Probability Math Quotes

You’re probably better at math than I am, because pretty much everyone’s better at math than I am, but it’s okay, I’m fine with it. See, I excel at other, more important things—guitar, sex, and consistently disappointing my dad, to name a few.

If nature has taught us anything it is that the impossible is probable

In our travels, we have come across many equations--math for understanding the universe, for making music, for mapping stars, and also for tipping, which is important. Here is our favorite equation: Us plus Them equals All of Us. It is very simple math. Try it sometime. You probably won’t even need a pencil.

I failed math twice, never fully grasping probability theory. I mean, first off, who cares if you pick a black ball or a white ball out of the bag? And second, if you’re bent over about the color, don’t leave it to chance. Look in the damn bag and pick the color you want.

Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean.

I am no friend of probability theory, I have hated it from the first moment when our dear friend Max Born gave it birth. For it could be seen how easy and simple it made everything, in principle, everything ironed and the true problems concealed. Everybody must jump on the bandwagon [Ausweg]. And actually not a year passed before it became an official credo, and it still is.

A person who spends eight years learning how to make a cake will probably make you a good cake, but a person who spends eight years as an aviator and a tailor and a math tutor and a trainer of bears in the circus will probably kill you in a plane he is flying very badly while wearing a shirt that doesn't fit and fighting off an ill-behaved bear, all the while insisting that seven times six is harmonica.

Another mistaken notion connected with the law of large numbers is the idea that an event is more or less likely to occur because it has or has not happened recently. The idea that the odds of an event with a fixed probability increase or decrease depending on recent occurrences of the event is called the gambler's fallacy. For example, if Kerrich landed, say, 44 heads in the first 100 tosses, the coin would not develop a bias towards the tails in order to catch up! That's what is at the root of such ideas as "her luck has run out" and "He is due." That does not happen. For what it's worth, a good streak doesn't jinx you, and a bad one, unfortunately , does not mean better luck is in store.

Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable, whereas most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity

Srinivasa Ramanujan was the strangest man in all of mathematics, probably in the entire history of science. He has been compared to a bursting supernova, illuminating the darkest, most profound corners of mathematics, before being tragically struck down by tuberculosis at the age of 33... Working in total isolation from the main currents of his field, he was able to rederive 100 years’ worth of Western mathematics on his own. The tragedy of his life is that much of his work was wasted rediscovering known mathematics.

When will it begin, anyway?" Sirus held his gaze for a moment, his eyes full of concern- a concern that Joss didn't understand. "Probably sooner than you're ready for." "When's that?" "Well." Sirus sighed, as if doing the math in his head."It'll take us about three minutes to gather this stuff and get to the cabin, and another two or three for Abraham to realize you're here. So I'd say you have about seven more minutes of freedom left.

It may be appropriate to quote a statement of Poincare, who said (partly in jest no doubt) that there must be something mysterious about the normal law since mathematicians think it is a law of nature whereas physicists are convinced that it is a mathematical theorem.

Jesus probably studied this same information, in his youth. The apostle Paul probably studied this same information. How can I make such a bold assertion? Because, without this knowledge, much of the New Testament would make no sense. Many of the idioms used in the New Testament are the result of lessons learned from this ancient Hebrew education system. Unfortunately, what was common in their day, has become forgotten in ours. For a Hebrew, math doesn’t get in the way. It blazes the way. Other languages are disconnected from this mathematical relationship . . . and it shows.

So what were your favorite subjects in school?" "School?" He leaned back in his chair as though he needed the extra space to think about it. "Probably math. It always made sense. Unlike English, economics, and girls." "And exactly how do you plan on taking over the free world if you don't understand economics?" "I'll hire advisers. I'll hire you, in fact." "Okay. Let me know when your army of junior high zombies is ready.

People enjoy inventing slogans which violate basic arithmetic but which illustrate “deeper” truths, such as “1 and 1 make 1” (for lovers), or “1 plus 1 plus 1 equals 1” (the Trinity). You can easily pick holes in those slogans, showing why, for instance, using the plus-sign is inappropriate in both cases. But such cases proliferate. Two raindrops running down a window-pane merge; does one plus one make one? A cloud breaks up into two clouds -more evidence of the same? It is not at all easy to draw a sharp line between cases where what is happening could be called “addition”, and where some other word is wanted. If you think about the question, you will probably come up with some criterion involving separation of the objects in space, and making sure each one is clearly distinguishable from all the others. But then how could one count ideas? Or the number of gases comprising the atmosphere? Somewhere, if you try to look it up, you can probably fin a statement such as, “There are 17 languages in India, and 462 dialects.” There is something strange about the precise statements like that, when the concepts “language” and “dialect” are themselves fuzzy.

The probability of an event is the reason we have to believe that it has taken place, or that it will take place. The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible.

Even if you’re bad at math, you’re probably much better at it than the smartest chimpanzee, whose genetic identity varies in only trifling ways from ours.

open a basic math book, and make sure you are really good at multiplying, dividing, compounding, probability, and statistics.

Conventional wisdom nor scientific, mathematical prove of randomness in life could do nothing to deter human's curiosity for the unknown, however small the chance of a positive outcome maybe.

You can be a rich person alone. You can be a smart person alone. But you cannot be a complete person alone. For that you must be part of, and rooted in, an olive grove. This truth was once beautifully conveyed by Rabbi Harold S. Kushner in his interpretation of a scene from Gabriel García Márquez’s classic novel One Hundred Years of Solitude: Márquez tells of a village where people were afflicted with a strange plague of forgetfulness, a kind of contagious amnesia. Starting with the oldest inhabitants and working its way through the population, the plague causes people to forget the names of even the most common everyday objects. One young man, still unaffected, tries to limit the damage by putting labels on everything. “This is a table,” “This is a window,” “This is a cow; it has to be milked every morning.” And at the entrance to the town, on the main road, he puts up two large signs. One reads “The name of our village is Macondo,” and the larger one reads “God exists.” The message I get from that story is that we can, and probably will, forget most of what we have learned in life—the math, the history, the chemical formulas, the address and phone number of the first house we lived in when we got married—and all that forgetting will do us no harm. But if we forget whom we belong to, and if we forget that there is a God, something profoundly human in us will be lost.

You’re probably better at math than I am, because pretty much everyone’s better at math than I am, but it’s okay, I’m fine with it. See, I excel at other, more important things—guitar, sex, and consistently disappointing my dad, to name a few. By the way, it’s apparently true that you’ll never use it in the real world. Math, I mean.

Math is a difficult thing to love, he said. It’s precise and unforgiving, it’s evasive and it will never love me back, but I don’t have much of a choice, do I? It’s the thing that I can do that other people can’t, or that other people lack the patience for. Are there worthier things, more rewarding things? Yes, probably. But I don’t know what they are, they never showed themselves to me. Only math did.

If scientific reasoning were limited to the logical processes of arithmetic, we should not get very far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability.

He'd met other prodigies in mathematical competitions. In fact he'd been thoroughly trounced by competitors who probably spent literally all day practising maths problems and who'd never read a science-fiction book and who would burn out completely before puberty and never amount to anything in their future lives because they'd just practised known techniques instead of learning to think creatively. (Harry was something of a sore loser.)

I probably asked her too many questions. It is a maths teacher trait. The value of the unknown variable must always be found.

