionaries look like the real thing. We always want to believe that the words we use are solidly defined and that the phrases we speak have a precise meaning. But once you scratch a bit below the surface, you find circular definitions. The error would be in believing that people who put together dictionaries aren’t doing their job correctly, and that there’s a better, smarter, and more rigorous way of defining the words we use. But there’s a deep structural reason why dictionary definitions are so deficient: it’s rigorously impossible to truly define words in our language, and our relationship to the world is much less solid than we would like to believe.
No hay metalenguaje
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The Austrian philosopher Ludwig Wittgenstein (1889–1951) perfectly summed up this frustration: “The more closely we examine actual language, the greater becomes the conflict between it and our requirement.” Logic doesn’t function unless words have a definition that is explicit, perfectly precise, and stable over time. Despite immense efforts, we’ve been unable to produce these kinds of definitions outside of mathematics. Wittgenstein affirms that it’s a quixotic quest: “We feel as if we had to repair a torn spider’s web with our fingers.” In acknowledging the intrinsic limitations of our language, Wittgenstein made one of the great philosophical breakthroughs of the twentieth century. This allowed him to break with a multi-millennial tradition dominated by metaphysics, in which philosophers believed that it was possible to attack, using rationality, problems that were strikingly similar to that of the chicken and the egg: problems so remote from our daily experience that they were occurring only as a result of our language losing its grip.
Wittgenstein resumido.
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To find out, to understand how math really works and what it can really do for us, we cannot continue to overlook its most direct practical aspect: math works on our brain and modifies how we see the world.
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Our brain automatically makes the connection between a drawn animal and a real animal. It’s in this way that drawings are true optical illusions.
Desarrollo cerebral que nos permite “proyectar” sentido en el mundo, como una capa superpuesta (que, a veces, es difícil de obviar).
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We’d like to define elephants as we perceive them, because that definition would have the most sense for us. But the method that our brain uses to recognize elephants is at the same time stunningly efficient and perfectly impossible to translate into words. In fact, it’s so efficient that it’s hardly believable.
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If the child understands right away, it’s because they already see the elephant. They noticed it immediately, well before you said what it was. The elephant stood out as something remarkable that deserved a name. They were probably getting ready to ask you what it was. Without this ability, our language simply wouldn’t exist. We wouldn’t be able to explain what words referred to.
La maquinaria cerebral prerrequisito para utilizar lenguaje.
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The conclusion to all of this is that our brain seems to automatically extract, from the raw visual data fed continuously into it by the optic nerve, a universal idea of what an elephant is. It then becomes able to recognize this abstract concept of an elephant through its multiple incarnations, in situations so remarkably varied it would be ridiculous to try to list them all. Without trying, as if by magic, we develop a curiously reliable sense of “elephantness” by mere exposure to scenes involving elephants.
Generación espontánea de categorías.
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Our brain is often compared to a computer. This metaphor is correct in two aspects: both the brain and computers are capable of accomplishing complex tasks of information processing, and they both make use of electrical signals. As for all the rest, the metaphor is catastrophically false. It ruins our chance of understanding what is going on. A computer is a perfect embodiment of System 2: it’s a machine capable of mechanically applying long sequences of logical instructions at breathtaking speeds without making mistakes—something our brain is entirely unable to accomplish. A computer is made of a central processing unit where calculations are made and memory units where information is stored. Between these distinct units, information circulates at high speeds along electrical circuits without being transformed. In our brains, it’s quite the contrary. Information circulates slowly and is transformed along each step of its circulation. Memory, processing, and circulation are indissoluble. A computer strings instructions one after the other, sequentially, paced by an internal clock that ticks billions of times per second. The time it takes to activate a connection between neurons is on the order of a thousandth of a second. The base operations of our brain are thus a million times slower than that of a computer. But our brain isn’t sequential: it processes in parallel billions and billions of these operations. The silicon circuits of computers are immutable, engraved in an inert material. Our brain is living tissue that constantly reconfigures itself.
Las falencias de la metáfora computacionalista de la mente.
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The mechanisms of our intelligence are impossible to understand as long as you try to locate them in a specific place in our brain. Intelligence is what is called an emergent property: individually our neurons are primitive and limited, but vast assemblies of neurons make incredibly sophisticated behaviors “emerge” that can’t be attributed to any neuron by itself—these large-scale behaviors are what we call intelligence. It’s a bit like traffic jams: 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.
