How to develop thinking: lessons from geniuses
What do Bill Gates, physicists Michael Faraday and Richard Feynman, mathematician Andrei Kolmogorov and poet Ezra Pound have in common? Their ideas changed the world, and they became innovators in their fields. But how did they do it? The author of the article, Nabeel Qureshi, discusses how the rejection of self-deception, the absence of haste and the fear of appearing stupid makes thinking flexible and judgments deep.
Read below to find out what thinking habits distinguish geniuses and how to develop these skills, because each of us has them to one degree or another.
I.
One of the habits of the smartest man I ever knew struck me as remarkable in my teenage years. After proving a theorem or solving a problem, he would go back and keep thinking about it, trying to find other ways to achieve the same result. Sometimes he would sit for hours on a problem, which has already been completed.
I did the opposite: having finished with the proof, I immediately stopped, because I “found the answer.”
He would eventually find three or four possible solutions, explaining why each of them was somehow connected to each other. And thus, he would penetrate the essence of the situation much deeper than I did.
I came to a conclusion: What we call “intelligence” is as much about “real intelligence” as it is about qualities like honesty, integrity, and courage.
Smart people just not ready to accept answers they don't understand – It doesn't matter how many people around you believe it or try to convince them; people won't accept the information if they can't convince themselves.
It is important to note that this property is of a “software” nature and does not depend on “hardware” characteristics, such as information processing speed, working memory, etc.
I've also noticed that these “hardware” settings vary widely among the smartest people I know – some are amazingly fast thinkers, others are slow. But they all have similar “software” traits that, with enough effort, can be mastered.
This means that you can develop useful intellectual habits that will actually “increase your intelligence.” Intelligence can change.
II.
The quality of “not stopping at an unsatisfactory answer” deserves special attention.
One of its aspects is energy: intensive thinking requires strength.It's much easier to settle on an answer that seems logical than to keep diving down an endless series of rabbit holes.
It is also very easy to think that you understand something when you really don't. So to make sure that you understand it, you need to check it and look at the issue from different angles.
It's hard. It requires serious internal motivation. And most people don't even try.
Nobel laureate William Shockley liked to talk about the “will to think”:
Shockley believed that motivation is no less important for a serious thinker than methodology… The basic element of successful work in any field is the “will to think.” He heard this phrase from nuclear physicist Enrico Fermi and never forgot it. “In these three words,” Shockley later wrote, [Ферми] expressed a significant point: a competent thinker will not be willing to put in the effort of tedious and careful thinking—will not find the “will to think”—without the certainty that the results of his work will be worthwhile.” The discipline of competent thinking is important throughout life… (source)
But it's not just about energy. You need the skill to motivate yourself to expend a lot of energy to solve problems. That is, in part, You should be very concerned about any misunderstanding or mental errorYou must have a desire, a desire to know.
Related to this honesty, or integrity: a sort of overwhelming reluctance or inability to lie to yourself. Richard Feynman said that the first rule of science is not to deceive yourself, and that is the easiest thing to do. Lying to yourself is very easy because there is no external force that enforces honesty; you can only continually ask yourself, “Do I really understand?”
(That's why writing is so important. It's hard to be fooled when you try to write about something and it comes out disjointed and confusing. Writing requires clarity.)
III.
Physicist Michael Faraday didn't believe it into nothing without the possibility of experimentally confirming this, no matter how tedious the demonstration.
It was not enough for him to simply hear or read. In assessing the work of others, he always repeated and often expanded on their experiments. This became a lifelong habit for Faraday, his way of claiming ownership of an idea. As he would do many times later under different circumstances, he set out to demonstrate the new phenomenon by his own experiment. Having saved enough money to buy materials, the physicist made a battery of seven copper half-pennies and seven zinc disks, interspersed with pieces of paper soaked in salt water. He attached a copper wire to each end plate, dipped the other ends of the wires in a solution of Epsom salts (magnesium sulfate), and began to observe (source).