Marcus couldn’t believe it. Dead. A dead duck. OK, he'd been trying to hit it on the head with a piece of sandwich, but he tried to do all sorts of things, and none of them had ever happened before. He'd tried to get te highest score on the Stargazer machine in the kebab shop on Hornsey Road--nothing. He's tried to read Nicky’s thoughts by staring at the back of his head every maths lesson for a week--nothing. It really annoyed him that the only thing he'd ever achieved was something he hadn't really wanted to do that much in the first place. And anyway, since when did hitting a bird with a sandwich kill it? Kids must spend half their lives throwing things at the ducks in Regent's Park. How come he managed to pick a duck that pathetic? There must have been something wrong with it. It was probably about to die from a heart attack or something; it was just a coincidence. But if it was, nobody would believe him. If there were any witnesses, they'd only have seen the bread hit the duck right on the back of the head, and then seen it keel over. saw it die. They'd put two and two together and make five, and he'd be imprisoned for a crime he never committed. ... "What's that floating next to it?" Will asked. "Is that the bread you threw at it?" Marcus nodded unhappily. "That's not a sandwich, that's a bloody french loaf. No wonder it keeled over. That would've killed me.

Why did math matter so much? Some reasons were practical: More and more jobs required familiarity with probability, statistics, and geometry. The other reason was that math was not just math. Math is a language of logic. It is a disciplined, organized way of thinking. There is a right answer; there are rules that must be followed. More than any other subject, math is rigor distilled. Mastering the language of logic helps to embed higher-order habits in kids’ minds: the ability to reason, for example, to detect patterns and to make informed guesses. Those kinds of skills had rising value in a

Habits can be good and bad. Habit, after all, is simply when our brain launches into a preprogrammed “zombie” mode. You will probably not be surprised to learn that chunking, that automatically connected neural pattern that arises from frequent practice, is intimately related to habit.1 Habit is an energy saver for us. It allows us to free our mind for other types of activities

A distinguished writer [Siméon Denis Poisson] has thus stated the fundamental definitions of the science: 'The probability of an event is the reason we have to believe that it has taken place, or that it will take place.' 'The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible' (equally like to happen). From these definitions it follows that the word probability, in its mathematical acceptation, has reference to the state of our knowledge of the circumstances under which an event may happen or fail. With the degree of information which we possess concerning the circumstances of an event, the reason we have to think that it will occur, or, to use a single term, our expectation of it, will vary. Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave neither room nor demand for a theory of probabilities.

We know next to nothing with any certainty about Pythagoras, except that he was not really called Pythagoras. The name by which he is known to us was probably a nickname bestowed by his followers. According to one source, it meant ‘He who spoke truth like an oracle’. Rather than entrust his mathematical and philosophical ideas to paper, Pythagoras is said to have expounded them before large crowds. The world’s most famous mathematician was also its first rhetorician.

A curious peculiarity of our memory is that things are impressed better by active than by passive repetition. I mean that in learning by heart (for example), when we almost know the piece, it pays better to wait and recollect by an effort from within, than to look at the book again. If we recover the words in the former way, we shall probably know them the next time; if in the latter way, we shall very likely need the book once more.” —William James, writing in 189012

The full impact of the Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky the Copernicus of Geometry [as did Clifford], for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought.

You’re probably better at math than I am, because pretty much everyone’s better at math than I am, but it’s okay, I’m fine with it. See, I excel at other, more important things—guitar, sex, and consistently disappointing my dad, to name a few. By the way, it's apparently true that you'll never use it in the real world. Math, I mean.

Great literature remains great when it says new things to new generations, and the loops of a knot quite nicely parallel the contours and convolutions of Carroll’s plot anyway.What’s more, he probably would have been delighted at how this whimsical branch of math invaded the real world and became crucial to understanding our biology.

Were he now still among the living, Dr. Incandenza would now describe tennis in the paradoxical terms of what’s now called ‘Extra-Linear Dynamics.’ And Schtitt, whose knowledge of formal math is probably about equivalent to that of a Taiwanese kindergartner, nevertheless seemed to know what Hopman and van der Meer and Bollettieri seemed not to know: that locating beauty and art and magic and improvement and keys to excellence and victory in the prolix flux of match play is not a fractal matter of reducing chaos to a pattern. Seemed intuitively to sense that it was a matter not of reduction at all, but — perversely — of expansion, the aleatory flutter of uncontrolled, metastatic growth — each well-shot ball admitting of n possible responses, n² responses to those responses, and on into what Incandenza would articulate to anyone who shared both his backgrounds as a Cantorian continuum of infinities of possible move and response, Cantorian and beautiful because infoliating, contained, this diagnate infinity of infinities of choice and execution, mathematically uncontrolled but humanly contained, bounded by the talent and imagination of self and opponent, bent in on itself by the containing boundaries of skill and imagination that brought one player finally down, that kept both from winning, that made it, finally, a game, these boundaries of self.

You know, Piggly Wiggly never could hang on to a night stock manager. Your math skills would be a plus, maybe even your Spanish, and you don't mind staying up late." "Piggly Wiggly, wow. I hadn't thought about that. I'll swing by, pick up an application tomorrow. But if it doesn't pan out maybe ... Never mind, it's a crazy idea." "No, tell me. I want to hear it." "Well, just as a backup plan, I did hear that Sony has an opening. They're, um, they're looking for a rock star. The hours suck, but it's no worse than night stock manager at Pigs. I bet it pays better too." Isabel stopped in her tracks, playfully slapping his arm. "Aidan, that's genius! That's what you should do! I've heard you sing, you can carry a decent tune." She looked him up and down. "With a little work, you can probably pull off the image." He tugged on her arms until she was in his. "Only if you're sure. Only if it's what we want." "Aidan, it's who you are. I've known it since the day we brought that first guitar here. I'd never want to take that away.

We're constantly judging and grading other parents, just to make sure that they aren't any better than us. I'm as guilty as anyone. I see some lady hand her kid a Nintendo DS at the supermarket and I instantly downgrade that lady to Shitty Parent status. I feel pressure to live up to a parental ideal that no one probably has ever achieved. I feel pressure to raise a group of human beings that will help America kick the shit out of Finland and South Korea in the world math rankings. I feel pressure to shield my kids from the trillion pages of hentai donkey porn out there on the Internet. I feel pressure to make the insane amounts of money needed for a supposedly 'middle-class' upbringing for the kids, an upbringing that includes a house and college tuition and health care and so many other expenses that you have to be a multimillionaire to afford it. PRESSURE PRESSURE PRESSURE.

If you're not thinking about numbers, you're probably not thinking.

Reporters go through four stages in a war zone. In the first stage, you’re Superman, invincible. In the second, you’re aware that things are dangerous and you need to be careful. In the third, you conclude that math and probability are working against you. In the fourth, you know you’re going to die because you’ve played the game too long. I was drifting into stage three.

probably heard that math is the language of science, or the language of Nature is mathematics. Well, it’s true. The more we understand the universe, the more we discover its mathematical connections. Flowers have spirals that line up with a special sequence of numbers (called Fibonacci numbers) that you can understand and generate yourself. Seashells form in perfect mathematical curves (logarithmic spirals) that come from a chemical balance. Star clusters tug on

Joanna gave me a leveling look. "You think we're the only major that's superinsular and clique-ish and gossips about everyone else in it? Believe me, if Nathan bought a new brand of toothpaste, those math nerds probably already know about it.

That night when I went to bed, I laid there in the dark and pictured a clothesline full of somebody's else's troubles. I knew for sure there were a lot of them I'd rather pluck off of that line than mine. I imagined what the other troubles might be. There would probably be toothaches and failed math tests. Lost cats and ugly hair. Cheating boyfriends and broken-down cars. But none of those could hold a candle to my troubles, weighing down that clothesline like a sack full of bricks.

