Solving Math Problems Quotes

The formulation of the problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill.

A basic rule of mathematical life: if the universe hands you a hard problem, try to solve an easier one instead, and hope the simple version is close enough to the original problem that the universe doesn’t object.

I'm not a math problem." "But I'll still solve you.

The steps to solving a problem, from elementary math to breaking out of a police station, remained the same.

One must acknowledge with cryptography no amount of violence will ever solve a math problem.

You can't fix people like you can solve a math problem.

It is better to solve one problem five different ways, than to solve five problems one way.

The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them.

You want your brain to become used to the idea that just knowing how to use a particular problem-solving technique isn’t enough—you also need to know when to use it.

When I was in high school, my math teacher Mr. Packwood used to say, "If you're stuck on a problem, don't sit there and think about it; just start working on it. Even if you don't know what you're doing, the simple act of working on it will eventually cause the right ideas to show up in your head".

So, if I'm no cheerleader of sports, why write a chapter about it? Sports do have some positive impact on society. They solve problems, such as how to get inner-city kids to spend $175 on shoes. They serve as a backdrop for some of our most memorable commercials. And they remain the one and only relevant application of math. Not only that, but we have sports to thank for most of the last century's advances in manliness. The system starts in school, where gym class separates the men from the boys. Then those men are taught to be winners, or at least, losers that hate themselves.

Focused problem solving in math and science is often more effortful than focused-mode thinking involving language and people.

Andrew said. "I can't decide. Your loose ends aren't adding up." "I'm not a math problem." "But I'll still solve you.

I liked maths because it meant solving problems, and these problems were difficult and interesting but there was always a straightforward answer at the end

The steps to solving a problem, from elementary math to breaking out of a police station, remained the same. A simple matter of understanding the problem, and selecting the best solution.

Some mathematics problems look simple, and you try them for a year or so, and then you try them for a hundred years, and it turns out that they're extremely hard to solve. There's no reason why these problems shouldn't be easy, and yet they turn out to be extremely intricate. [Fermat's] Last Theorem is the most beautiful example of this.

Internalizing problem-solving techniques enhances the neural activity that allows you to more easily hear the whispers of your growing intuition. When you know—really know—how to solve a problem just by looking at it, you’ve created a commanding chunk that sweeps like a song through your mind.

MR JEAVONS SAID THAT I liked maths because it was safe. He said I liked maths because it meant solving problems, and these problems were difficult and interesting, but there was always a straightforward answer at the end. And what he meant was that maths wasn’t like life because in life there are no straightforward answers at the end. I know he meant this because this is what he said. This is because Mr Jeavons doesn’t understand numbers. Here

... I succeeded at math, at least by the usual evaluation criteria: grades. Yet while I might have earned top marks in geometry and algebra, I was merely following memorized rules, plugging in numbers and dutifully crunching out answers by rote, with no real grasp of the significance of what I was doing or its usefulness in solving real-world problems. Worse, I knew the depth of my own ignorance, and I lived in fear that my lack of comprehension would be discovered and I would be exposed as an academic fraud -- psychologists call this "imposter syndrome".

I tell my students they can procrastinate as long as they follow three rules: 1. No going onto the computer during their procrastination time. It’s just too engrossing. 2. Before procrastinating, identify the easiest homework problem. (No solving is necessary at this point.) 3. Copy the equation or equations that are needed to solve the problem onto a small piece of paper and carry the paper around until they are ready to quit procrastinating and get back to work. “I have found this approach to be helpful because it allows the problem to linger in diffuse mode—students are working on it even while they are procrastinating.” —Elizabeth Ploughman, Lecturer of Physics, Camosun College, Victoria, British Columbia

A Mind for Numbers is an excellent book about how to approach mathematics, science, or any realm where problem solving plays a prominent role.

Life is not a math problem, Charlotte.” If it was, I’d have been a lot better at it. I’d often wished I could work out people as easily as I did arithmetic: simply break them down to their common denominators and solve. Numbers didn’t lie; there was always an answer, and the answer was either right or it was wrong. Simple. But nothing in life was simple, and there was no answer here to solve for.

One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.

I carried this problem around in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day, and I would be thinking about it when I went to sleep. Without distraction I would have the same thing going round and round in my mind. (Recalling the degree of focus and determination that eventually yielded the proof of Fermat's Last Theorem.)

When I was up there, stranded by myself, did I think I was going to die? Yes. Absolutely, and that’s what you need to know going in because it’s going to happen to you. This is space. It does not cooperate. At some point everything is going to go south on you. Everything is going to go south and you’re going to say 'This is it. This is how I end.' Now you can either accept that or you can get to work. That’s all it is. You just begin. You do the math, you solve one problem. Then you solve the next one, and then the next and if you solve enough problems you get to come home.

women live the lie from birth on, and then one day they realize that it's too late for them, they're too old to write a book or solve a difficult problem in math, they'll never learn to sing or play the piano, they showed such promise early on. so they run to the priest, their voices take on a hysterical edge, like the one mine has right now, and the priest tells them they have lived righteously and their reward will be in heaven, and he could certainly use someone in the kitchen for the potluck on Sunday night.

