How Not to Be Wrong Quotes

“if we now feel comfortable rejecting the conclusions of the Witztum study, what does that say about the reliability of our standard statistical tests?”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“impossible to walk across the street,”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“Math is a science of not being wrong about things,”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“Ділення одного числа на інше - це просто обчислення; з'ясування того, що на що ділити, - це математика.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“A real-world problem is something like “Has the recession and its aftermath been especially bad for women in the workforce, and if so, to what extent is this the result of Obama administration policies?”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“There is real danger that, by strengthening our abilities to analyze some questions mathematically, we acquire a general confidence in our beliefs, which extends unjustifiably to those things we’re still wrong about. We become like those pious people who, over time, accumulate a sense of their own virtuousness so powerful as to make them believe the bad things they do are virtuous too. I’ll do my best to resist that temptation. But watch me carefully.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“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”
― Jordan Ellenberg, How Not To Be Wrong: The Hidden Maths of Everyday

“Every mathematician creates new things, some big, some small. All mathematical writing is creative writing. And the entities we can create mathematically are subject to no physical limits; they can be finite or infinite, they can be realizable in our observable universe or not. This sometimes leads outsiders to think of mathematicians as voyagers in a psychedelic realm of dangerous mental fire, staring straight at visions that would drive lesser beings mad, sometimes indeed being driven mad themselves.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“....when you're working hard on a theorem you should try to prove it by day and disprove it by night.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“But it didn't really happen in the space of a footstep, Poincare explains. That moment of inspiration is the product of weeks of work, both conscious and unconscious, which somehow prepare the mind to make the necessary connection of ideas. Sitting around waiting for inspiration leads to failure, no matter how much of a whiz kid you are.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, btu I felt a perfect certainty. On my return to Caen, for conscience's sake I verified the result at my leisure.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“The cult of the genius also tends to undervalue hard work. When I was starting out, I thought "hardworking" was a kind of veiled insult-something to say about a student when you can't honestly say they're smart. But the ability to work hard-to keep one's whole attention and energy focused on a problem, systematically turning it over and over and pushing at everything that looks like a crack, despite the lack of outward signs of progress-is not a skill everybody has. Psychologists nowadays call it "grit," and it's impossible to do math without it. It's easy to lose sight of the importance of work, because mathematical inspiration, when it finally does come, can feel effortless and instant.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“Formalism has an austere elegance. It appeals to people like G.H. Hardy, Antonin Scalia, and me, who relish that feeling of a nice rigid theory shut tight against contradiction. But it's not easy to hold to principles like this consistently, and it's not clear it's even wise. Even Justice Scalia has occasionally conceded that when the literal words of the law seem to require an absurd judgment, the literal words have to be set aside in favor of a reasonable guess as to what Congress must have meant. In just the same way, no scientist really wants to be bound strictly by the rules of significance, no matter what they say their principles are. When you run two experiments, one testing a clinical treatment that seems theoretically promising and the other testing whether dead salmon respond emotionally to romantic photos, and both experiments succeed with p-values of .03, you don't really want to treat the two hypotheses the same. You want to approach absurd conclusions with an extra coat of skepticism, rules be damned.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“And this is Pearson's formula, in geometric language. The correlation between the two variables is determined by the angle between the two vectors. If you want to get all trigonometric about it, the correlation is the cosine of the angle. It doesn't matter if you remember what cosine means; you just need to know that the cosine of an angle is 1 when the angle is 0 (i.e. when the two vectors are pointing in the same direction) and -1 when the angle is 180 degrees (vectors pointing in opposite directions). Two variables are positively correlated when the corresponding vectors are separated by an acute angle-that is, an angle smaller than 90 degrees- and negatively correlated when the angle between the vectors is larger than 90 degrees, or obtuse. It makes sense: vectors at an acute angle to one another are, in some loose sense, "Pointed in the same direction," while vectors that form an obtuse angle seem to be working at cross purposes.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“Why study geometry that isn't realized in the universe?