Perfect moments, what did they even mean? They were blind luck, that was all. Coincidences. Statistical anomalies. I did some Googling and it turned out somebody had actually bothered to do the math on this, a real actual Cambridge University mathematician named John Littlewood (1885-1977; thank you, Wikipedia). He proposed that if you define a miracle as something with a probability of one in a million, and if you’re paying close attention to the world around you eight hours a day, every day, and little things happen around you at a rate of one per second, then you’d observe about thirty thousand things every day, which means about a million things a month. So, on average, you should witness one miracle every month (or every thirty-three-and-one-third days, if we’re being strictly accurate). It’s called Littlewood’s law. So, there you have it, a miracle a month. They’re not even that special.

To the average mathematician who merely wants to know his work is securely based, the most appealing choice is to avoid difficulties by means of Hilbert's program. Here one regards mathematics as a formal game and one is only concerned with the question of consistency ... . The Realist position is probably the one which most mathematicians would prefer to take. It is not until he becomes aware of some of the difficulties in set theory that he would even begin to question it. If these difficulties particularly upset him, he will rush to the shelter of Formalism, while his normal position will be somewhere between the two, trying to enjoy the best of two worlds.

To the average mathematician who merely wants to know his work is securely based, the most appealing choice is to avoid difficulties by means of Hilbert's program. Here one regards mathematics as a formal game and one is only concerned with the question of consistency ... . The Realist position is probably the one which most mathematicians would prefer to take. It is not until he becomes aware of some of the difficulties in set theory that he would even begin to question it. If these difficulties particularly upset him, he will rush to the shelter of Formalism, while his normal position will be somewhere between the two, trying to enjoy the best of two worlds.

Who else in Spearcrest—in this world, probably—would make me feel the way she makes me feel? The thrill of her expression when I solve a problem first in maths class? The slight pinch of her lips when my name gets called out before hers as our teacher hands us our marked essays back? The satisfaction of being invited to the sixth form lectures when she’s not?

Do you remember that day in Senior Goals when you said it wasn't strange that I wanted to go to the moon?" "I remember." "I think that was the first day I really noticed you." "Took you that long?" Kath teased her. "Maybe I'm a late bloomer," Lily said tartly. "Why, when did you notice me?" Kath shot her a grin. "You really want to know?" "Yes!" "Well . . . last year, you helped me with a geometry proof. You probably don't remember. You do this thing where you . . ." Kath trailed off, looking a little shy. "What? What do I do?" "You chew on your lip when you do a difficult math problem," Kath said. "It's cute." Lily's face went red, and she laughed. "I'd better stop that in college, or no one with take me seriously.

That assumption—that labeling and sorting children based on gender doesn’t really matter as long as everyone is treated fairly—would hold true if children only paid attention to the more overt, obvious messages we adults send. If children only listened to our purposeful messages, parenting would be easy. Most (but not all) parents and teachers take great effort in treating their children fairly, regardless of gender. Parents don’t need to say to their daughters, “You probably won’t enjoy math” or say to their sons, “Real boys don’t play with dolls.” Most parents wouldn’t dream of saying these blatant stereotypes to their kids. But research has shown that when we label (and sort and color-code) by gender, children do notice. And it matters—children are learning whether you mean to be teaching them or not.

Math is a difficult thing to love,” he said. “It’s precise and unforgiving, it’s evasive and it will never love me back, but I don’t have much of a choice, do I? It’s the thing that I can do that other people can’t, or that other people lack the patience for. Are there worthier things, more rewarding things? Yes, probably. But I don’t know what they are, they never showed themselves to me. Only math did .

Further, the same Arguments which explode the Notion of Luck, may, on the other side, be useful in some Cases to establish a due comparison between Chance and Design: We may imagine Chance and Design to be, as it were, in Competition with each other, for the production of some sorts of Events, and many calculate what Probability there is, that those Events should be rather be owing to the one than to the other.

Luck is not some esoteric, godlike phenomenon. Luck is countable but undefinable. Luck easily can be explained by a number of factors acting in favour of a person. These factors' behaviour could be statistically proved, and the probability of such a result is possible. So, it is not an unexplainable event. Actually, the miracle would be if these events (luck) are not in presence in our life. So, make your luck!"

Probably nothing,” Tom ventured, wanting it to be true. Prophet didn’t argue, but they both decided that school was off-limits for a bit. Remy, of course, didn’t mind a damned bit. Mal offered to tutor him in the interim and that actually worked out quite well. Then again, for all Tom knew, math was a code for bomb building. When he mentioned that to Prophet, all he got in response was, “They’re both useful life skills.

Monstrous Sea Private Message 2:54 p.m. 28 - Oct -16 rainmaker: Hey, it’s Wallace. Please tell me I blew your mind again. You make the best face when your mind is being blown. MirkerLurker: Whoa that sounded dirty. rainmaker: Too much? MirkerLurker: Ummmmmmmmmm rainmaker: Too much. Noted. MONSTROUS SEA FORUMS USER PROFILE rainmaker * Fanfiction Moderator AGE: Not telling you LOCATION: NO INTERESTS: MS. Writing things.Campfires. Sweaters. Sleeping in. Dogs. Followers 1,350,199 | Following 54 | Posts 9,112 [Unique Works 144] UPDATES View earlier updates Oct 20 2016 The next chapter of the Auburn Blue fanfic will probably be a little late. Just started at the new school. So, that’s fun. Oct 21 2016 Thanks to @joojooboogee for my new avatar! #DallasRainerForever Oct 23 2016 If math homework were a real person, I’d be doing 25 to life. #Mathslaughter Oct 24 2016 There might actually be other MS fans at this school. THANK JESUS I’M SAVED. Oct 26 2016 Life is destroying me today. No time to write. Stupid math. #Mathslaughter Oct 27 2016 Definitely another MS fan at this school. Pros: Awesome; Not alone; Pretty girl. Cons: Pretty girl. #Fuuuuuuuuck Oct 28 2016 Heyyyy let’s not talk about the pretty girl anymore okay she’s probably looking at this.

Education is one of the Grand Christianson Obsessions. They’ve been whole years my mother’s kept us home for intensive private study. As a result of that, Paul will perform the first brain transplant, James will someday build a bridge across the Atlantic Ocean, Charlie – who is an actual musical genius – will probably end up writing the Great American Symphony, and I – I know a little bit about a lot of things. I can tell you the chemical composition of the stuff your stick in your hair; how long it would take you, at just under the speed of light, to get to Alpha Centauri – and how old your body would be when you finally got there; the middle name of the third president of the United States; the amount of the present budget deficit; the author of the Brothers Karamazov, and how many feet there are in a line of trochaic heptameter. The Little Girl Who Had to Know Why, Paul used to call me. But even my mother couldn’t reconcile me and math.

When one day Lagrange took out of his pocket a paper which he read at the Académe, and which contained a demonstration of the famous Postulatum of Euclid, relative to the theory of parallels. This demonstration rested on an obvious paralogism, which appeared as such to everybody; and probably Lagrange also recognised it such during his lecture. For, when he had finished, he put the paper back in his pocket, and spoke no more of it. A moment of universal silence followed, and one passed immediately to other concerns.

How did you even get in here?” I asked him. “Would you believe they leave the door open all night?” Gus asked. “Um, no,” I said. “As well you shouldn’t.” Gus smiled. “Anyway, I know it’s a bit self-aggrandizing.” “Hey, you’re stealing my eulogy,” Isaac said. “My first bit is about how you were a self-aggrandizing bastard.” I laughed. “Okay, okay,” Gus said. “At your leisure.” Isaac cleared his throat. “Augustus Waters was a self-aggrandizing bastard. But we forgive him. We forgive him not because he had a heart as figuratively good as his literal one sucked, or because he knew more about how to hold a cigarette than any nonsmoker in history, or because he got eighteen years when he should have gotten more.” “Seventeen,” Gus corrected. “I’m assuming you’ve got some time, you interrupting bastard. “I’m telling you,” Isaac continued, “Augustus Waters talked so much that he’d interrupt you at his own funeral. And he was pretentious: Sweet Jesus Christ, that kid never took a piss without pondering the abundant metaphorical resonances of human waste production. And he was vain: I do not believe I have ever met a more physically attractive person who was more acutely aware of his own physical attractiveness. “But I will say this: When the scientists of the future show up at my house with robot eyes and they tell me to try them on, I will tell the scientists to screw off, because I do not want to see a world without him.” I was kind of crying by then. “And then, having made my rhetorical point, I will put my robot eyes on, because I mean, with robot eyes you can probably see through girls’ shirts and stuff. Augustus, my friend, Godspeed.” Augustus nodded for a while, his lips pursed, and then gave Isaac a thumbs-up. After he’d recovered his composure, he added, “I would cut the bit about seeing through girls’ shirts.” Isaac was still clinging to the lectern. He started to cry. He pressed his forehead down to the podium and I watched his shoulders shake, and then finally, he said, “Goddamn it, Augustus, editing your own eulogy.” “Don’t swear in the Literal Heart of Jesus,” Gus said. “Goddamn it,” Isaac said again. He raised his head and swallowed. “Hazel, can I get a hand here?” I’d forgotten he couldn’t make his own way back to the circle. I got up, placed his hand on my arm, and walked him slowly back to the chair next to Gus where I’d been sitting. Then I walked up to the podium and unfolded the piece of paper on which I’d printed my eulogy. “My name is Hazel. Augustus Waters was the great star-crossed love of my life. Ours was an epic love story, and I won’t be able to get more than a sentence into it without disappearing into a puddle of tears. Gus knew. Gus knows. I will not tell you our love story, because—like all real love stories—it will die with us, as it should. I’d hoped that he’d be eulogizing me, because there’s no one I’d rather have…” I started crying. “Okay, how not to cry. How am I—okay. Okay.” I took a few breaths and went back to the page. “I can’t talk about our love story, so I will talk about math. I am not a mathematician, but I know this: There are infinite numbers between 0 and 1. There’s .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities. A writer we used to like taught us that. There are days, many of them, when I resent the size of my unbounded set.

How is she already asleep?” Sully whispers. “At home she stays up until like two a.m.” “She probably was tired,” Church whispers back. “What, from climbing a hill?” Church doesn’t respond. They get into their sleeping bags and whisper for half an hour about the outdoor soccer season about to start. I hadn’t even realized the indoor season was over—Mom and Dad just told me when I needed to take them to practice or pick them up. I didn’t know how they’d done. Were there any tournaments? Trophies? After a long stretch of silence, Sully says, “So did you really try out for the spring musical?” Church doesn’t respond for a second. “Yes. Why?” “Just wondering. Why didn’t you tell me?” “Because you would have made it about Macy Garrison.” “It—it’s not?” “No.” “Oh. But you’re not going to try out forchoir?” “Maybe.” “Why?” Just the smallest bit of mocking enters Sully’s tone. “Because I like it,” Church snaps back. “We don’t have to do all the same things. Try out for mathletes or something. You like math. You’d be good at it.” “Mathletes is for nerds.” “Sull, there’s something you should know.” “Don’t say it.” “You are a nerd.” “I’m not a nerd. Eliza’s a nerd.” “Actually, I think Eliza’s a geek. I’ve seen her grades. Compared to us, she’s horrible at school.” “You’re a nerd for knowing the difference.” “That’s fine.” Sully makes no sound, but I can feel him fuming in the darkness. I didn’t know Church could get under Sully’s skin so easily. I didn’t know Sully liked math. I didn’t know either of them were that good at school. I didn’t know Church already knew he was good at singing . . . or that he was interested in musical theater. I’ve been living with them their whole lives, but until right now, they’ve felt like strangers

Mr. Schu nodded, because he knew magic could happen in the library - all kinds of magic, not just book magic. And he was like the magician who helped it happen. Libraries and good librarians are wonder-filled like that. If you don't believe me, it's probably because you haven't discovered a magical library or librarian...yet. I hope you do, though. They can make a life-changing difference. You just don't always realize it at the time. Libraries can help you fall in love with science or travel or math or...imagination from so many people across so many times. Quite a wondrous thing, if you stop and think about it.

It's specifically this Z = 2^(Aleph0) that he couldn't prove. Ever. Despite years of unimaginable doodling. Whether it's what unhinged him or not is an unanswerable question, but it is true that his inability to prove the C.H. caused Cantor pain for the rest of his life; he considered it his great failure. This too, in hindsight, is sad, because professional mathematicians now know exactly why G. Cantor could neither prove nor disprove the C.H. The reasons are deep and important and go corrosively to the root of axiomatic set theory's formal Consistency, in rather the same way that K. Godel's Incompleteness proofs deracinate all math as a formal system. Once again, the issues here can be only sketched or synopsized (although this time Godel is directly involved, so the whole thing is probably fleshed out in the Great Discoveries Series' Godel booklet).

You’re really nice,” I slur. We’re waiting for the valet to bring Gavin’s truck around, and it feels like the fresh Colorado air has increased my alcohol level from drunk to trashed . . . and I still haven’t cracked open my wine.   “You’re pretty nice too.” He’s watching me closely, and I’m trying to watch him closely. His eyes are crinkled with amusement; mine are struggling to focus.   “I really wish you were an investment banker.”   Oh no. The loose lips part of the night has arrived.   “Besides my mom, you’re probably the only person in the world who does.”   “Because everyone else would miss their superstar quarterback in his super-hot pants throwing the ball every Sunday?” Sober me hates drunk me so hard right now.   “Because I’m terrible with numbers. I had three different tutors trying to get me to pass my math courses in college. And I’m not sure most of the fans focus on my pants, but I’m glad you do.” His body is shaking with laughter as he nudges me with his shoulder.

The core physics relies on a process known as quantum tunneling. Imagine a particle, an electron for instance, encountering a solid barrier, say a slab of steel ten feet think, that classical physics predicts it can't penetrate. A hallmark of quantum mechanics is that the rigid classical notion of "can't penetrate" often translates into the softer quantum declaration of "has a small but nonzero probability of penetrating." The reason is that the quantum jitters of a particle allow it, every so often, to suddenly materialize on the other side of an otherwise impervious barrier. The moment at which such quantum tunneling happens is random; the best we can do is predict the likelihood that it will take place during one interval or another. But the math says that if you wait long enough, penetration through just about any barrier will happen. And it does happen. If it didn't, the sun wouldn't shine: for hydrogen nuclei to get close enough to fuse, they must tunnel through the barrier created by the electromagnetic repulsion of their protons.