Deep understanding of a subject is closely related to our physical intuition. Simple understanding “in words” has its limits. Three-dimensional visualization can help your brain to latch on to a specific “hook” and use it as a model; thus, understanding acquires a physical context.
This is why Jesus spoke in parables in the New Testament – they stick in your mind long after you read them, unlike abstract principles. The phrase “Are not two sparrows sold for a farthing? And not one of them will fall to the ground without your Father's will.” can linger in your mind much longer than “God watches over all living things.”
Faraday had this quality in spades – the book explains that this was due to his lack of knowledge of mathematics and his use of experiments to understand things; this contrasts with French scientists (such as Ampère) who constructed ideas in the abstract.
Faraday's physical intuition led him to some of the most significant scientific discoveries:
Despite his admiration for Ampère's work, he began to develop his own ideas about the nature of the force between a current-carrying wire and the magnetic needle it deflected. Ampère's mathematics (which Faraday saw no reason to doubt) showed that the motion of the magnetic needle was caused by its repulsion and attraction with the wire. But Faraday considered this to be a mistake, or at least the wrong approach. He believed that the wire formed a force field around itself, and that everything else followed from there. The next step clearly demonstrates Faraday's genius. Calling Sarah's fourteen-year-old brother George into the lab as a companion, he poured hot wax into the bottom of a basin, placed an iron bar magnet in it, waited for it to harden, and then poured in mercury, leaving only the top visible. I hung a short piece of wire with an insulated fastener so that its lower end was dipped into the basin, and then connected one pole of the battery to the upper end of the wire, and the other to the mercury. Now the mercury formed part of a circuit with the wire, which remained intact even if it moved. And the wire began to move – quickly circling the magnet! (source)
The ability to create such concrete examples, even without physically conducting experiments, is extremely important.
I recently saw a stunning “word cloud” image of an NLP model. If you read it from a purely mathematical standpoint and then forced yourself to create a visualization like this, you would gain a lot of insight.
Conversely, without creating visualizations and limited perception at the level of formulas and abstract concepts, you most likely do not understand the concept and need to delve deeper into it.
Another quality I have noticed in very smart people is that lack of fear of appearing stupid.
Malcolm Gladwell on his father, a mathematics professor:
My father has no intellectual complexes at all… It has never occurred to him to worry that the world might think him a fool. He doesn't play that game. So if he doesn't understand something, he just asks. He doesn't care if the words sound stupid. He'll ask the most obvious question and not worry.… So he asks a lot of stupid things, in a good way. My father will say to someone, “I don't understand. Explain it to me.” He won't stop until he understands, and throughout my life I've heard his questions in every conceivable circumstance. If he met Bernie Madoff, he would never invest money in him because he would say a hundred times, “I don't understand.”“I don't understand how this works,” he would say in a slow, dull voice. “I don't understand, sir. What's going on?”
Most people don't want to look stupid – it takes courage, and sometimes it's easier to just let things slide. It's amazing how many times I've started asking basic questions, feeling guilty for slowing down the group, only to find that no one understood what was going on (people often write to me directly about the relief they felt when I said something), but I was the only one who spoke up.
It's a habit. It's easy to learn. And it will make you smarter.
IV.
I remember when I was in math class at school and we were learning calculus, I got stuck on the “dy/dx” notation (also known as Leibniz notation).
The “dy/dx” sign looked like a fraction, as if we were dividing, even though we weren't. “dy/dx” means “the value of an infinitesimal change in y relative to an infinitesimal change in x,” and I didn't see how that could be divided as a simple fraction.
At one time, the proof of the fundamental theorem of analysis involved multiplying polynomials, and in the process one could discard “dy*dx” because “both of these quantities are infinitesimal, so in fact they can be eliminated.” This reasoning didn't make sense.
