Walter Pitts was used to being bullied. He’d been born into a tough family in Prohibition-era Detroit, where his father, a boiler-maker, had no trouble raising his fists to get his way. The neighborhood boys weren’t much better. One afternoon in 1935, they chased him through the streets until he ducked into the local library to hide. The library was familiar ground, where he had taught himself Greek, Latin, logic, and mathematics—better than home, where his father insisted he drop out of school and go to work. Outside, the world was messy. Inside, it all made sense. ... Not wanting to risk another run-in that night, Pitts stayed hidden until the library closed for the evening. Alone, he wandered through the stacks of books until he came across Principia Mathematica, a three-volume tome written by Bertrand Russell and Alfred Whitehead between 1910 and 1913, which attempted to reduce all of mathematics to pure logic. Pitts sat down and began to read. For three days he remained in the library until he had read each volume cover to cover—nearly 2,000 pages in all—and had identified several mistakes. Deciding that Bertrand Russell himself needed to know about these, the boy drafted a letter to Russell detailing the errors. Not only did Russell write back, he was so impressed that he invited Pitts to study with him as a graduate student at Cambridge University in England. Pitts couldn’t oblige him, though—he was only 12 years old. But three years later, when he heard that Russell would be visiting the University of Chicago, the 15-year-old ran away from home and headed for Illinois. He never saw his family again. ... Though they started at opposite ends of the socioeconomic spectrum, McCulloch and Pitts were destined to live, work, and die together. Along the way, they would create the first mechanistic theory of the mind, the first computational approach to neuroscience, the logical design of modern computers, and the pillars of artificial intelligence. But this is more than a story about a fruitful research collaboration. It is also about the bonds of friendship, the fragility of the mind, and the limits of logic’s ability to redeem a messy and imperfect world. ... “He was absolutely incomparable in the scholarship of chemistry, physics, of everything you could talk about history, botany, etc. When you asked him a question, you would get back a whole textbook … To him, the world was connected in a very complex and wonderful fashion.”
You wouldn’t see it in most classrooms, you wouldn’t know it by looking at slumping national test-score averages, but a cadre of American teenagers are reaching world-class heights in math—more of them, more regularly, than ever before. The phenomenon extends well beyond the handful of hopefuls for the Math Olympiad. The students are being produced by a new pedagogical ecosystem—almost entirely extracurricular—that has developed online and in the country’s rich coastal cities and tech meccas. In these places, accelerated students are learning more and learning faster than they were 10 years ago—tackling more-complex material than many people in the advanced-math community had thought possible. ... The change is palpable at the most competitive colleges. At a time when calls for a kind of academic disarmament have begun echoing through affluent communities around the nation, a faction of students are moving in exactly the opposite direction. ... lately, dozens of new math-enrichment camps with names like MathPath, AwesomeMath, MathILy, Idea Math, sparc, Math Zoom, and Epsilon Camp have popped up, opening the gates more widely to kids who have aptitude and enthusiasm for math, but aren’t necessarily prodigies. ... In New York City last fall, it was easier to get a ticket to the hit musical Hamilton than to enroll your child in certain math circles. Some circles in the 350-student New York Math Circle program run out of New York University filled up in about five hours. ... The pedagogical strategy at the heart of the classes is loosely referred to as “problem solving,” a pedestrian term that undersells just how different this approach to math can be. The problem-solving approach has long been a staple of math education in the countries of the former Soviet Union and at elite colleges such as MIT and Cal Tech. It works like this: Instructors present small clusters of students, usually grouped by ability, with a small number of open-ended, multifaceted situations that can be solved by using different approaches.
Unlike engineers and chemists, economists cannot point to concrete objects – cell phones, plastic – to justify the high valuation of their discipline. Nor, in the case of financial economics and macroeconomics, can they point to the predictive power of their theories. ... In the hypothetical worlds of rational markets, where much of economic theory is set, perhaps. But real-world history tells a different story, of mathematical models masquerading as science and a public eager to buy them, mistaking elegant equations for empirical accuracy. ... take the extraordinary success of Evangeline Adams, a turn-of-the-20th-century astrologer whose clients included the president of Prudential Insurance, two presidents of the New York Stock Exchange, the steel magnate Charles M Schwab, and the banker J P Morgan. To understand why titans of finance would consult Adams about the market, it is essential to recall that astrology used to be a technical discipline, requiring reams of astronomical data and mastery of specialised mathematical formulas. ... ‘An astrologer’ is, in fact, the Oxford English Dictionary’s second definition of ‘mathematician’. For centuries, mapping stars was the job of mathematicians, a job motivated and funded by the widespread belief that star-maps were good guides to earthly affairs. The best astrology required the best astronomy, and the best astronomy was done by mathematicians – exactly the kind of person whose authority might appeal to bankers and financiers. ... Romer believes that macroeconomics, plagued by mathiness, is failing to progress as a true science should, and compares debates among economists to those between 16th-century advocates of heliocentrism and geocentrism. Mathematics, he acknowledges, can help economists to clarify their thinking and reasoning. But the ubiquity of mathematical theory in economics also has serious downsides: it creates a high barrier to entry for those who want to participate in the professional dialogue, and makes checking someone’s work excessively laborious. Worst of all, it imbues economic theory with unearned empirical authority.