There are several different frameworks one could use to get a handle on the indeterminate vs. determinate question. The math version is calculus vs. statistics. In a determinate world, calculus dominates. You can calculate specific things precisely and deterministically. When you send a rocket to the moon, you have to calculate precisely where it is at all times. It’s not like some iterative startup where you launch the rocket and figure things out step by step. Do you make it to the moon? To Jupiter? Do you just get lost in space? There were lots of companies in the ’90s that had launch parties but no landing parties. “But the indeterminate future is somehow one in which probability and statistics are the dominant modality for making sense of the world. Bell curves and random walks define what the future is going to look like. The standard pedagogical argument is that high schools should get rid of calculus and replace it with statistics, which is really important and actually useful. There has been a powerful shift toward the idea that statistical ways of thinking are going to drive the future.” —PETER THIEL

And Schtitt, whose knowledge of formal math is probably about equivalent to that of a Taiwanese kindergartner, nevertheless seemed to know what Hopman and van der Meer and Bollettieri seemed not to know: that locating beauty and art and magic and improvement and keys to excellence and victory in the prolix flux of match play is not a fractal matter of reducing chaos to pattern. Seemed intuitively to sense that it was a matter not of reduction at all, but — perversely — of expansion, the aleatory flutter of uncontrolled, metastatic growth — each well-shot ball admitting of n possible responses, 2n possible responses to those responses, and on into what Incandenza would articulate to anyone who shared both his backgrounds as a Cantorian 35 continuum of infinities of possible move and response, Cantorian and beautiful because infoliating, contained, this diagnate infinity of infinities of choice and execution, mathematically uncontrolled but humanly contained, bounded by the talent and imagination of self and opponent, bent in on itself by the containing boundaries of skill and imagination that brought one player finally down, that kept both from winning, that made it, finally, a game, these boundaries of self.

How to live (forty pieces of advice I feel to be helpful but which I don’t always follow) 1. Appreciate happiness when it is there 2. Sip, don’t gulp. 3. Be gentle with yourself. Work less. Sleep more. 4. There is absolutely nothing in the past that you can change. That’s basic physics. 5. Beware of Tuesdays. And Octobers. 6. Kurt Vonnegut was right. “Reading and writing are the most nourishing forms of meditation anyone has so far found.” 7. Listen more than you talk. 8. Don’t feel guilty about being idle. More harm is probably done to the world through work than idleness. But perfect your idleness. Make it mindful. 9. Be aware that you are breathing. 10. Wherever you are, at any moment, try to find something beautiful. A face, a line out of a poem, the clouds out of a window, some graffiti, a wind farm. Beauty cleans the mind. 11. Hate is a pointless emotion to have inside you. It is like eating a scorpion to punish it for stinging you. 12. Go for a run. Then do some yoga. 13. Shower before noon. 14. Look at the sky. Remind yourself of the cosmos. Seek vastness at every opportunity, in order to see the smallness of yourself. 15. Be kind. 16. Understand that thoughts are thoughts. If they are unreasonable, reason with them, even if you have no reason left. You are the observer of your mind, not its victim. 17. Do not watch TV aimlessly. Do not go on social media aimlessly. Always be aware of what you are doing and why you are doing it. Don’t value TV less. Value it more. Then you will watch it less. Unchecked distractions will lead you to distraction. 18. Sit down. Lie down. Be still. Do nothing. Observe. Listen to your mind. Let it do what it does without judging it. Let it go, like Snow Queen in Frozen. 19. Don’t’ worry about things that probably won’t happen. 20. Look at trees. Be near trees. Plant trees. (Trees are great.) 21. Listen to that yoga instructor on YouTube, and “walk as if you are kissing the earth with your feet”. 22. Live. Love. Let go. The three Ls. 23. Alcohol maths. Wine multiplies itself by itself. The more you have, the more you are likely to have. And if it is hard to stop at one glass, it will be impossible at three. Addition is multiplication. 24. Beware of the gap. The gap between where you are and where you want to be. Simply thinking of the gap widens it. And you end up falling through. 25. Read a book without thinking about finishing it. Just read it. Enjoy every word, sentence, and paragraph. Don’t wish for it to end, or for it to never end. 26. No drug in the universe will make you feel better, at the deepest level, than being kind to other people. 27. Listen to what Hamlet – literature’s most famous depressive – told Rosencrantz and Guildenstern. “There is nothing either good or bad, but thinking makes it so.” 28. If someone loves you, let them. Believe in that love. Live for them, even when you feel there is no point. 29. You don’t need the world to understand you. It’s fine. Some people will never really understand things they haven’t experienced. Some will. Be grateful. 30. Jules Verne wrote of the “Living Infinite”. This is the world of love and emotion that is like a “sea”. If we can submerge ourselves in it, we find infinity in ourselves, and the space we need to survive. 31. Three in the morning is never the time to try and sort out your life. 32. Remember that there is nothing weird about you. You are just a human, and everything you do and feel is a natural thing, because we are natural animals. You are nature. You are a hominid ape. You are in the world and the world is in you. Everything connects. 33. Don’t believe in good or bad, or winning and losing, or victory and defeat, or ups and down. At your lowest and your highest, whether you are happy or despairing or calm or angry, there is a kernel of you that stays the same. That is the you that matters.

Marcus couldn’t believe it. Dead. A dead duck. OK, he'd been trying to hit it on the head with a piece of sandwich, but he tried to do all sorts of things, and none of them had ever happened before. He'd tried to get te highest score on the Stargazer machine in the kebab shop on Hornsey Road--nothing. He's tried to read Nicky’s thoughts by staring at the back of his head every maths lesson for a week--nothing. It really annoyed him that the only thing he'd ever achieved was something he hadn't really wanted to do that much in the first place. And anyway, since when did hitting a bird with a sandwich kill it? Kids must spend half their lives throwing things at the ducks in Regent's Park. How come he managed to pick a duck that pathetic? There must have been something wrong with it. It was probably about to die from a heart attack or something; it was just a coincidence. But if it was, nobody would believe him. If there were any witnesses, they'd only have seen the bread hit the duck right on the back of the head, and then seen it keel over. saw it die. They'd put two and two together and make five, and he'd be imprisoned for a crime he never committed. ... "What's that floating next to it?" Will asked. "Is that the bread you threw at it?" Marcus nodded unhappily. "That's not a sandwich, that's a bloody french loaf. No wonder it keeled over. That would've killed me.

Before I walked into the door, the room got shades darker as a cloud did a summersault in front of the sun. I turned my head up to the sky and saw Gauss in the glass smirking down at me. In that moment I was reminded of a story about Gauss. 
 When he was in the fifth grade, his teacher wanted some quiet, so he asked his class to add up all the numbers from 1-100. Thinking he had plenty of time to relax, he was shocked that within minutes Gauss had an answer. Gauss had cleverly noticed that the numbers 1 and 100 added up to 101, and 2 and 99 also added up to 101 and on down until you hit 50 and 51. So there are 50 pairs of 101, and a simple multiplication problem by Gauss left his teacher perplexed.
 The recollection of this story reminded me about my own fifth grade experience. Thor was the volunteer at my school for the “Math Superstar” program. After each assignment, stars of various colors signifying degrees of excellence were stuck on all the papers handed in. Like the Olympics, gold was the highest honor. 
 Wendy, the girl who sat next to me, was baffled that no matter how many wrong answers I got (usually all of them), I consistently had gold stars on my papers. She thought Thor was showing a personal bias towards me, but the truth is that I knew where he kept his boxes of stars, so I simply awarded myself what I thought I deserved. Hey, Gauss, how’s that for clever?

So the way to think about the problem is this. You and I and computers and bacteria and viruses and everything else material are made of molecules and atoms, which are themselves composed of particles like electrons and quarks. Schrodinger's equation works for electrons and quarks, and all evidence points to its working for things made of these constituents, regardless of the number of particles involved. This means that Schrodinger's equation should continue to apply during a measurement. After all, a measurement is just one collection of particles (the person, the equipment, the computer...) coming into contact with another (the particle or particles being measured). But if that's the case, if Schrodinger's math refuses to bow down, then Bohr is in trouble. Schrodinger's equation doesn't allow waves to collapse. An essential element of the Copenhagen approach would therefore be undermined. So the third question is this: If the reasoning just recounted is right and probability waves don't collapse, how do we pass from the range of possible outcomes that exist before a measurement to the single outcome the measurement reveals? Or to put it in more general terms, what happens to a probability wave during a measurement that allows a familiar, definite, unique reality to take hold? Everett pursued this question in his Princeton doctoral dissertation and came to an unforeseen conclusion.