No matter how limited their powers of reason might have been. still they must have understood that living like that was just murder, a capital crime - except it was slow, day-by-day murder. The government (or humanity) could not permit capital punishment for one man, but they permitted the murder of millions a little at a time. To kill one man - that is, to subtract 50 years from the sum of all human lives - that was a crime; but to subtract from the sum of all human lives 50,000,000 years - that was not a crime! No, really, isn't it funny? This problem in moral math could be solved in half a minute by any ten-year-old Number today, but they couldn't solve it. All their Kant's together couldn't solve it (because it never occurred to one of their Kant's to construct a system of scientific ethics - that is, one based on subtraction, addition, division, and multiplication).

Dear math, please grow up and solve your own problems im tired of solving them for you

This approach [solving easiest problems first, during the test] works for some people, mostly because anything works for some people.

With math, a question doesn’t exist without an answer. It’s a guarantee. If I work on a problem long enough, I will solve it.

It is foolish to answer a question that you do not understand. It is sad to work for an end that you do not desire.

Your loose ends aren't adding up." "I'm not a math problem." "But I'll still solve you.

Learning is often paradoxical. The very thing we need in order to learn impedes our ability to learn. We need to focus intently to be able to solve problems—yet that focus can also block us from accessing the fresh approach we may need. Success is important, but critically, so is failure. Persistence is key—but misplaced persistence causes needless frustration.

At some point, everything's gonna go south on you... everything's going to go south and you're going to say, this is it. This is how I end. Now you can either accept that, or you can get to work. That's all it is. You just begin. You do the math. You solve one problem... and you solve the next one... and then the next. And If you solve enough problems, you get to come home.

Tengo's lectures took on uncommon warmth, and the students found themselves swept up in his eloquence. He taught them how to practically and effectively solve mathematical problems while simultaneously presenting a spectacular display of the romance concealed in the questions it posed. Tengo saw admiration in the eyes of several of his female students, and he realized that he was seducing these seventeen- or eighteen-year-olds through mathematics. His eloquence was a kind of intellectual foreplay. Mathematical functions stroked their backs; theorems sent warm breath into their ears.

Truth in life is broad and nuanced; you can make all kinds of arguments, such as whether a president or person is fantastic or awful,” he says. “That’s why I love math problems—they have clear answers.

I have been able to solve a few problems of mathematical physics on which the greatest mathematicians since Euler have struggled in vain ... But the pride I might have held in my conclusions was perceptibly lessened by the fact that I knew that the solution of these problems had almost always come to me as the gradual generalization of favorable examples, by a series of fortunate conjectures, after many errors. I am fain to compare myself with a wanderer on the mountains who, not knowing the path, climbs slowly and painfully upwards and often has to retrace his steps because he can go no further—then, whether by taking thought or from luck, discovers a new track that leads him on a little till at length when he reaches the summit he finds to his shame that there is a royal road by which he might have ascended, had he only the wits to find the right approach to it. In my works, I naturally said nothing about my mistake to the reader, but only described the made track by which he may now reach the same heights without difficulty.

Well, regular math, or applied math, is what I suppose you could call practical math," he said. "It's used to solve problems, to provide solutions, whether it's in the realm of economics, or engineering, or accounting, or what have you. But pure math doesn't exist to provide immediate, or necessarily obvious, practical applications. It's purely an expression of form, if you will - the only thing it proves is the almost infinite elasticity of mathematics itself, within the accepted set of assumptions by which we define it, of course.

Being surrounded by great people isn't a fluke. It's almost like solving a math problem, finding variables, adding and subtracting to figure out a formula that works. Being surrounded by people who fuel you is intentional.

For all the time schools devote to the teaching of mathematics, very little (if any) is spent trying to convey just what the subject is about. Instead, the focus is on learning and applying various procedures to solve math problems. That's a bit like explaining soccer by saying it is executing a series of maneuvers to get the ball into the goal. Both accurately describe various key features, but they miss the \what?" and the \why?" of the big picture.

Have you any idea what it takes to catch a ball, or raise a cup to your lips, or make immediate sense of a word, a phrase or an ambiguous sentence? We didn't, not at first. Solving maths problems is the tiniest fraction of what human intelligence does. We learned from a new angle just how wondrous a thing the brain is. A one-litre, liquid-cooled, three-dimensional computer. Unbelievable processing power, unbelievably compressed, unbelievable energy efficiency, no overheating. The whole thing running on twenty-five watts -- one dim light bulb.

There are chunks related to both concepts and procedures that reinforce one another. Solving a lot of math problems provides an opportunity to learn why the procedure works the way it does or why it works at all. Understanding the underlying concept makes it easier to detect errors when you make them.

There is much creativity underlying math and science problem solving. Many people think that there’s only one way to do a problem, but there are often a number of different solutions, if you have the creativity to see them. For example, there are more than three hundred different known proofs of the Pythagorean theorem.