- Also: Janus - Zeno’s Paradox < 5min
We are exposed to possible events all the time: some of them probable, but many of them highly improbable. Each rare event—by itself—is unlikely. But by the mere act of living, we constantly draw cards out of decks. Because something must happen when a card is drawn, so to speak, the highly improbable does appear from time to time. ... It is the repetitiveness of the experiment that makes the improbable take place. The catch is that you can’t tell beforehand which of a very large set of improbable events will transpire. The fact that one out of many possible rare outcomes does happen should not surprise us because of the number of possibilities for extraordinary events to occur. The probabilities of these singly unlikely happenings compound statistically, so that the chance of at least one of many highly improbable events occurring becomes quite high. ... Persi Diaconis, professor of statistics at Stanford University, describes extremely unlikely coincidences as embodying the “blade of grass paradox.” If you were to stand in a meadow and reach down to touch a blade of grass, there are millions of grass blades that you might touch. But you will, in fact, touch one of them. The a priori fact that the blade you touch will be any particular one has an extremely tiny probability, but such an occurrence must take place if you are going to touch a blade of grass. ... The devil is in the details of how we interpret what we see in life. And here, psychology—more so than mathematics or logic—plays a key role. We tend to remember coincidences such as the one I experienced with my editor Scott and conveniently forget the thousands of times we may have met someone and had a conversation finding absolutely nothing in common.
Grosjean specializes in finding vulnerable games like the one in Shawnee. He uses his programming skills to divine the odds in various situations and then develops strategies for exploiting them. Only two questions seemed to temper his confidence in taking on this particular game. How long would they be allowed to play before being asked to leave? How much money would they be able to win? ... Many casino executives despise gamblers like Grosjean. They accuse him of cheating. Yet what he does is entirely legal. ... because regulated casino gambling now takes place in at least 40 states, casinos compete for customers in part by introducing new games, some of which turn out to be vulnerable. ... Common advantage-play techniques include “hole carding,” in which sharp-eyed players profit from careless dealers who unwittingly reveal tiny portions of the cards; “shuffle tracking,” or memorizing strings of cards in order to predict when specific cards will be dealt after they are next shuffled; and counting systems that monitor already dealt cards in order to estimate the value of those that remain in the deck. ... Teams of advantage players — which usually require one person to bet and another to spot dealers’ hole cards (those turned down and not supposed to be seen), track shuffles or count cards — have become so prevalent that they often find themselves in the same casino, at the same time, targeting the same game.
Scholze’s key innovation — a class of fractal structures he calls perfectoid spaces — is only a few years old, but it already has far-reaching ramifications in the field of arithmetic geometry, where number theory and geometry come together. ... his unnerving ability to see deep into the nature of mathematical phenomena. Unlike many mathematicians, he often starts not with a particular problem he wants to solve, but with some elusive concept that he wants to understand for its own sake. ... “I understood nothing, but it was really fascinating,” ... Scholze worked backward, figuring out what he needed to learn to make sense of the proof. ... Despite the complexity of perfectoid spaces, Scholze is known for the clarity of his talks and papers. ... Scholze makes a point of trying to explain his ideas at a level that even beginning graduate students can follow
Governments around the world have sought to incorporate elements of the “Singapore model” into their own approach to teaching maths and science. The latest is the UK, which earlier this month announced that half of England’s primary schools would adopt the style of maths teaching that is used in Singapore, with up to £41m in funding over four years to train teachers and provide new textbooks. But what is it about Singapore’s system that enables its children to outperform their international peers? And how easy will it be for other countries to import its success? ... A sense of being dwarfed by vast neighbours runs deep in the national psyche, inspiring both fear and pride. In a speech to trade union activists on May Day last year, prime minister Lee Hsien Loong told citizens: “To survive, you have to be exceptional.” ... Aiming to move away from simple rote-learning and to focus instead on teaching children how to problem solve, the textbooks the group produced were influenced by educational psychologists such as the American Jerome Bruner, who posited that people learn in three stages: by using real objects, then pictures, and then through symbols. That theory contributed to Singapore’s strong emphasis on modelling mathematical problems with visual aids; using coloured blocks to represent fractions or ratios, for example. ... The Singapore curriculum is more stripped down at primary level than in many western countries, covering fewer topics but doing so in far greater depth — a crucial factor in its effectiveness