One answer comes from the study of data, currently in extreme vogue. Remember the digital photo from the four-megapixel camera: it's described by 4 million numbers, one for each pixel. (And that's before we take color into account!) So that image is a 4-million-dimensional vecotr; or if you like, a point in 4-million-dimensional space. And an image that changes with time is represented by a point that's moving around in a 4-million dimensional space, which traces out a curve in 4-million dimensional space, and before you know it you're doing 4-million-dimensional calculus, and then the fun can really start.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“That's the dirty little secret of advanced geometry. It may sound impressive that we can do geometry in ten dimensions (or a hundred, or a million...), but the mental pictures we keep in our mind are two-or at most three-dimensional. That's all our brains can handle. Fortunately, this impoverished vision is usually enough.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“What Secrist's findings really show is that businesses are much more like the cities in Wisconsin. Superior management and business insight play a role, but so does plain luck, in equal measure.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“A photograph, which used to be a pattern of pigment on a sheet of chemically coated paper, is not a string of numbers, each one representing the brightness and color of a pixel. An image captured on a 4-megapixel camera is a list of 4 million numbers-no small commitment of memory for the device shooting the picture. But these numbers are highly correlated with each other. If one pixel is bright green, the next one over likely to be as well. The actual information contained in the image is much less than 4 million numbers' worth-and it's precisely this fact that makes it possible to have compression, the critical mathematical technology that allows images, videos, music, and text to be stored in much smaller spaces than you'd think. The presence of correlation makes compression possible; actually doing it involves much more modern ideas, like the theory of wavelets developed in the 1970s and 80s by Jean Morlet, Stephane Mallat, Yves Meyer, Ingrid Daubechies, and others; and the rapidly developing area of compressed sensing, which started with a 2005 paper by Emmanuel Candes, Justin Romberg, and Terry Tao, and has quickly become its own active subfield of applied math.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“In modern terms we would say that the more strongly correlated the measurements, the less information, in Shannon's precise sense, a Bertillon card conveys.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“My colleague Michael Harris, a distinguished number theorist at the Institut de Mathematiques de Jussieu in Paris, has a theory that three of Thomas Pynchon's major novels are governed by the three conic sections: Gravity's Rainbow is about paraboloas (all those rockets, launching, dropping!), Mason & Dixon about ellipses, and Against the Day about hyperbolas. This seems as good to me as any other organizing theory of these novels I've encountered; certainly Pynchon, a former physics major who likes to drop references to Mobius strips and the quaternions in his novels, knows very well what the conic sections are.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“John Leech, in the 1960s, used one of Golay's codes to build an incredibly dense packing of twenty-four dimensional spheres, in a configuration now known as the Leech lattice. It's a crowded place, the Leech lattice, where each of the twenty-four-dimensional spheres touches 196,560 of its neighbors. We still don't know whether it's the tightest possible twenty-four-dimensional packing, but in 2003, Henry Cohn and Abhinav Kumar proved that if a denser lattice exists, it beats Leech by a factor of at most

1.00000000000000000000000000000165.

In other words: close enough”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“Money must not be estimated by its numerical quantity: if the metal, that is merely the sign of wealth, was wealth itself, that is, if the happiness or the benefits that result from wealth were proportional to the quantity of money, men would have reason to estimate it numerically and by its quantity, but it is barely necessary that the benefits that one derives from money are in just proportions with its quantity; a rich man of one hundred thousand ecus income is not ten times happier than the man of only ten thousand ecus; there is more than that what money is, as soon as one passes certain limits it has almost no real value, and cannot increase the well-being of its possessor; a man that discovered a mountain of gold would not be richer than the one that found only one cubic fathom.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“When we say that there's a 5% chance that RED is true, we are making a statement not about the global distribution of biased roulette wheels (how could we know?) but rather about our own mental state. Five percent is the degree to which we believe that a roulette wheel we encounter is weighed toward the red.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“Zhang's success, along with related work of other contemporary big shots like Ben Green and Terry Tao, points to a prospect even more exciting than any individual result about primes: that we might, in the end, be on our way to developing a richer theory of randomness. Say, a way of specifying precisely what we mean when we say that numbers act as if randomly scattered with no governing structure, despite arising from completely deterministic processes. How wonderfully paradoxical: what helps us break down the final mysteries about prime numbers may be new mathematical ideas that structure the concept of structurelessness itself.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“Inference is a hard thing, maybe the hardest thing. From the shape of the clouds and the way they move we struggle to go backward, to solve for x, the system that made them.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“Frederick Mosteller, who would later found Harvard’s statistics department, was there. So was Leonard Jimmie Savage, the pioneer of decision theory and great advocate of the field that came to be called Bayesian statistics.* Norbert Wiener, the MIT mathematician and the creator of cybernetics, dropped by from time to time. This was a group where Milton Friedman, the future Nobelist in economics, was often the”
― Jordan Ellenberg, How Not To Be Wrong: The Hidden Maths of Everyday

“(Revised Sherlock Holmes quote) 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.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“Math is a science of not being wrong about things, its techniques and habits hammered out by centuries of hard work and argument.”
― Jordan Ellenberg, How Not To Be Wrong: The Hidden Maths of Everyday

“A player who made a layup was no more likely to shoot from distance than a player who just missed a layup. Layups are easy and shouldn’t give the player a strong sense of being hot. But a player is much more likely to try a long shot after a three-point basket than after a three-point miss. In other words, the hot hand might “cancel itself out”—players, believing themselves to be hot, get overconfident and take shots they shouldn’t. The nature of the analogous phenomenon in stock investment is left as an exercise for the reader.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

“And you see, both of us were right, though nothing Has somehow come to nothing; the avatars Of our conforming to the rules and living Around the home have made—well, in a sense, “good citizens” of us, Brushing the teeth and all that, and learning to accept The charity of the hard moments as they are doled out, For this is action, this not being sure, this careless Preparing, sowing the seeds crooked in the furrow, Making ready to forget, and always coming back To the mooring of starting out, that day so long ago.”
― Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking