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.

Statement: A girl and a boy jump into a river. The boy swims over to the girl and says, "God, it's cold." Question: What's the probability they will kiss?

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.

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.

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.

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.

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.

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.

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’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.

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.

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.

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.)

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.

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.

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.

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.

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.

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.

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.

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.

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.

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?

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.

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.

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:

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.)

Daimon had been out of prison for about six years. Do the math—two days per weekend, times fifty-two weekends per year, times six years—the numbers are staggering. I think a person could conservatively say Daimon had gotten away with committing three hundred burglaries. The actual number will never be known but is probably closer to six hundred. “Oh,

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.

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.

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.

Broadly speaking, it seems that teachers help you know and understand and coaches help you practise and get better. Returning to the notion of how we are useful to our students, which do your students need most from you? The following questionnaire can help you think about this further.   Your turn 2 Underlying philosophies Grade each statement from 1 (don’t agree) to 10 (agree completely). Make a note of your answers, if you wish, to discuss with a colleague. a) Learning a language means learning words and rules. b) Learning a language means repeatedly using it. c) People learn best by noticing and working things out. d) They learn best by communicating with other people. e) Mistakes show that students have not understood the grammar properly. f) They show that they have not practised enough. g) Speaking is a conscious, cognitive process. h) Speaking is an automatic habit.   Perhaps all of these statements are true for you. Still, you may have found that you favour some more than others. Notice how a), c), e) and g) promote the idea of language as knowledge to learn, a bit like maths or music theory, whereas b), d), f) and h) reflect the side of learning that involves practising a skill, more like tennis or playing the piano. If you think that the skill side of language learning is particularly important, you will probably feel comfortable thinking of yourself as a language coach, someone whose main job is to get students practising and improving. If you see language as knowledge, the question to ask yourself is how your students can best acquire that knowledge, from you teaching it to them or from other sources of reference and input?

Autistic people are individuals. We are not all maths geniuses, we don’t all like trains. I am hopeless with technology and much prefer painting. There is no ‘typical Autistic.’ But I think we probably all like being respected and validated

I found I missed the compelling riddle-like quality of especially tricky math problems, that tight flourish of logic unfolding step by step. This is probably the same quality that also attracts me to particularly tricky poems. In both cases, a sense of wonder animates the premise: how can these constructed symbols mean something true about the world?

In discussing California's "Whole Math" (as it is called) with one such leading critic (math professor Abby Thompson of the University of California at Davis), I once suggested that the program seemed designed to brake the progress of bright kids. Thompson replied that I underestimated its anti-elitist fervor: "I don't think this is quite correct. The idea [espoused by 'Whole' Math advocates] is more precisely that there are no smart kids, that all children are equally talented at math, and that any differences that appear are due to societal failure, and probably racism and sexism to boot."-'(

Probably the greatest intellectual accomplishment in adolescence is the ability to think hypothetically, to engage in “if/ then” thinking. “If I blow off my math test, then I may fail the course and have to go to summer school.” All of a sudden, blowing off the math test is about a whole set of consequences.

they were between the ages of sixty and eight-thousand months. Whatever. Math is stupid. I'm assuming Nosferatu is somewhere around that age, and since these are clearly his offspring, my math is probably right

Skin in the game can make boring things less boring. When you have skin in the game, dull things like checking the safety of the aircraft because you may be forced to be a passenger in it cease to be boring. If you are an investor in a company, doing ultra-boring things like reading the footnotes of a financial statement (where the real information is to be found) becomes, well, almost not boring. But there is an even more vital dimension. Many addicts who normally have a dull intellect and the mental nimbleness of a cauliflower—or a foreign policy expert—are capable of the most ingenious tricks to procure their drugs. When they undergo rehab, they are often told that should they spend half the mental energy trying to make money as they did procuring drugs, they are guaranteed to become millionaires. But, to no avail. Without the addiction, their miraculous powers go away. It was like a magical potion that gave remarkable powers to those seeking it, but not those drinking it. A confession. When I don’t have skin in the game, I am usually dumb. My knowledge of technical matters, such as risk and probability, did not initially come from books. It did not come from lofty philosophizing and scientific hunger. It did not even come from curiosity. It came from the thrills and hormonal flush one gets while taking risks in the markets. I never thought mathematics was something interesting to me until, when I was at Wharton, a friend told me about the financial options I described earlier (and their generalization, complex derivatives). I immediately decided to make a career in them. It was a combination of financial trading and complicated probability. The field was new and uncharted. I knew in my guts there were mistakes in the theories that used the conventional bell curve and ignored the impact of the tails (extreme events). I knew in my guts that academics had not the slightest clue about the risks. So, to find errors in the estimation of these probabilistic securities, I had to study probability, which mysteriously and instantly became fun, even gripping. When there was risk on the line, suddenly a second brain in me manifested itself, and the probabilities of intricate sequences became suddenly effortless to analyze and map. When there is fire, you will run faster than in any competition. When you ski downhill some movements become effortless. Then I became dumb again when there was no real action. Furthermore, as traders the mathematics we used fit our problem like a glove, unlike academics with a theory looking for some application—in some cases we had to invent models out of thin air and could not afford the wrong equations. Applying math to practical problems was another business altogether; it meant a deep understanding of the problem before writing the equations.

Day Four: A Fellow Mystery Lover Dear Diary, Today I went to school, and nothing extremely exciting happened. Things went along pretty normally, so I usually wouldn’t write about it. I mean, I could glue my math homework in here if I really wanted to, but that would hardly make for a thrilling diary entry. No, I’m writing today because I think I made a new friend. I don’t know if it was clear up until now, but I don’t really have many friends. I mean, my mom is probably my best friend. I talk to some people around class and at recess and stuff, but for the most part, I like to keep to myself. Sure, it’s fun to understand people from the gossip that my mom brings home, but for some reason, I never really got the hang of making friends and getting to know people on a personal level. Today something changed. While I was at lunch, I had my tray of food in front of me, and a book in front of my nose. This was a pretty normal thing for me to do, especially on meatloaf surprise day. The surprise was that no one knew what kind of meat was in the loaf. I picked at it with my fork for a little while, but once it started to move I decided that I was done. I started to read my book, but that’s when someone sat across from me.

Can I ask you somethin’?” Wes asked his brother. Hunter’s response was immediate. “Anything. Well…except probably math questions. Or anything about history or naming the bones in the body. Or anything about constellations or what my favorite book is. Or—” “Shut it,” Wes demanded. “I don’t need a bunch of nerd answers. I need to know…” He swallowed hard and cleared his throat, tried again. “I need to know how to shop for a woman. What do those she-critters need?” “What do you mean, what do they need? You just feed ’em and hug ’em sometimes and tell them they’re pretty when they’re on their periods.

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

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

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.

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.

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

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.

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.

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.

In Zadeh’s fuzzy logic, we can not only estimate degrees of probability a la Korzybski but also degrees of presence or remaining presence. In other words, where Aristotle only lets us say “The U.S. Senate consists of imbeciles” or “The U.S. Senate does not consist of imbeciles,” and von Neumann and Korzybski allow us to estimate the probability that the Senate consists of imbeciles, Zadeh’s math allows us, if we know the facts, to state precisely how many imbeciles we will find present among the 100 persons in the Senate . . . 1, 10, 25, 53, 90 or whatever . . .

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.

The social psychologist Claude Steele demonstrated the power of what he calls “stereotype threat” in the U.S. context: Women do better on math tests when they are explicitly told that the stereotype that women are worse in math does not apply to this particular test; African Americans do worse on tests if they have to start by indicating their race on the cover sheet.33 Following Steele’s work, two researchers from the World Bank had lower-caste children in the Indian state of Uttar Pradesh compete against high-caste children in solving mazes.34 They found that the low-caste children compete well against the high-caste children as long as caste is not salient, but once low-caste children are reminded that they are low castes competing with high-caste children (by the simple contrivance of asking them their full names before the game starts), they do much worse. The authors argue that this may be driven in part by a fear of not being evaluated fairly by the obviously elite organizers of the game, but it could just as well be the internalization of the stereotype. A child who expects to find school difficult will probably blame herself and not her teachers when she can’t understand what is being taught, and may end up deciding she’s not cut out for school—“stupid,” like most of her ilk—and give up on education altogether, daydreaming in class or, like Shantarama’s children, just refusing to go

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.

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

Like most of the giants who revolutionize math or science, Cantor was 100% a man of his time and place, and his accomplishments were the usual conjunction of extraordinary personal brilliance and courage* and just the right context of general problems and conditions that, in hindsight, tend to make intellectual advances seem inevitable and their authors almost incidental. *Naturally, the significant results that legitimize a mathematical theory take time to derive, and then even more time to be fully accepted, and of course throughout this time the Insanity-v.-Genius question remains undecided, probably even for the mathematician himself, so that he’s developing his theory and cooking his proofs under conditions of enormous personal stress and doubt, and sometimes isn’t even vindicated in his own lifetime, etc.

Nevertheless, here’s the point for even a nontechnical reader: the Bitcoin blockchain gives a history that’s hard to falsify. Unless there’s an advance in quantum computing, a breakthrough in pure math, a heretofore unseen bug in the code, or a highly expensive 51% attack that probably only China could muster, it is essentially infeasible to rewrite the history of the Bitcoin blockchain — or anything written to it. And even if such an event does happen, it wouldn’t be an instantaneous burning of Bitcoin’s Library of Alexandria. The hash function could be replaced with a quantum-safe version, or another chain robust to said attack could take Bitcoin’s place, and back up the ledger of all historical Bitcoin transactions to a new protocol.

Haese is a money grubbing goondah trying to exploit teachers by making them buy an online subscription every 12 months. Boycott Haese!!!

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.