Consider the most familiar random shape, the random walk, which shows up everywhere from the movement of financial asset prices to the path of particles in quantum physics. These walks are described as random because no knowledge of the path up to a given point can allow you to predict where it will go next. ... Beyond the one-dimensional random walk, there are many other kinds of random shapes. There are varieties of random paths, random two-dimensional surfaces, random growth models that approximate, for example, the way a lichen spreads on a rock. All of these shapes emerge naturally in the physical world, yet until recently they’ve existed beyond the boundaries of rigorous mathematical thought. Given a large collection of random paths or random two-dimensional shapes, mathematicians would have been at a loss to say much about what these random objects shared in common. ... have shown that these random shapes can be categorized into various classes, that these classes have distinct properties of their own, and that some kinds of random objects have surprisingly clear connections with other kinds of random objects. Their work forms the beginning of a unified theory of geometric randomness. ... “You take the most natural objects — trees, paths, surfaces — and you show they’re all related to each other,” Sheffield said. “And once you have these relationships, you can prove all sorts of new theorems you couldn’t prove before.” ... incoherent is not the same as incomprehensible. ... In practical terms, the results by Sheffield and Miller can be used to describe the random growth of real phenomena like snowflakes, mineral deposits, and dendrites in caves, but only when that growth takes place in the imagined world of random surfaces.
Learning math and then science as an adult gave me passage into the empowering world of engineering. But these hard-won, adult-age changes in my brain have also given me an insider’s perspective on the neuroplasticity that underlies adult learning. ... In the current educational climate, memorization and repetition in the STEM disciplines (as opposed to in the study of language or music), are often seen as demeaning and a waste of time for students and teachers alike. Many teachers have long been taught that conceptual understanding in STEM trumps everything else. And indeed, it’s easier for teachers to induce students to discuss a mathematical subject (which, if done properly, can do much to help promote understanding) than it is for that teacher to tediously grade math homework. What this all means is that, despite the fact that procedural skills and fluency, along with application, are supposed to be given equal emphasis with conceptual understanding, all too often it doesn’t happen. Imparting a conceptual understanding reigns supreme—especially during precious class time. ... The problem with focusing relentlessly on understanding is that math and science students can often grasp essentials of an important idea, but this understanding can quickly slip away without consolidation through practice and repetition. Worse, students often believe they understand something when, in fact, they don’t. ... Chunking was originally conceptualized in the groundbreaking work of Herbert Simon in his analysis of chess—chunks were envisioned as the varying neural counterparts of different chess patterns. Gradually, neuroscientists came to realize that experts such as chess grand masters are experts because they have stored thousands of chunks of knowledge about their area of expertise in their long-term memory. ... As studies of chess masters, emergency room physicians, and fighter pilots have shown, in times of critical stress, conscious analysis of a situation is replaced by quick, subconscious processing as these experts rapidly draw on their deeply ingrained repertoire of neural subroutines—chunks. ... Understanding doesn’t build fluency; instead, fluency builds understanding.
It was in an earlier work, 1759’s The Theory of Moral Sentiments, that Smith put his finger on the social and psychological impulses that push people to accumulate objects and gadgets. People, he observed, were stuffing their pockets with “little conveniences,” and then buying coats with more pockets to carry even more. By themselves, tweezer cases, elaborate snuff boxes, and other “baubles” might not have much use. But, Smith pointed out, what mattered was that people looked at them as “means of happiness." It was in people’s imagination that these objects became part of a harmonious system and made the pleasures of wealth “grand and beautiful and noble." ... This moral assessment was a giant step towards a more sophisticated understanding of consumption, for it challenged the dominant negative mindset that went back to the ancients. ... Rather than being passive, the consumer is now celebrated for actively adding value and meaning to media and products. ... there have been many prophecies and headlines that predict “peak stuff” and the end of consumerism. ... Such forecasts sound nice but they fail to stand up to the evidence. After all, a lot of consumption in the past was also driven by experiences, such as the delights of pleasure gardens, bazaars, and amusement parks. In the world economy today, services might be growing faster than goods, but that does not mean the number of containers is declining—far from it.