The “proof” of the chain rule that was presented to us looked like this:
(Funnily enough, even using incorrect math, you can get correct results, as in this case. Although the example is clearly wrong, it is not that far from the “correct” proof of the chain rule that I was taught.)
It turns out that my fears were justified: Leibniz's notation is indeed just a convenient shorthand, and these elements can be treated “as if” fractions, although the proof is too complicated. Moreover, Leibniz's shorthand is actually much more efficient and easier to work with than Newton's functional shorthand, which is why continental Europe was far ahead of England (which adhered to Newton's notation) in the development of mathematical analysis. All the logical problems were not solved until Riemann came along 200 years later and formulated analysis in terms of limits. But all this flew past me in my school days.
At the time, I was annoyed by these inadequate proofs, but I was under time pressure – I just had to learn the operations to answer the exam questions because the class had to move on to the next topic.
And since you can take the exam and do the calculus operations mechanically, without going into detail, it is much easier to pass the exam without thinking about the concepts – which is really what happens to most people.
How many people actually come back to this or any other similar topic and try to delve deeper? Very few. Besides, the unspoken lesson is: don't ask too deep questions or you'll fall behind. Just learn the algorithm, plug in the numbers, and pass the exams. Speed is everything. This is how school kills people's desire to understand.
My advice to those who are trying to understand something: do not hurry. Read slowly, think slowly, take time to reflect on the question. Before you read a bunch of information, start with your own reflections. A week or a month of continuous thought work can move you forward significantly.
Your brain will form a semantic structure on which you can then “hang” all that wonderful information from books, increasing the likelihood of memorization. I read somewhere that Bill Gates organizes his famous “reading weeks” based on a list of important questions that he thinks through and breaks down into parts. For example, he thinks about “water shortage”, breaks this question down into such sub-questions as “how much water is there in the world?”, “where does drinking water come from?”, “how to turn ocean water into drinking water?” etc., and then selects material to search for answers.
This method much more effective than just reading random sources, allowing them to not linger in your head.
V.
The best thing I have read about truly understanding things is “Sequences“, especially the section “Notice confusion”.
There are several mantra-like questions that are useful to ask during reflection. Some examples:
But what exactly is X? What is it? (link to fast Laura Deming)
Why should X be true? Why should there be a reason for this? What is the single, fundamental reason?
Do I really believe in my heart that this is true? Would I bet a friend a large sum of money on it?
VI.
Two parables:
First, Ezra Pound's parable about Agassiz from his “ABCs of reading” (one of the most underrated books on literature, by the way). I've kept its eccentric formatting.
No one will be ready for modern thinking until they understand the joke about Agassiz and the fish:
The graduate student, awarded his diploma with honors, went to Agassiz for final instructions.
The great scientist showed him a small fish and asked him to describe it.
Graduate student: “It's just a sunfish.”
Agassiz: “I know. Make a description of it.”
A few minutes later the student returned with a description of Ichthus Heliodiplodokus, or whatever term was used to conceal the understanding of the common sunfish, the family Heliichterinkus, etc., as in the relevant textbooks.
Agassiz again asked the student to describe the fish. The student wrote a four-page essay.
Agassiz then asked him to look at the fish. After three weeks the fish was in a state of advanced decomposition, but the student knew something about it.
The second is one of my favorite passages from Zen and the Art of Motorcycle Maintenance:
He had problems with students who had nothing to say. At first he thought it was laziness, but then it became obvious that it wasn't. They just couldn't think of anything to say.
One of the students, a girl with thick glasses, wanted to write a five-hundred-word essay about the United States. He was used to the apprehension that such statements evoked, but he did not dismiss the student and suggested that she focus on Bozeman.
By the time it was due, the girl hadn't completed the assignment and was very upset. She tried her best, but couldn't think of anything to say.
He had already discussed this student's abilities with previous teachers and his impressions were confirmed. She was very serious, disciplined and hard-working, but extremely boring. There was not a spark of creativity in her. Behind the thick lenses of her glasses were the eyes of a hard worker. She was not bluffing, she genuinely could not think of anything to say and was frustrated by her inability to complete the task.