Why do you hate this game so much?" Andrew sighed as if Neil was being purposefully obtuse. "I don't care enough about Exy to hate it. It's just slightly less boring than living is, so I put up with it for now." "I don't understand." "That's not my problem." "Isn't it fun?" Neil asked. "Someone else asked me that same thing two years ago. Should I tell you what I told him? I said no. Something as pointless as this game is can never be fun." "Pointless," Neil echoed. "But you have real talent." "Flattery is uninteresting and gets you nowhere." "I'm just stating facts. You're selling yourself short. You could be something if only you'd try." Andrew's smile was small and cold. "You be something. Kevin says you'll be a champion. Four years and you'll go pro. Five years and you'll be Court. He promised Coach. He promised the school board. He argued until they signed off on you. [...] Then Kevin finally got the okay to sign you and you hit the ground running," Andrew said. "Curious that a man with so much potential, who has so much fun, who could be something wouldn't want any of it. Why is that?" [...] "You're lying," Neil said at last, because he needed that to be the truth. "Kevin hates me." "Or you hate him," Andrew said. "I can't decide. Your loose ends aren't adding up." "I'm not a math problem." "But I'll still solve you.

If you’re having trouble with a math problem, plug the equation into WolframAlpha.com and it will solve it for you.

It is easy to have love affairs than to solve math problems but it is easy to learn if you trust God

give a reasonable priority to your sleep time, since it will greatly boost your brain skills, your ability to focus and even your problem solving and stress management abilities.

Tengo has an innate knack for precision in all realms, including correct punctuation and discovering the simplest possible formula necessary to solve a math problem.

On the other hand, data smart marketers look beyond data and do not go around chasing KPI. They focus on solving their customers’ problems, one at a time.

merely changing how a math problem is presented on a page can change how your brain goes about solving it,

I can always tell which ones are serious and which aren’t. There’s something in their voices that communicates passion and conviction when they’re really excited about getting out of debt. But if they’re just playing around with the idea, if they’re simply curious about it, then their voices are flat. If I don’t hear any passion behind what they’re saying, I know they aren’t ready to cut up the credit cards and dump their debt for good. That’s because getting out of debt isn’t about solving a math problem; it’s about changing your life—and that requires a change of heart.

I'm really good with problems. I can solve a differential equation in my head. I chew through trig angles like candy. I know this, and it just makes it worse. Because I don't know how to solve this one.

The Goober was beautiful when he ran. His long arms and legs moved flowingly and flawlessly, his body floating as if his feet weren’t touching the ground. When he ran, he forgot about his acne and his awkwardness and the shyness that paralyzed him when a girl looked his way. Even his thoughts became sharper, and things were simple and uncomplicated—he could solve math problems when he ran or memorize football play patterns. Often he rose early in the morning, before anyone else, and poured himself liquid through the sunrise streets, and everything seemed beautiful, everything in its proper orbit, nothing impossible, the entire world attainable. When he ran, he even loved the pain, the hurt of the running, the burning in his lungs and the spasms that sometimes gripped his calves. He loved it because he knew he could endure the pain, and even go beyond it. He had never pushed himself to the limit but he felt all this reserve strength inside of him: more than strength actually—determination. And it sang in him as he ran, his heart pumping blood joyfully through his body.

School is often based not on problem solving, which perforce involves actions and goals, but on learning information, facts, and formulas that one has read about in texts or heard about in lectures. It is not surprising, then, that research has long shown that a student’s doing well in school, in terms of grades and tests, does not correlate with being able to solve problems in the areas in which the student has been taught (e.g., math, civics, physics).

I’ll be honest with you. The variables that construct my existence are confusing. Like handwritten math equations jammed together on a sloppy page of homework. They don’t make any sense. One math problem leads to another, and then another and so it goes. One day you realize that your life is one whole page of problems and nothing ever gets solved. One ongoing equation with no equal sign at the end. But it occurred to me, beneath the canopy of a starlight heaven, that I’d been looking at my life all wrong. It wasn’t a math equation. Things weren’t supposed to add up. There was no solution. In fact, there was no problem. Life’s variables and numbers and pages of chicken scratch weren’t mathematical marks. They were art. A drawing. An abstract painting. It was meant to be beautiful, not sensical. And embedded within the mess of it all were miracles. Small ones. I’d never paid attention to them because I was too busy, but it didn’t make them less real.

Psychologist Robert Zajonc takes this claim one step further: “For most decisions, it is extremely difficult to demonstrate that there has actually been any prior cognitive process whatsoever.”28 It isn’t that the decisions people make are irrational; it’s that the process by which decisions are made are utterly unlike the step-by-step rational process that might be used to solve, say, a math problem. Decisions are typically made in the unconscious mind, by means of some unknown process. Indeed,

In fact, people who are extremely adept at mental tasks that demand cognitive control and a roaring working memory—like solving complex math problems—can struggle with creative insights if they have trouble switching off their fully concentrated focus.5