As the longest-running longitudinal survey of intellectually talented children, SMPY has for 45 years tracked the careers and accomplishments of some 5,000 individuals, many of whom have gone on to become high-achieving scientists. The study's ever growing data set has generated more than 400 papers and several books and provided key insights into how to spot and develop talent in science, technology, engineering, mathematics (STEM), and beyond. ... the work to identify and support academically talented students has raised troubling questions about the risks of labeling children and the shortfalls of talent searches and standardized tests as a means of identifying high-potential students, especially in poor and rural districts.
Statistics were designed to give an understanding of a population in its entirety, rather than simply to pinpoint strategically valuable sources of power and wealth. In the early days, this didn’t always involve producing numbers. In Germany, for example (from where we get the term Statistik) the challenge was to map disparate customs, institutions and laws across an empire of hundreds of micro-states. What characterised this knowledge as statistical was its holistic nature: it aimed to produce a picture of the nation as a whole. Statistics would do for populations what cartography did for territory. ... the aspiration to depict a society in its entirety, and to do so in an objective fashion, has meant that various progressive ideals have been attached to statistics. The image of statistics as a dispassionate science of society is only one part of the story. The other part is about how powerful political ideals became invested in these techniques: ideals of “evidence-based policy”, rationality, progress and nationhood grounded in facts, rather than in romanticised stories.
A mathematical prodigy, he worked out how to “beat the dealer” at blackjack while a postdoctoral student at MIT. After he published a book in 1962 revealing how to count cards, he became so famous that casinos banned him from playing — he says one even resorted to drugging him. Many changed their rules to thwart people using his counting system. ... Next came an attempt to beat roulette, using a contraption tied to his foot that is now described as the world’s first wearable computer; after that, an expedition into Wall Street that netted hundreds of millions of dollars. ... Thorp’s then revolutionary use of mathematics, options-pricing and computers gave him a huge advantage. ... “Adam Smith’s market is a whole lot different from our markets. He imagined a market with lots of buyers and sellers of things, nobody had market dominance or could impose things on the market, and there was a lot of competition. The market we have now is nothing like that. The players are so big that they control the levers of financial policy.” ... “One of the things that’s served me very well in life is having an extraordinary bullsh*t detector.”
Statcheck had read some 50,000 published psychology papers and checked the maths behind every statistical result it encountered. In the space of 24 hours, virtually every academic active in the field in the past two decades had received an email from the program, informing them that their work had been reviewed. Nothing like this had ever been seen before: a massive, open, retroactive evaluation of scientific literature, conducted entirely by computer. ... Statcheck’s method was relatively simple, more like the mathematical equivalent of a spellchecker than a thoughtful review, but some scientists saw it as a new form of scrutiny and suspicion, portending a future in which the objective authority of peer review would be undermined by unaccountable and uncredentialed critics. ... When it comes to fraud – or in the more neutral terms he prefers, “scientific misconduct” ... Despite its professed commitment to self-correction, science is a discipline that relies mainly on a culture of mutual trust and good faith to stay clean. Talking about its faults can feel like a kind of heresy. ... Even in the more mundane business of day-to-day research, scientists are constantly building on past work, relying on its solidity to underpin their own theories. If misconduct really is as widespread as Hartgerink and Van Assen think, then false results are strewn across scientific literature, like unexploded mines that threaten any new structure built over them.
Marion Tinsley—math professor, minister, and the best checkers player in the world—sat across a game board from a computer, dying. ... Tinsley had been the world’s best for 40 years, a time during which he'd lost a handful of games to humans, but never a match. It's possible no single person had ever dominated a competitive pursuit the way Tinsley dominated checkers. But this was a different sort of competition, the Man-Machine World Championship. ... His opponent was Chinook, a checkers-playing program programmed by Jonathan Schaeffer, a round, frizzy-haired professor from the University of Alberta, who operated the machine. Through obsessive work, Chinook had become very good. It hadn't lost a game in its last 125—and since they’d come close to defeating Tinsley in 1992, Schaeffer’s team had spent thousands of hours perfecting his machine. ... The two men were slated to play 30 matches over the next two weeks. The year was 1994, before Garry Kasparov and Deep Blue or Lee Sedol and AlphaGo. ... With Tinsley gone, the only way to prove that Chinook could have beaten the man was to beat the game itself. The results would be published July 19, 2007, in Science with the headline: Checkers Is Solved. ... At the highest levels, checkers is a game of mental attrition. Most games are draws. In serious matches, players don’t begin with the standard initial starting position. Instead, a three-move opening is drawn from a stack of approved beginnings, which give some tiny advantage to one or the other player. They play that out, then switch colors. The primary way to lose is to make a mistake that your opponent can jump on.