Many of the really great, famous proofs in the history of math have been reduction proofs. Here's an example. It is Euclid's proof of Proposition 20 in Book IX of the Elements. Prop. 20 concerns the primes, which-as you probably remember from school-are those integers that can't be divided into smaller integers w/o remainder. Prop. 20 basically states that there is no largest prime number. (What this means of course is that the number of prime numbers is really infinite, but Euclid dances all around this; he sure never says 'infinite'.) Here is the proof. Assume that there is in fact a largest prime number. Call this number Pn. This means that the sequence of primes (2,3,5,7,11,...,Pn) is exhaustive and finite: (2,3,5,7,11,...,Pn) is all the primes there are. Now think of the number R, which we're defining as the number you get when you multiply all the primes up to Pn together and then add 1. R is obviously bigger than Pn. But is R prime? If it is, we have an immediate contradiction, because we already assumed that Pn was the largest possible prime. But if R isn't prime, what can it be divided by? It obviously can't be divided by any of the primes in the sequence (2,3,5,...,Pn), because dividing R by any of these will leave the remainder 1. But this sequence is all the primes there are, and the primes are ultimately the only numbers that a non-prime can be divided by. So if R isn't prime, and if none of the primes (2,3,5,...,Pn) can divide it, there must be some other prime that divides R. But this contradicts the assumption that (2,3,5,...,Pn) is exhaustive of all the prime numbers. Either way, we have a clear contradiction. And since the assumption that there's a largest prime entails a contradiction, modus tollens dictates that the assumption is necessarily false, which by LEM means that the denial of the assumption is necessarily true, meaning there is no largest prime. Q.E.D.

There are many things the Chinese do differently from Westerners. There’s the question of extra credit, for example. One time, Lulu came home and told me about a math test she’d just taken. She said she thought it had gone extremely well, which is why she didn’t feel the need to do the extra-credit problems. I was speechless for a second, uncomprehending. “Why not?” I asked. “Why didn’t you do them?” “I didn’t want to miss recess.” A fundamental tenet of being Chinese is that you always do all of the extra credit all of the time. “Why?” asked Lulu, when I explained this to her. For me this was like asking why I should breathe. “None of my friends do it,” Lulu added. “That’s not true,” I said. “I’m 100% sure that Amy and Junno did the extra credit.” Amy and Junno were the Asian kids in Lulu’s class. And I was right about them; Lulu admitted it. “But Rashad and Ian did the extra credit too, and they’re not Asian,” she added. “Aha! So many of your friends did do the extra credit! And I didn’t say only Asians do extra credit. Anyone with good parents knows you have to do the extra credit. I’m in shock, Lulu. What will the teacher think of you? You went to recess instead of doing extra credit?” I was almost in tears. “Extra credit is not extra. It’s just credit. It’s what separates the good students from the bad students." "Aww - recess is so fun," Lulu offered as her final sally. But after that, Lulu, like Sophia. always did the extra credit. Sometimes the girls got more points on extra credit than on the test itself - an absurdity that would never happen in China. Extra credit is one reason that Asian kids get such notoriously good grades in the United States. Rote drilling is another. Once Sophia came in second on a multiplication speed test, which her fifth grade teacher administered every Friday. She lost to a Korean boy named Yoon-seok. Over the next week, I made Sophia do twenty practice tests (of 100 problems each) every night, with me clocking her with a stopwatch. After that, she came in first every time. Poor Yoon-seok. He went back to Korea with his family, but probably not because of the speed test.

It’s relaxing to think that knowledge has no sell-by date: that once you have learned something, it stays fresh forever and you never have to learn it again. In the sciences like math and physics, and in the arts, that is often true. In those subjects, what we all learned at school (2 + 2 = 4) is probably still good. But in the social sciences, even the most basic knowledge goes off very quickly. As with milk or vegetables, you have to keep getting it fresh. Because everything changes.

options pushed me to study the math of probability.

Look at stocks as part ownership of a business. 2. Look at Mr. Market—volatile stock price fluctuations—as your friend rather than your enemy. View risk as the possibility of permanent loss of purchasing power, and uncertainty as the unpredictability regarding the degree of variability in the possible range of outcomes. 3. Remember the three most important words in investing: “margin of safety.” 4. Evaluate any news item or event only in terms of its impact on (a) future interest rates and (b) the intrinsic value of the business, which is the discounted value of the cash that can be taken out during its remaining life, adjusted for the uncertainty around receiving those cash flows. 5. Think in terms of opportunity costs when evaluating new ideas and keep a very high hurdle rate for incoming investments. Be unreasonable. When you look at a business and get a strong desire from within saying, “I wish I owned this business,” that is the kind of business in which you should be investing. A great investment idea doesn’t need hours to analyze. More often than not, it is love at first sight. 6. Think probabilistically rather than deterministically, because the future is never certain and it is really a set of branching probability streams. At the same time, avoid the risk of ruin, when making decisions, by focusing on consequences rather than just on raw probabilities in isolation. Some risks are just not worth taking, whatever the potential upside may be. 7. Never underestimate the power of incentives in any given situation. 8. When making decisions, involve both the left side of your brain (logic, analysis, and math) and the right side (intuition, creativity, and emotions). 9. Engage in visual thinking, which helps us to better understand complex information, organize our thoughts, and improve our ability to think and communicate. 10. Invert, always invert. You can avoid a lot of pain by visualizing your life after you have lost a lot of money trading or speculating using derivatives or leverage. If the visuals unnerve you, don’t do anything that could get you remotely close to reaching such a situation. 11. Vicariously learn from others throughout life. Embrace everlasting humility to succeed in this endeavor. 12. Embrace the power of long-term compounding. All the great things in life come from compound interest.

You cannot read mathematics the way you read a novel. If you zip through a page in less than an hour, you are probably going too fast.

You cannot read mathematics the way you read a novel. If you zip through a page in less than an hour, you are probably going too fast.

To miss the forest for the trees – that phrase is strange to me. How can you miss the forest when there’s one in every tree? –Robert Frost, probably

Probably you are discouraged by this solution because it seems impossible to invent it. The authors share your feeling.

That night when I went to bed, I laid there in the dark and pictured a clothesline full of somebody else's troubles. I knew for sure there were a lot of them I'd rather pluck off of that line than mine. I imagined what the other troubles might be. There would probably be toothaches and failed math tests. Lost cats and ugly hair. Cheating boyfriends and broken-down cars. But none of those could hold a candle to my troubles, weighing down that clothesline like a sack full of bricks.

The following rules are the fundamental differentiators to keep in mind throughout this book. 1. Retirement Is Worst-Case-Scenario Insurance. Retirement planning is like life insurance. It should be viewed as nothing more than a hedge against the absolute worst-case scenario: in this case, becoming physically incapable of working and needing a reservoir of capital to survive. Retirement as a goal or final redemption is flawed for at least three solid reasons: a. It is predicated on the assumption that you dislike what you are doing during the most physically capable years of your life. This is a nonstarter—nothing can justify that sacrifice. b. Most people will never be able to retire and maintain even a hotdogs-for-dinner standard of living. Even one million is chump change in a world where traditional retirement could span 30 years and inflation lowers your purchasing power 2–4% per year. The math doesn’t work.3 The golden years become lower-middle-class life revisited. That’s a bittersweet ending. c. If the math does work, it means that you are one ambitious, hardworking machine. If that’s the case, guess what? One week into retirement, you’ll be so damn bored that you’ll want to stick bicycle spokes in your eyes. You’ll probably opt to look for a new job or start another company. Kinda defeats the purpose of waiting, doesn’t it? I’m not saying don’t plan for the worst case—I have maxed out 401(k)s and IRAs I use primarily for tax purposes—but don’t mistake retirement for the goal.