This stumped him. He himself was at a loss for words. There was a silence, and then came the unusual answer: “Narrow it down to the main street of Bozeman.” A real revelation had occurred.
The student nodded obediently and left. But before her next class, she returned in a state of disarray, this time in tears. The girl had clearly been crying for a long time. She still hadn't figured out what to say, and she didn't understand why, if she couldn't think of anything about all of Bozeman, she couldn't even write about one street.
He was furious. “You just don’t look!” he said. He remembered how he himself had been expelled from college for talking too much. For every fact there were infinite hypotheses. The more you looked, the more you saw. She wasn’t really looking, and somehow she didn’t realize it. He suggested angrily, “Then limit yourself to the front of one of the buildings on Bozeman’s main street. The Opera House. Start with the top left brick.”
Her eyes widened behind her thick glasses. She showed up at the next class, looking puzzled, and handed him a five-thousand-word essay about the facade of the Opera House on Main Street in Bozeman, Montana. “I was sitting across the street at the diner,” she said, “and I started writing about the first brick, and then the second, and when I got to the third, it all became clear to me and I couldn’t stop. They thought I was crazy and they made fun of me the whole time, but here’s the result. I don’t understand it.”
He didn't understand either, but he thought about it as he walked through the city streets and came to the conclusion that she had obviously been stopped by the same obstacle that had paralyzed him on his first day of teaching. She was stuck because she was trying to repeat in her writing what she had already heard, just as he had tried to repeat on the first day what he had already decided to say. She couldn't think of anything to write about Bozeman because she couldn't remember anything worth repeating. Strangely, she hadn't realized that she could look and see for herself, write without relying on someone else's words. Concentrating on one brick destroyed the block, as it became clear that a fresh and direct look was needed.
The point of both parables is that there is no substitute for personal experience. Collect the data yourself. That is why, for example, I wanted to personally analyze coronavirus genomeYou build a foundation for understanding reality by getting first-hand data and reasoning from it, rather than relying on someone else's simplified version of a complex and changing phenomenon and then being continually surprised by the unexpected.
Unless people have personally experienced the phenomenon, they are unlikely to make truthful statements. They are more likely to simply repeat memorized thoughts and ideas. Reading popular science books or news is no substitute for understanding the subject and can even worsen your awareness by filling your mind with other people's stories and anecdotes that do not reflect the results. own analysis and reflection.
Even if you can't experience something firsthand, turn to information-rich sources with lots of detail and facts, and then draw conclusions from them. On foreign policy, read books from university presses rather than articles in The Atlantic or The Economist. Come to such publications after you've formed your own understanding of the subject, so you can evaluate the popular narrative.
Another lesson that the parable of the bricks teaches us: Understanding is not a simple yes or no. It has different levels. My friend understood the Pythagorean theorem much more deeply than I did: he could prove it in six different ways and thought about it much longer.
With careful study, even the simplest things can give you a lot of new and useful information.
Michael Nielsen gives a great example – the equal sign:
I first came to appreciate it after reading an essay by the mathematician Andrey Kolmogorov. You might expect such a great mathematician to write about something very complex, but the topic was simple: the humble equals sign, its virtues and vices. Kolmogorov lovingly explored the subject in detail, noting many important aspects, such as how the equals sign allowed for the development of concepts such as equations and algebraic operations on them.
Before reading the essay, I thought I understood the equal sign. And I would even be offended if someone suggested otherwise. But the essay convincingly showed that my understanding of the equal sign can be much deeper. (link)
Photographer Robert Capa gave advice to aspiring photographers: “If your photos aren't good enough, you're not close enough” (By the way, this advice is also great for fiction writers.)
This is valuable advice for training your understanding of the essence of the issue. When in doubt, move closer.
Thanks to Jose-Luis Ricon for reading a draft of this essay.