If you have a library of concepts and solutions internalized as chunked patterns, you can more easily skip to the right solution to a problem by listening to the whispers from your diffuse mode. Your diffuse mode can also help you connect two or more chunks together in new ways to solve unusual problems. There are two ways to solve problems—first, through sequential, step-by-step reasoning, and second, through more holistic intuition. Sequential thinking, where each small step leads deliberately toward the solution, involves the focused mode. Intuition, on the other hand, often seems to require a creative, diffuse mode linking of several seemingly different focused mode thoughts. Most difficult problems are solved through intuition, because they make a leap away from what you are familiar with.24 Keep in mind that the diffuse mode’s semi-random way of making connections means that the solutions it provides with should be carefully verified using the focused mode. Intuitive insights aren’t always correct!25

Researchers at the Heidelberg University Hospital conducted a study in which they subjected a young doctor to a job interview, which they made even more stressful by forcing him to solve complex math problems for thirty minutes. Afterward, they took a blood sample. What they discovered was that his antibodies had reacted to stress the same way they react to pathogens, activating the proteins that trigger an immune response. The problem is that this response not only neutralizes harmful agents, it also damages healthy cells, leading them to age prematurely.

Math. It’s your favorite subject. Which surprises you. Last year your teacher tried to convince you that you had a real “aptitude” for math, but all you got in the end was a B minus. The truth is you weren’t even trying. But then you got low Cs and Ds in all your other classes and you weren’t trying there, either, so maybe you are good at math after all. You like it because either you’re right or you’re wrong. Not like social studies and definitely not like English, where you always have to explain your answers and support your opinions. With math it’s right or it’s wrong and you’re done with it. But even that’s changing, my teacher said now you have to explain how you solved the problem and support your answer, saying that having the right answer isn’t as important as explaining how you got it and bam, just like that, you hate math.

As Albert Einstein said years ago, “The significant problems we face today cannot be solved at the same level of thinking we were at when we created them.” Developing the capacity to deal with the challenge of living in a stressful, ever-changing world is now more important than ever.

So: in the 1800s you wouldn’t steal because God would smite you, in the 1900s you didn’t steal because the State would punish you, but in the 2000s you can’t steal because the Network won’t let you.40 Either the social network will mob you, or the cryptocurrency network won’t let you steal because you lack the private key, or (eventually) the networked AI will detect you, or all of the above. Put another way, what’s the most powerful force on earth? In the 1800s, God. In the 1900s, the US military. And by the mid-2000s, encryption. Because as Assange put it, no amount of violence can solve certain kinds of math problems.

An extreme representative of this view is Ted Kaczynski, infamously known as the Unabomber. Kaczynski was a child prodigy who enrolled at Harvard at 16. He went on to get a PhD in math and become a professor at UC Berkeley. But you’ve only ever heard of him because of the 17-year terror campaign he waged with pipe bombs against professors, technologists, and businesspeople. In late 1995, the authorities didn’t know who or where the Unabomber was. The biggest clue was a 35,000-word manifesto that Kaczynski had written and anonymously mailed to the press. The FBI asked some prominent newspapers to publish it, hoping for a break in the case. It worked: Kaczynski’s brother recognized his writing style and turned him in. You might expect that writing style to have shown obvious signs of insanity, but the manifesto is eerily cogent. Kaczynski claimed that in order to be happy, every individual “needs to have goals whose attainment requires effort, and needs to succeed in attaining at least some of his goals.” He divided human goals into three groups: 1. Goals that can be satisfied with minimal effort; 2. Goals that can be satisfied with serious effort; and 3. Goals that cannot be satisfied, no matter how much effort one makes. This is the classic trichotomy of the easy, the hard, and the impossible. Kaczynski argued that modern people are depressed because all the world’s hard problems have already been solved. What’s left to do is either easy or impossible, and pursuing those tasks is deeply unsatisfying. What you can do, even a child can do; what you can’t do, even Einstein couldn’t have done. So Kaczynski’s idea was to destroy existing institutions, get rid of all technology, and let people start over and work on hard problems anew. Kaczynski’s methods were crazy, but his loss of faith in the technological frontier is all around us. Consider the trivial but revealing hallmarks of urban hipsterdom: faux vintage photography, the handlebar mustache, and vinyl record players all hark back to an earlier time when people were still optimistic about the future. If everything worth doing has already been done, you may as well feign an allergy to achievement and become a barista.

Lao-tzu advised, “As soon as you have a thought, laugh at it,” because reality is not what we think. We perceive the world through a window colored by beliefs, interpretations, and associations. We see things not as they are but as we are. The same brain that enables us to contemplate philosophy, solve math equations, and create poetry also generates a stream of static known as discursive thoughts, which seem to arise at random, bubbling up into our awareness. Such mental noise is a natural phenomenon, no more of a problem than the dreams that appear in the sleep state. Therefore, our schooling aims not to struggle with random thoughts but to transcend them in the present moment, where no thoughts exist, only awareness. Our mind’s liberation awaits not in some imagined future but here and now.