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Since the 1950s psychologists and computer scientists have looked for inspiration in the functioning of our neurons and the anatomy of our cerebral cortex to construct systems of artificial intelligence. Because they imitate the architecture of our brain, the behavior of these systems sheds light on what’s going on in our heads. Frank Rosenblatt (1928–1971), one of the pioneers of this approach, helped construct the first mathematical model of a neuron and fashioned a computing device that implemented this model. But modeling the behavior of complex neural networks capable of simulating our ability to see was a problem of an entirely different scale. The technology stumbled along for decades and went through numerous ups and downs. At some point, the AI community grew so disillusioned that artificial neural networks were seen as a technological dead end. Three scientists, Geoffrey Hinton, Yann LeCun, and Yoshua Bengio, continued to believe in the approach. History proved them right. Toward the end of the 2000s their “deep-learning” algorithms had made so much progress that they had become capable of resolving advanced problems in the recognition of images, such as the detection of the presence of elephants.
Redes neuronales como una técnica antigua que no logró resultados por falta de desarrollo tecnológico, hasta el 2000 en que fue posible desarrollar redes neuronales profundas.
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With deep learning, the process of understanding could be made tangible and concrete. It had finally become possible to speak of it without invoking some kind of black magic.
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To explain the nature of our intelligence and the mechanisms of our thought, deep learning offers the best metaphor I know of.
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The first mystery that deep learning allows you to dissipate is that of the emergence of concepts. In other words, what had been for millennia one of the liveliest debates in metaphysics was suddenly reincarnated in the realm of software, confronting us with an undisputable experimental reality: conceptual thought spontaneously emerges in vast assemblies of artificial neurons subjected to unstructured data, for example, a flood of images. Roughly speaking, here’s how it works in the context of vision. Deep-learning algorithms model our cortex as a neural network with multiple layers. The first layer is the raw image: a matrix of neurons that represent pixels. The second layer is formed of neurons whose dendrites are linked with neurons in the first layer. The third layer is formed of neurons whose dendrites are linked to the neurons in the second layer, and so on. It’s because the network is made up of many superimposed layers that it’s called “deep” learning. In my description of how neurons function, I omitted one important detail: when a neuron runs a poll of its dendrites to decide whether it should fire up, the poll isn’t democratic. Each connection in a neural network carries a certain “weight” that determines how much it counts toward the decision.
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It’s through this process of adjusting weights that the network “learns” and “becomes intelligent.” 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.”
Ese ajuste o actualización en base a la experiencia es la acomodación Piagetana. Estas redes también dan cuenta de la jerarquía en su profundidad, que permite entender efectos no lineares.
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It remains to describe the process of learning itself: what mechanisms do the neurons rely on to determine the “weight” of their connections with other neurons? Let’s go back to the example of your elephant neuron. It is constantly analyzing the state of its upstream neurons to decide whether or not it should fire up. You’re scrutinizing the world in real time, on the watch for elephants. (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.) In chapter 11, we called this System 1, instantaneous intuitive thinking, that which gives you the impression of thinking as fast as lightning. In parallel to this, another phenomenon takes place in the background. It happens at a much slower pace, and is so discrete that we can’t perceive it. The correct metaphor isn’t lightning, but organic growth. It’s the process through which we learn. It’s the basis of what we have called System 3, our ability to gradually modify the way we represent the world to ourselves. If one day you come across an elephant without a trunk, you’d be surprised. What does it mean “to be surprised”? A trunkless elephant surprises you because your vision of the world hadn’t anticipated it. That still doesn’t keep you from understanding. You’d almost certainly still see that it’s an elephant, while having the disturbing feeling that something’s terribly wrong. 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.
Cómo describir el sistema 1 y 3 en base a las redes neuronales y la plasticidad cerebral.
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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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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. Conceptual thought isn’t a human privilege. It doesn’t arise from our language or our culture. When saying this, I’m using the word thought in a very broad sense, to designate the neurological processes that constitute the substrate of our intelligence. Any lion thinks in a conceptual manner, and has an elephant neuron in its head.
La conceptualización es una característica de los sistemas nerviosos en general.
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Presenting mathematics as an external tool is the surest way to make us hate it. Official math, with its sharp edges, its cold logic, its unbearable air of superiority, is impossible to fall in love with.
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Math is first and foremost an inner tool. Its main purpose is to enhance human cognition. With the correct exercises of imagination, we have the ability to develop an intuitive and familiar understanding of mathematical notions. We can appropriate them and make them an extension of our bodies. The true math is the secret math, the one that extends our intuitive understanding of the world that surrounds us.
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In other words, from a purely practical standpoint, math is indistinguishable from fiction. Learning math is an activity of pure imagination. We bring mathematical objects into our heads through the power of thought and keep them together there through the cohesive effect of a mysterious ingredient, which in a way is the true hero of the fiction: mathematical truth.