First, there’s mathematics. Obviously, you’ve got to be able to handle numbers and quantities—basic arithmetic. And the great useful model, after compound interest, is the elementary math of permutations and combinations. That was taught in my day in the sophomore year in high school. I suppose by now, in great private schools, it’s probably down to the eighth grade or

she happens to be daydreaming about some Lovecraft meta that she read on Tumblr, and looking outside, in the moment of the city’s rebirth. The meta wasn’t so much interesting as funny, science-side Tumblr arguing with fantasy-side over the comical notion that non-Euclidean geometry could somehow be sinister, and concluding that Lovecraft was probably just scared of math.

Lucas would be home at any minute and would probably destroy something on the way. I went to my room and closed the door, then sat down on my bed and ate the cookies. The chocolate melted in my mouth, and I began to relax. At least if I was taking the following day off, I didn’t have to rush through all my homework. That idea quickly disappeared. I’d have to try double as hard on my homework to make sure each answer was correct. I didn’t want a repeat of the Math exam. Sighing, I opened my backpack, tossing my books beside me. I organized them by subject and started with my English assignment. Thoughts of Ali filtered through my head, but I was able to push them away for the time being. As the next day was Friday, I wouldn’t have to see her until the following week, and I wasn’t going to have annoying thoughts of her ruin my day off.

We intuitively know that the heart is the center of love and empathy, and studies are showing this to be true. In fact, empathy manifests in the electromagnetic field (EMF), which is generated by the heart in amounts greater than anywhere else in the body. The heart’s EMF emits fifty thousand femtoteslas (a measure of EMF), in contrast to the ten generated by the brain.37 Other research shows that when separated from the magnetic field, the heart’s electrical field is sixty times greater in amplitude than the brain’s field.38 Through this field, a person’s nervous system tunes in to and responds to the magnetic fields produced by the hearts of other people.39 The heart’s field is therefore one of the means by which a practitioner affects patients. This effect leads to the question, What do you want to share? To generate positive outcomes for a patient, a practitioner must hold positive feelings in his or her own heart. Not only does good will profit the client, but it also benefits the practitioner as a person. A set of studies by researcher Dr. Rollin McCraty of the HeartMath Institute in California, and described in his e-book, The Energetic Heart, helps explain the importance of positive energy.40 For decades, scientists have known that information is encoded in the nervous system in the time intervals between activities or in the pattern of electrical activity. Recent studies also reveal that information is captured in hormone pulses. Moreover, there is a hormone pulse that coincides with heart rhythms, which means that information is also shared in the interbeat intervals of the pressure and electromagnetic waves produced by the heart. Negative emotions such as anger, frustration, or anxiety disturb the heart rhythm. Positive emotions such as appreciation, love, or compassion produce coherent or functional patterns. Feelings, distributed throughout the body, produce chemical changes within the entire system. Do you want to be a healthy person? Be sincerely positive as often as you can. You thus “increase the probability of maintaining coherence and reducing stress, even during challenging situations.”41 What you as a practitioner believe will be shared—everywhere and with everyone you meet.

So I went down to register at De Anza and saw that the classes for chemistry, physics, and calculus were all full. What? I couldn’t believe it. Here I was—the star science and math student at my high school and all set to be an engineer—and the three most important courses I needed were locked out. It was horrible. I called the chemistry teacher on the phone, who said if I showed up I could probably get in, but I couldn’t shake this terrible feeling that my future was shutting down. I could see it shutting down right in front of me. I felt my whole academic life was going to be messed up right from the start. And it was right then that I changed my mind, and decided to see if it was still possible to go to Colorado.

Like that long talk about “mathematical Platonism.” We lay in bed on a Sunday morning because we had nowhere to be and talked about the belief that mathematical truths are discovered, not invented, because they already exist, which, if correct, means reality must extend beyond the physical world. We talked about the idea that math is a sense like sight and touch, and if that is true, well, then, was it also possible my mother possessed another sense that gave her the ability to access another reality? I wondered if my love of math and her love of the spiritual could therefore coalesce in a way far more beautiful and mysterious than my cynical belief that she used data and probability to make her predictions, no different from an actuarial table.

The evidence suggested that many American parents treated their children as if they were delicate flowers. In one Columbia University study, 85 percent of American parents surveyed said that they thought they needed to praise their children’s intelligence in order to assure them they were smart. However, the actual research on praise suggested the opposite was true. Praise that was vague, insincere, or excessive tended to discourage kids from working hard and trying new things. It had a toxic effect, the opposite of what parents intended. To work, praise had to be specific, authentic, and rare. Yet the same culture of self-esteem boosting extended to many U.S. classrooms. In the survey of exchange students conducted for this book, about half of U.S. and international students said that American math teachers were more likely to praise their work than math teachers abroad. (Fewer than 10 percent said that their international teachers were more likely to praise.) That finding was particularly ironic, given that American students scored below average for the developed world in math. It also suggested that whatever the intent of American teachers, their praise was probably not always specific, authentic, and rare.

She says Ariel is going to interview me after she's done and he's going to ask me how many golf balls can fit into a stretch limo, and the right answer is to make reasonable estimates on the spot, maybe say, "It's probably like 100 golf balls high by 60 golf balls wide by 1,000 golf balls long," and to look like I'm thinking really hard, and then just do the math in my head and give him the answer. I ask, "Out of curiosity, what would a wrong answer be?" She says, "Freaking out about the question.

Big data is not about trying to “teach” a computer to “think” like humans. Instead, it’s about applying math to huge quantities of data in order to infer probabilities: the

Win sidled up next to him. “Did you give them the full-wattage smile?” “I’d say a good eighty, ninety watts.” Win studied the young women before making a declaration: “Lesbians,” he said. “Must be.” “A lot of that going around, isn’t there?” Myron did the math in his head. He probably had fifteen to twenty years on them. When it comes to young girls, you just never want to feel it.

There was an older guy in the photos on the bookshelf who was probably Isaiah’s brother, who was most likely dead and that was no doubt the reason Isaiah was so messed up. His face was either blank as Dodson’s math assignment or his eyes were tight and his jaw hard-set like he was about to smack somebody.

Today’s equivalent is probably ‘get an engineering degree’, but it will not necessarily be as lucrative. A third of Americans who graduated in STEM subjects (science, technology, engineering and maths) are in jobs that do not require any such qualification.52 They must still pay off their student debts. Up and down America there are programmers working as office temps and even fast-food servers. In the age of artificial intelligence, more and more will drift into obsolescence. On the evidence so far, this latest technological revolution is different in its dynamics from earlier ones. In contrast to earlier disruptions, which affected particular sectors of the economy, the effects of today’s revolution are general-purpose. From janitors to surgeons, virtually no jobs will be immune. Whether you are training to be an airline pilot, a retail assistant, a lawyer or a financial trader, labour-saving technology is whittling down your numbers – in some cases drastically so. In 2000, financial services employed 150,000 people in New York. By 2013 that had dropped to 100,000. Over the same period, Wall Street’s profits have soared. Up to 70 per cent of all equity trades are now executed by algorithms.53 Or take social media. In 2006, Google bought YouTube for $1.65 billion. It had sixty-five employees, so the price amounted to $25 million per employee. In 2012 Facebook bought Instagram, which had thirteen employees, for $1 billion. That came to $77 million per employee. In 2014, it bought WhatsApp, with fifty-five employees, for $19 billion, at a staggering $345 million per employee.54 Such riches are little comfort to the thousands of engineers who cannot find work. Facebook’s data servers are now managed by Cyborg, a software program. It requires one human technician for every twenty thousand computers.