Most of us do not like not being able to see what others see or make sense of something new. We do not like it when things do not come together and fit nicely for us. That is why most popular movies have Hollywood endings. The public prefers a tidy finale. And we especially do not like it when things are contradictory, because then it is much harder to reconcile them (this is particularly true for Westerners). This sense of confusion triggers in a us a feeling of noxious anxiety. It generates tension. So we feel compelled to reduce it, solve it, complete it, reconcile it, make it make sense. And when we do solve these puzzles, there's relief. It feels good. We REALLY like it when things come together. What I am describing is a very basic human psychological process, captured by the second Gestalt principle. It is what we call the 'press for coherence.' It has been called many different things in psychology: consonance, need for closure, congruity, harmony, need for meaning, the consistency principle. At its core it is the drive to reduce the tension, disorientation, and dissonance that come from complexity, incoherence, and contradiction. In the 1930s, Bluma Zeigarnik, a student of Lewin's in Berlin, designed a famous study to test the impact of this idea of tension and coherence. Lewin had noticed that waiters in his local cafe seemed to have better recollections of unpaid orders than of those already settled. A lab study was run to examine this phenomenon, and it showed that people tend to remember uncompleted tasks, like half-finished math or word problems, better than completed tasks. This is because the unfinished task triggers a feeling of tension, which gets associated with the task and keeps it lingering in our minds. The completed problems are, well, complete, so we forget them and move on. They later called this the 'Zeigarnik effect,' and it has influenced the study of many things, from advertising campaigns to coping with the suicide of loved ones to dysphoric rumination of past conflicts.

George Dantzig was a graduate student in math at Berkeley. One day, as usual, he rushed in late to his math class and quickly copied the two homework problems from the blackboard. When he later went to do them, he found them very difficult, and it took him several days of hard work to crack them open and solve them. They turned out not to be homework problems at all. They were two famous math problems that had never been solved.

Part of what makes the Focus Question work so well are those two final words, “for you.” A 1997 study involving a fairly convoluted series of math problems focused on the impact of having the word “you” as part of a math problem’s description. The researchers found that when the word “you” was present, the questions needed to be repeated fewer times, and the problems were solved in a shorter amount of time and with more accuracy.

There needs to be a revolution in education. We must encourage schools to train our students in the art of living in peace and harmony. It isn’t easy to learn to read, write, or solve math problems, but children manage to do it. Learning how to breathe, smile, and transform anger can also be difficult, but I have seen many young people succeed. If we teach children properly, by the time they are around twelve, they will know how to live harmoniously with others.

The thing is, a bad feeling doesn’t always mean something is wrong. It just means you’re taxing your body budget. When people exercise to the point of labored breathing, for example, they feel tired and crappy well before they run out of energy. When people solve math problems and perform difficult feats of memory, they can feel hopeless and miserable, even when they are performing well. Any graduate student of mine who never feels distress is clearly doing something wrong.

The researchers found that when students were given problems to solve, and they did not know methods to solve them, but they were given opportunity to explore the problems, they became curious, and their brains were primed to learn new methods, so that when teachers taught the methods, students paid greater attention to them and were more motivated to learn them. The researchers published their results with the title “A Time for Telling,” and they argued that the question is not “Should we tell or explain methods?” but “When is the best time do this?

The distance between my lips and yours cannot be deciphered from the square root of the sum of the days we have spent wondering what to do with three minutes and ten seconds. The distance between my lips and yours cannot be deduced by the difference in the circumferences of our necks or in how many minutes we can sit in the noon sun. The distance between my lips and yours can only be measured in poems. Tell me, how many are there? Were there? Will there be? (But who knows what to call a poem and what to call a conversation? And who knows whether to call at all?)

Intelligence isn’t just about how many levels of math courses you’ve taken, how fast you can solve an algorithm, or how many vocabulary words you know that are over 6 characters. It’s about being able to approach a new problem, recognize its important components, and solve it—then take that knowledge gained and put it towards solving the next, more complex problem. It’s about innovation and imagination, and about being able to put that to use to make the world a better place. This is the kind of intelligence that is valuable, and this is the type of intelligence we should be striving for and encouraging.

Honestly,” Gray Man continued, “why would these people engage in these ritual activities if those activities didn’t work? They were doing stuff that other people don’t do and in fact wouldn’t dare do. They engaged in these activities because they most likely worked for them. What’s the alternative? That somehow they were irrational, delusional people, while at the same time being some of the best problem solvers on the planet, requiring a high degree of critical thinking, an excellent understanding of math, and a very good working knowledge of other various fields of science to solve the problems they were working on?

Pure math,” he replied. “How is that different from”—she laughed—“regular math?” Gillian asked. “Well, regular math, or applied math, is what I suppose you could call practical math,” he said. “It’s used to solve problems, to provide solutions, whether it’s in the realm of economics, or engineering, or accounting, or what have you. But pure math doesn’t exist to provide immediate, or necessarily obvious, practical applications. It’s purely an expression of form, if you will—the only thing it proves is the almost infinite elasticity of mathematics itself, within the accepted set of assumptions by which we define it, of course.

you could be something if only you’d try.” “you be something. kevin says you’ll be a champion. four years and you’ll go pro. five years and you’ll be court. he promised coach. he promised the school board. he argued until they signed off on you.” “he— what?” “then kevin finally got the okay to sign you and you hit the ground running. curious that a man with so much potential, who has so much fun, who could ‘be something’ wouldn’t want any of it. why is that?” “you’re lying. kevin hates me.” “or you hate him. i can’t decide. your loose ends aren’t adding up.” “i’m not a math problem.” “but i’ll still solve you.” - neil & andrew