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Cantor was labeled a “scientific charlatan,” a “renegade,” a “corruptor of youth” for having talked about infinity calmly and precisely. What people really reproached him for was having made tangible what should have stayed evanescent. From a theological perspective, mathematics is unfair competition. “The essence of mathematics is its freedom,” declared Cantor. The freedom of mathematicians is to treat “imaginary” things as “real” things from the moment they are “true.” In the end they even see them as being “obvious.” It happens that this approach works remarkably well. Mathematicians obviously aren’t going to stop when things are going so well. They continue to amuse themselves with the supernatural or miraculous nature of their constructions. They manipulate “ideals” and “vanishing spectra.”
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If the process of understanding math is already quite bizarre, the discovery process is even more so. The experience is so singular and disconcerting that most accounts look like they were written by mystics. One of the most baffling aspects is the abrupt manner in which ideas come to you, without effort and almost always inconveniently. They emerge, as Grothendieck puts it, “as if summoned from the void.” In an influential research article by Bob Thomason and Tom Trobaugh, we’re told that the second author contributed only after he was dead, by means of appearing in a dream of the first author. Not only did he suggest the right approach, he stopped the first author from readily dismissing it as hopeless: “Tom’s simulacrum had been so insistent, I knew he wouldn’t let me sleep undisturbed until I had worked out the argument.” One of my close friends, an excellent mathematician whose name I won’t disclose, recently told me that he had the distinct impression (which he never dared share with others) that the greatest ideas in his career had been directly suggested by God (even though he’s an avowed atheist). For my part, I’ve never felt anything along those lines. I’ve simply had the impression of being able to levitate and pass through walls.
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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.
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Each time I found myself in a situation of teaching or explaining my work, I tried to bring together two levels of discourse: a formal level made of rigorous definitions and precise statements, and an intuitive level, with the right metaphors, the right drawings, the right inflection of my voice, the right way of waving my hands.
cómo desarrollar nuevas representaciones mentales en otros cuando no puedes apuntar a nada visibe que se asemeje?
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To understand math is to reprogram your intuition. It is, above all, a matter of neuroplasticity. The secret techniques of mathematicians are neither more nor less paranormal than those that allowed Ben Underwood to see the world by clicking his tongue.
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we learn precisely when we force ourselves to imagine things that we don’t yet understand, which unfortunately is the same exact thing that most people run away from. Yes, paying attention to the small details that trouble us is of the utmost importance, and the fastest way to learn is to follow the path of maximum perplexity.
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Learning math should be like learning any other motor skill, like learning to swim or ride a bike, and it should be accessible to everyone. Our false beliefs about the nature of our language and the functioning of our thought are obstacles to this simple and direct learning. They instill fears and inhibitions that block the unseen actions without which no mathematical learning can take place. How do you teach math to someone who believes that their intuition and perception of reality are given and impossible to reprogram? It’s exactly like teaching swimming to someone who is convinced their body is as dense as rock and will sink. A prelude to any successful teaching is getting rid of such beliefs.
Mindset.
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When mathematics was just about proving theorems, it was entirely natural to treat the subjective experience as a second-rate citizen, to be covered only informally and anecdotally, if time allowed. But as soon as we realize that mathematics is ultimately about human understanding, this ceases to be an option. Understanding is, in essence, a subjective experience.
Mi intuición de lo que pasa en el satori
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My way of looking at things, which has served me well throughout my career, was to imagine that creative mathematicians were hackers who had found ways to unlock “hidden modes” of our cognition. Most of the time, they’d done so unwittingly, and were entirely incapable of explaining how.
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Grothendieck attributes the singularity of his work to his transgression of a taboo: “It would seem that among all the natural sciences, it is only in mathematics that what I call ‘the dream’ or ‘the daydream’ is struck with an apparently absolute interdiction, more than two millennia old.” Thurston puts it in a less grandiose but equally impactful fashion: “I have decided that daydreaming is not a bug but a feature.”
El poder del fantaseo para crear nuevas categorías, lógicas, perspectivas y realidades.
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using our imagination isn’t navigating an ethereal layer of the cosmos. Nor is it a parasitical activity that we should seek to suppress. It is, instead, a genuine physical activity that is central to human cognition. What we see and do in our heads contributes to neural learning every bit as much as what we see and do for real. If we feel the urge to perform unseen actions in our heads, if we dream and if we daydream, it’s because this allows us to fabricate understanding. What we imagine modifies the actual wiring of our brain and literally changes the way we see the world.
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we construct our intelligence on our own, with ordinary human means, with our imagination, curiosity, and sincerity.
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I know now that my candor and my sensitivity are my most powerful intellectual weapons. The mathematical approach is one of integrity and being in tune with oneself.