Jake,” I murmur. He opens his eyes. “Are you absolutely sure this is what you want? The baby, I mean.” “I’m sure.” His gaze drops to my stomach. “This baby will be made of everything I have loved my whole life.” “I’m gonna get fat,” I mumble. “No, you’re going to get even sexier.” Coming close again, he wraps his arms around me tightly, rubbing the tip of his nose against mine. “How could I not want something made up of Trudy Wethers’s DNA?” “Still Bennett.” I grin. “You haven’t made an honest woman of me yet.” “You ready to hop that plane to Vegas now?” “A shotgun wedding. My folks would be so proud.” I laugh. “What do you want to do about the wedding?” he asks. “Move it forward?” “That would give me a matter of weeks to plan it. Why don’t we just wait until after the baby is born?” I see him quickly do the math in his head. “We wouldn’t be able to get married July twenty-first. You okay with that?” “I’m going to have a mini-Jake soon. Of course I’m okay with that.” “Or a mini-Tru,” he says. Then his expression suddenly changes. “Fuck, a girl. We might have to lock her up, Tru.” I scrunch up my face. “Why?” “Because, if she looks anything like you, I’m one day going to be fighting off horny teenage boys left, right, and centre. I’ll probably end up in jail for beating one to death if I find him with his hands on my baby girl.” He shudders comically. I let out a laugh. “Let’s hope if we have a boy, he’s doesn’t grow up to be one of those horny teenagers…or God forbid, as horny as you are. Otherwise we’ll have some girl’s dad round here kicking his ass.” “Then I’ll end up in jail for beating the shit out of the dad—fuck, this is a no-win, sweetheart,” he groans, dropping his head back against the rest. “I’m doomed to a future behind bars.” Laughing softly, I say, “Don’t worry, baby, we’ll figure a way to keep you out of prison.” I kiss the tip of his nose, then open the door, ready to get out of the car and into the house to bed.

Why do things happen the way they do? Is there some kind of order in all this chaos that we just don't see, or is it all, as the mathematically minded people would like us to believe, just random coincidence? If you put one hundred apes in a room, they'll tell you, with one hundred typewriters, and given an infinite amount of time and bananas, one of them would eventually churn out the complete Oxford dictionary. It's all statistical math and probability. The odds of winning the lottery are greater than the odds of getting struck by lightning, but someone wins, don't they? And people get hit by lightning disturbingly more often that you would think. Their point is, eventually all things happen. No matter how philosophically unprejudiced you are, you can't argue with statistical probabilities. But you can certainly give the mathematicians some substantial cud to chew on, can't you? For instance, sure, everything may be eventual from a statistical point of view, but what happens to the formula if you plug in when a particular thing happens? The fortuitousness of the timing? Or combine a particular coincidence with other seemingly non-related coincidences that might have occurred within the same general time frame? We've all had it happen. It's one of our favorite phrases: "Why me? Why now?" Well, when you take the "when" into account, all kinds of very interesting and un-mathematical things begins to happen. The coincidence becomes too coincidental to be a coincidence.

You probably know what Sherlock Holmes had to say about inference, the most famous thing he ever said that wasn’t “Elementary!”: “It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth.” Doesn’t that sound cool, reasonable, indisputable? But it doesn’t tell the whole story. What Sherlock Holmes should have said was: “It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth, unless the truth is a hypothesis it didn’t occur to you to consider.” Less pithy, more correct. The

Here was my first lesson: This type of skill development is hard. When I got to the first tricky gap in the paper’s main proof argument, I faced immediate internal resistance. It was as if my mind realized the effort I was about to ask it to expend, and in response it unleashed a wave of neuronal protest, distant at first, but then as I persisted increasingly tremendous, crashing over my concentration with mounting intensity. To combat this resistance, I deployed two types of structure. The first type was time structure: “I am going to work on this for one hour,” I would tell myself. “I don’t care if I faint from the effort, or make no progress, for the next hour this is my whole world.” But of course I wouldn’t faint and eventually I would make progress. It took, on average, ten minutes for the waves of resistance to die down. Those ten minutes were always difficult, but knowing that my efforts had a time limit helped ensure that the difficulty was manageable. The second type of structure I deployed was information structure—a way of capturing the results of my hard focus in a useful form. I started by building a proof map that captured the dependencies between the different pieces of the proof. This was hard, but not too hard, and it got me warmed up in my efforts to understand the result. I then advanced from the maps to short self-administered quizzes that forced me to memorize the key definitions the proof used. Again, this was a relatively easy task, but it still took concentration, and the result was an understanding that was crucial for parsing the detailed math that came next. After these first two steps, emboldened by my initial successes in deploying hard focus, I moved on to the big guns: proof summaries. This is where I forced myself to take each lemma and walk through each step of its proofs—filling in missing steps. I would conclude by writing a detailed summary in my own words. This was staggeringly demanding, but the fact that I had already spent time on easier tasks in the paper built up enough momentum to help push me forward. I returned to this paper regularly over a period of two weeks. When I was done, I had probably experienced fifteen hours total of deliberate practice–style strain, but due to its intensity it felt like much more. Fortunately, this effort led to immediate benefits. Among other things, it allowed me to understand whole swaths of related work that had previously been mysterious. The researchers who wrote this paper had enjoyed a near monopoly on solving this style of problem—now I could join them.

In practice, Bacon’s method doesn’t bother scientists, or most reasonable people, because the chances of being wrong, while present, are usually in a practical sense very small. It is, for example, theoretically possible that chemical processes taking place in your body could cause you to spontaneously combust, but we don’t live our lives worrying about it because the probability is extremely small. That is why math and statistics have become such important parts of science: they quantify the relative probability that a conclusion is true or false.

There are C(n, k) ways to distribute a given event k-times in a string of n events. Hence, and now comes the grand moment, the probability that a certain event, occurring with the probability p, occurs k times within n trials is thus: P = C(n, k) · pk · (1-p)n-k And that! Ladies and gentleman, is the formula for the binomial distribution in all its glory. If

PageRank, the algorithm that gave rise to Google, is itself a Markov chain. Larry Page’s idea was that web pages with many incoming links are probably more important than pages with few, and links from important pages should themselves count for more. This sets up an infinite regress, but we can handle it with a Markov chain. Imagine a web surfer going from page to page by randomly following links: the states of this Markov chain are web pages instead of characters, making it a vastly larger problem, but the math is the same. A page’s score is then the fraction of the time the surfer spends on it, or equivalently, his probability of landing on the page after wandering around for a long time. Markov chains turn up everywhere and are one of the most intensively studied topics in mathematics, but they’re still a very limited kind of probabilistic model. We can go one step further with a model like this: The states form a Markov chain, as before, but we don’t get to see them; we have to infer them from the observations. This is called a hidden Markov model, or HMM for short. (Slightly misleading, because it’s the states that are hidden, not the model.)

At its core, big data is about predictions. Though it is described as part of the branch of computer science called artificial intelligence, and more specifically, an area called machine learning, this characterization is misleading. Big data is not about trying to “teach” a computer to “think” like humans. Instead, it’s about applying math to huge quantities of data in order to infer probabilities:

He thought about explaining math's beauty to her, the elegance of an equation, the simplicity within the complexity. The thrill of touching truth and knowing it as ancient and unassailable, as permanent and profound. But he was a math guy - he didn't have the words to do it - not in English. Probably not in Russian either.

It is written nowhere in the math of probability theory that one may have no fun.