Despite their ubiquity on the number line, transcendentals are surprisingly hard to pin down. It took until 1873 to prove that e was transcendental, making it the first number we knew for definite was. The poster-child of maths, pi, didn't join the transcendental fold until 1882. Even today, we know that at least one of e + pi and e × pi is transcendental, but we have no idea which. On David Hilbert's 1900 list of important maths problems to solve, one of them involved checking if e^pi is transcendental, and since 1934 we have known that it is. However, e^e, pi^pi, and pi^e are still open problems. Transcendentals are really hard to find in the wild.

The fundamental problem with learning mathematics is that while the number sense may be genetic, exact calculation requires cultural tools—symbols and algorithms—that have been around for only a few thousand years and must therefore be absorbed by areas of the brain that evolved for other purposes. The process is made easier when what we are learning harmonizes with built-in circuitry. If we can’t change the architecture of our brains, we can at least adapt our teaching methods to the constraints it imposes. For nearly three decades, American educators have pushed “reform math,” in which children are encouraged to explore their own ways of solving problems. Before reform math, there was the “new math,” now widely thought to have been an educational disaster. (In France, it was called les maths modernes and is similarly despised.) The new math was grounded in the theories of the influential Swiss psychologist Jean Piaget, who believed that children are born without any sense of number and only gradually build up the concept in a series of developmental stages. Piaget thought that children, until the age of four or five, cannot grasp the simple principle that moving objects around does not affect how many of them there are, and that there was therefore no point in trying to teach them arithmetic before the age of six or seven.

When an official report in the UK was commissioned to examine the mathematics needed in the workplace, the investigator found that estimation was the most useful mathematical activity. Yet when children who have experienced traditional math classes are asked to estimate, they are often completely flummoxed and try to work out exact answers, then round them off to look like an estimate. This is because they have not developed a good feel for numbers, which would allow them to estimate instead of calculate, and also because they have learned, wrongly, that mathematics is all about precision, not about making estimates or guesses. Yet both are at the heart of mathematical problem solving.

With the threat of failure looming, students with the growth mindset set instead mobilized their resources for learning. They told us that they, too, sometimes felt overwhelmed, but their response was to dig in and do what it takes. They were like George Danzig. Who? George Danzig was a graduate student in math at Berkeley. One day, as usual, he rushed in late to his math class and quickly copied the two homework problems from the blackboard. When he later went to do them, he found them very difficult, and it took him several days of hard work to crack them open and solve them. They turned out not to be homework problems at all. They were two famous math problems that had never been solved.

Why? This is our world of habituation, where nothing is ever as good as that first time. Unfortunately, things have to work this way because of our range of rewards.86 After all, reward coding must accommodate the rewarding properties of both solving a math problem and having an orgasm. Dopaminergic responses to reward, rather than being absolute, are relative to the reward value of alternative outcomes. In order to accommodate the pleasures of both mathematics and orgasms, the system must constantly rescale to accommodate the range of intensity offered by particular stimuli. The response to any reward must habituate with repetition, so that the system can respond over its full range to the next new thing.

Naomi doubted that any human being really understood math, they simply all pretended to have it down pat, when in truth they were every bit as confused by it as she was. Math was nothing but a giant hoax, and everyone participated in it, everyone faked belief in math so they could be done with hideous classes and the drudgery of the hateful homework and get on with life. The sun came up every morning, so the sun was real, and every time you inhaled you got the air you needed, so the atmosphere was obviously real, but half the time when you tried to use math to solve the simplest problem, the math absolutely would not work, which meant that it couldn't be real like the sun and the atmosphere. Math was a waste of time.

If you had asked Dan during that period whether he still loved his wife, he would have looked at you in total confusion and said, “Of course!” Although his wife was at that very moment wallowing in despair over his treatment of her, he perceived things to be fine between them. This isn’t because he is dense; it’s just that after a lifetime of having people mad at or disappointed with him, Dan weathers periods of anger and criticism by mostly ignoring them. And, because people with ADHD don’t receive and process information in a hierarchical way, Maria’s suffering enters his mind at about the same level as everything else he perceives—the lights on the radio clock, the dog barking, the computer, the worrisome project he has at work. “But wait!” you say. “It doesn’t matter—she’s still alone!” You would be right. Regardless of whether Dan was intentionally ignoring his wife or just distracted, actions speak louder than words. She becomes lonely and unhappy, and her needs must be addressed. But recognizing and then identifying the correct underlying problem is critical to finding the right solution. In marriage, just like in middle school math, if you pick the wrong problem to solve, you generally don’t end up with a satisfactory result. Furthermore, the hurt caused by the incorrect interpretation that he no longer loves her elicits a series of bad feelings and behaviors that compound the problem. This is the critical dynamic of symptom–response–response at work.

He would have to keep her, somehow, and that would mean solving her. That would mean making her his impossible problem. Time travel no longer held any interest for him, only Regan and whatever it would take to make her a fixture in his life. Knowing her would mean knowing everything, not just her thoughts or her truths or the way she liked to be fucked. Knowing her would mean knowing her future, having it for himself. It was knowing what her children would look like, and what she would look like someday, when the youth was gone from her face and replaced by something else; by what? A mystery. It was a fucking mystery and Aldo couldn’t sit idly by while there were mysteries afoot. Uncertainty was something he lived with, yes, but not anymore. Frustration and restraint, she had said, equating his love of math with his love of her. I am Atlas, he thought, holding up the heavens. I will be endurance, I will have to endure.

If you're now noticing a certain family resemblance among this no-successive-instant problem, Zeno's Paradoxes, and some of the Real Line crunchers described in Paragraph 2c and -e, be advised that this is not a coincidence. They are all facets of the great continuity conundrum for mathematics, which is that (Infinity)-related entities can apparently be neither handled nor eliminated. Nowhere is this more evident than with 1/(Infinity)s. They're riddled with paradox and can't be defined, but if you banish them from math you end up having to posit an infinite density to any interval, in which the idea of succession makes no sense and no ordering of points in the interval can ever be complete, since between any two points there will be not just some other points but a whole infinity of them. Overall point: However good calculus is at quantifying motion and change, it can do nothing to solve the real paradoxes of continuity. Not without a coherent theory of (Infinity), anyway.

In both cultures, wealth is no longer a means to get by. It becomes directly tied to personal worth. A young suburbanite with every advantage—the prep school education, the exhaustive coaching for college admissions tests, the overseas semester in Paris or Shanghai—still flatters himself that it is his skill, hard work, and prodigious problem-solving abilities that have lifted him into a world of privilege. Money vindicates all doubts. They’re eager to convince us all that Darwinism is at work, when it looks very much to the outside like a combination of gaming a system and dumb luck. In both of these industries, the real world, with all of its messiness, sits apart. The inclination is to replace people with data trails, turning them into more effective shoppers, voters, or workers to optimize some objective. This is easy to do, and to justify, when success comes back as an anonymous score and when the people affected remain every bit as abstract as the numbers dancing across the screen. More and more, I worried about the separation between technical models and real people, and about the moral repercussions of that separation. In fact, I saw the same pattern emerging that I’d witnessed in finance: a false sense of security was leading to widespread use of imperfect models, self-serving definitions of success, and growing feedback loops. Those who objected were regarded as nostalgic Luddites.

I work in theoretical computer science: a field that doesn’t itself win Fields Medals (at least not yet), but that has occasions to use parts of math that have won Fields Medals. Of course, the stuff we use cutting-edge math for might itself be dismissed as “ivory tower self-indulgence.” Except then the cryptographers building the successors to Bitcoin, or the big-data or machine-learning people, turn out to want the stuff we were talking about at conferences 15 years ago—and we discover to our surprise that, just as the mathematicians gave us a higher platform to stand on, so we seem to have built a higher platform for the practitioners. The long road from Hilbert to Gödel to Turing and von Neumann to Eckert and Mauchly to Gates and Jobs is still open for traffic today. Yes, there’s plenty of math that strikes even me as boutique scholasticism: a way to signal the brilliance of the people doing it, by solving problems that require years just to understand their statements, and whose “motivations” are about 5,000 steps removed from anything Caplan or Bostrom would recognize as motivation. But where I part ways is that there’s also math that looked to me like boutique scholasticism, until Greg Kuperberg or Ketan Mulmuley or someone else finally managed to explain it to me, and I said: “ah, so that’s why Mumford or Connes or Witten cared so much about this. It seems … almost like an ordinary applied engineering question, albeit one from the year 2130 or something, being impatiently studied by people a few moves ahead of everyone else in humanity’s chess game against reality. It will be pretty sweet once the rest of the world catches up to this.

When I first started coming to the seminar, Gelfand had a young physicist, Vladimir Kazakov, present a series of talks about his work on so-called matrix models. Kazakov used methods of quantum physics in a novel way to obtain deep mathematical results that mathematicians could not obtain by more conventional methods. Gelfand had always been interested in quantum physics, and this topic had traditionally played a big role at his seminar. He was particularly impressed with Kazakov’s work and was actively promoting it among mathematicians. Like many of his foresights, this proved to be golden: a few years later this work became famous and fashionable, and it led to many important advances in both physics and math. In his lectures at the seminar, Kazakov was making an admirable effort to explain his ideas to mathematicians. Gelfand was more deferential to him than usual, allowing him to speak without interruptions longer than other speakers. While these lectures were going on, a new paper arrived, by John Harer and Don Zagier, in which they gave a beautiful solution to a very difficult combinatorial problem.6 Zagier has a reputation for solving seemingly intractable problems; he is also very quick. The word was that the solution of this problem took him six months, and he was very proud of that. At the next seminar, as Kazakov was continuing his presentation, Gelfand asked him to solve the Harer–Zagier problem using his work on the matrix models. Gelfand had sensed that Kazakov’s methods could be useful for solving this kind of problem, and he was right. Kazakov was unaware of the Harer–Zagier paper, and this was the first time he heard this question. Standing at the blackboard, he thought about it for a couple of minutes and immediately wrote down the Lagrangian of a quantum field theory that would lead to the answer using his methods. Everyone in the audience was stunned.

One way to try to answer the question “What makes us human?” is to ask “What makes us different from great apes?” or, to be more precise, from nonhuman apes, since, of course, humans are apes. As just about every human by now knows—and as the experiments with Dokana once again confirm—nonhuman apes are extremely clever. They’re capable of making inferences, of solving complex puzzles, and of understanding what other apes are (and are not) likely to know. When researchers from Leipzig performed a battery of tests on chimpanzees, orangutans, and two-and-a-half-year-old children, they found that the chimps, the orangutans, and the kids performed comparably on a wide range of tasks that involved understanding of the physical world. For example, if an experimenter placed a reward inside one of three cups, and then moved the cups around, the apes found the goody just as often as the kids—indeed, in the case of chimps, more often. The apes seemed to grasp quantity as well as the kids did—they consistently chose the dish containing more treats, even when the choice involved using what might loosely be called math—and also seemed to have just as good a grasp of causality. (The apes, for instance, understood that a cup that rattled when shaken was more likely to contain food than one that did not.) And they were equally skillful at manipulating simple tools. Where the kids routinely outscored the apes was in tasks that involved reading social cues. When the children were given a hint about where to find a reward—someone pointing to or looking at the right container—they took it. The apes either didn’t understand that they were being offered help or couldn’t follow the cue. Similarly, when the children were shown how to obtain a reward, by, say, ripping open a box, they had no trouble grasping the point and imitating the behavior. The apes, once again, were flummoxed. Admittedly, the kids had a big advantage in the social realm, since the experimenters belonged to their own species. But, in general, apes seem to lack the impulse toward collective problem-solving that’s so central to human society. “Chimps do a lot of incredibly smart things,” Michael Tomasello, who heads the institute’s department of developmental and comparative psychology, told me. “But the main difference we’ve seen is 'putting our heads together.' If you were at the zoo today, you would never have seen two chimps carry something heavy together. They don’t have this kind of collaborative project.

Complex Math Problem: How do you solve the messes you can’t see? In other words—Is fitting in just covering up who you really are in hopes that people don’t notice?

Problem Solve by Making a Table

whose students are willing to concentrate and sit still long enough and focus on answering every single question in an endless questionnaire are the same countries whose students do the best job of solving math problems.

Creating problems. Solving problems. Which are you truly good at? Things are the way they are. Things happen the way they happen. Only mathematics equates to solving problems? Since when? What ever happened to common sense and honesty?

Knowledge of a subject doesn’t necessarily mean you can do anything with it. That would require skill. Likewise, a student can commit something to memory with no knowledge whatsoever. A grade-schooler, for example, might be able to recite the multiplication table perfectly, but still be unable to solve simple math problems. Coming to know something implies the goal of being able to use the knowledge, while being aware of facts or figures does not. Of

Mathematics works only with computationally reducible systems. Axioms and deductive logic are intended to provide shortcuts, to give general results that can compress a problem and provide insight into its workings so that it can be solved without having to take on the tedious task of running things through step by step. For example, the math behind ballistics tables allows an artillery gunner to calculate where the shell will land before it is fired. By contrast, there is no precalculated table to consult to determine the best path through rush hour traffic.

TEACHER: How many times did you play it? STUDENT: Ten or twenty. TEACHER: How many times did you play it correctly? STUDENT: Umm, I dunno . . . Once or twice . . . TEACHER: Hmm . . . How did you practice it? STUDENT: I dunno. I just played it. This is naive practice in a nutshell: I just played it. I just swung the bat and tried to hit the ball. I just listened to the numbers and tried to remember them. I just read the math problems and tried to solve them.

Grief, Mom discovered, was not a problem you could fix, a loose screw you could tighten, a math problem you could solve, a child whose pain you could comfort. It just sat there in your stomach and didn’t move. Sometimes it grew, sometimes it shrank, but it was always, always there. That was the hardest part, she said, harder than anything else, before or after. The grief doesn’t leave. It becomes a part of you. Either you learn to live with it or you die.

A young suburbanite with every advantage—the prep school education, the exhaustive coaching for college admissions tests, the overseas semester in Paris or Shanghai—still flatters himself that it is his skill, hard work, and prodigious problem-solving abilities that have lifted him into a world of privilege. Money vindicates all doubts.

If you have N toy blocks, are there any values of N for which you can only form one rectangle?

Even as a young boy, Axel loved math. Dani instinctively sensed that rational equations gave him a feeling of control in a world that had become chaotic. Math problems could be solved. They made sense, while feelings often did not.

populations at scale. There is greater upheaval still to come. In 2019, Google announced it had achieved “quantum supremacy”—a computing breakthrough so powerful that a math problem a supercomputer would need ten thousand years to solve could be cracked by its machine in just three minutes and twenty seconds.