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The Drunkard's Walk: How Randomness Rules Our Lives by Leonard Mlodinow
Albert Einstein, Alfred Russel Wallace, Antoine Gombaud: Chevalier de Méré, Atul Gawande, Brownian motion, butterfly effect, correlation coefficient, Daniel Kahneman / Amos Tversky, Donald Trump, feminist movement, forensic accounting, Gerolamo Cardano, Henri Poincaré, index fund, Isaac Newton, law of one price, pattern recognition, Paul Erdős, Pepto Bismol, probability theory / Blaise Pascal / Pierre de Fermat, RAND corporation, random walk, Richard Feynman, Ronald Reagan, Stephen Hawking, Steve Jobs, The Wealth of Nations by Adam Smith, The Wisdom of Crowds, Thomas Bayes, V2 rocket, Watson beat the top human players on Jeopardy!
You don’t need calculus, geometry, algebra, or even amphetamines, which Erdös was reportedly fond of taking.8 (As legend has it, once after quitting for a month, he remarked, “Before, when I looked at a piece of blank paper my mind was filled with ideas. Now all I see is a blank piece of paper.”) All you need is a basic understanding of how probability works and the law of the sample space, that framework for analyzing chance situations that was first put on paper in the sixteenth century by Gerolamo Cardano. GEROLAMO CARDANO was no rebel breaking forth from the intellectual milieu of sixteenth-century Europe. To Cardano a dog’s howl portended the death of a loved one, and a few ravens croaking on the roof meant a grave illness was on its way. He believed as much as anyone else in fate, in luck, and in seeing your future in the alignment of planets and stars. Still, had he played poker, he wouldn’t have been found drawing to an inside straight.
In the late summer of that year he sat at his desk and wrote his final words, an ode to his favorite son, his oldest, who had been executed sixteen years earlier, at age twenty-six. The old man died on September 20, a few days shy of his seventy-fifth birthday. He had outlived two of his three children; at his death his surviving son was employed by the Inquisition as a professional torturer. That plum job was a reward for having given evidence against his father. Before his death, Gerolamo Cardano burned 170 unpublished manuscripts.1 Those sifting through his possessions found 111 that survived. One, written decades earlier and, from the looks of it, often revised, was a treatise of thirty-two short chapters. Titled The Book on Games of Chance, it was the first book ever written on the theory of randomness. People had been gambling and coping with other uncertainties for thousands of years.
Cardano’s work also transcended the primitive state of mathematics in his day, for algebra and even arithmetic were yet in their stone age in the early sixteenth century, preceding even the invention of the equal sign. History has much to say about Cardano, based on both his autobiography and the writings of some of his contemporaries. Some of the writings are contradictory, but one thing is certain: born in 1501, Gerolamo Cardano was not a child you’d have put your money on. His mother, Chiara, despised children, though—or perhaps because—she already had three boys. Short, stout, hot tempered, and promiscuous, she prepared a kind of sixteenth-century morning-after pill when she became pregnant with Gerolamo—a brew of wormwood, burned barleycorn, and tamarisk root. She drank it down in an attempt to abort the fetus.
Human Compatible: Artificial Intelligence and the Problem of Control by Stuart Russell
3D printing, Ada Lovelace, AI winter, Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, Alfred Russel Wallace, Andrew Wiles, artificial general intelligence, Asilomar, Asilomar Conference on Recombinant DNA, augmented reality, autonomous vehicles, basic income, blockchain, brain emulation, Cass Sunstein, Claude Shannon: information theory, complexity theory, computer vision, connected car, crowdsourcing, Daniel Kahneman / Amos Tversky, delayed gratification, Elon Musk, en.wikipedia.org, Erik Brynjolfsson, Ernest Rutherford, Flash crash, full employment, future of work, Gerolamo Cardano, ImageNet competition, Intergovernmental Panel on Climate Change (IPCC), Internet of things, invention of the wheel, job automation, John Maynard Keynes: Economic Possibilities for our Grandchildren, John Maynard Keynes: technological unemployment, John Nash: game theory, John von Neumann, Kenneth Arrow, Kevin Kelly, Law of Accelerating Returns, Mark Zuckerberg, Nash equilibrium, Norbert Wiener, NP-complete, openstreetmap, P = NP, Pareto efficiency, Paul Samuelson, Pierre-Simon Laplace, positional goods, probability theory / Blaise Pascal / Pierre de Fermat, profit maximization, RAND corporation, random walk, Ray Kurzweil, recommendation engine, RFID, Richard Thaler, ride hailing / ride sharing, Robert Shiller, Robert Shiller, Rodney Brooks, Second Machine Age, self-driving car, Shoshana Zuboff, Silicon Valley, smart cities, smart contracts, social intelligence, speech recognition, Stephen Hawking, Steven Pinker, superintelligent machines, Thales of Miletus, The Future of Employment, Thomas Bayes, Thorstein Veblen, transport as a service, Turing machine, Turing test, universal basic income, uranium enrichment, Von Neumann architecture, Wall-E, Watson beat the top human players on Jeopardy!, web application, zero-sum game
This plan “easily and best” produces the end, but it’s not very likely to succeed. The difference between this harebrained house-buying plan and my sober and sensible airport plan is, however, just a matter of degree. Both are gambles, but one seems more rational than the other. It turns out that gambling played a central role in generalizing Aristotle’s proposal to account for uncertainty. In the 1560s, the Italian mathematician Gerolamo Cardano developed the first mathematically precise theory of probability—using dice games as his main example. (Unfortunately, his work was not published until 1663.13) In the seventeenth century, French thinkers including Antoine Arnauld and Blaise Pascal began—for assuredly mathematical reasons—to study the question of rational decisions in gambling.14 Consider the following two bets: A: 20 percent chance of winning $10 B: 5 percent chance of winning $100 The proposal the mathematicians came up with is probably the same one you would come up with: compare the expected values of the bets, which means the average amount you would expect to get from each bet.
Here I am pointing to the roots of our present-day concept of intelligence, rather than describing the ancient Greek concept of nous, which had a variety of related meanings. 12. The quotation is taken from Aristotle, Nicomachean Ethics, Book III, 3, 1112b. 13. Cardano, one of the first European mathematicians to consider negative numbers, developed an early mathematical treatment of probability in games. He died in 1576, eighty-seven years before his work appeared in print: Gerolamo Cardano, Liber de ludo aleae (Lyons, 1663). 14. Arnauld’s work, initially published anonymously, is often called The Port-Royal Logic: Antoine Arnauld, La logique, ou l’art de penser (Chez Charles Savreux, 1662). See also Blaise Pascal, Pensées (Chez Guillaume Desprez, 1670). 15. The concept of utility: Daniel Bernoulli, “Specimen theoriae novae de mensura sortis,” Proceedings of the St. Petersburg Imperial Academy of Sciences 5 (1738): 175–92.
The Golden Ticket: P, NP, and the Search for the Impossible by Lance Fortnow
Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, Albert Einstein, Andrew Wiles, Claude Shannon: information theory, cloud computing, complexity theory, Donald Knuth, Erdős number, four colour theorem, Gerolamo Cardano, Isaac Newton, Johannes Kepler, John von Neumann, linear programming, new economy, NP-complete, Occam's razor, P = NP, Paul Erdős, Richard Feynman, Rubik’s Cube, smart grid, Stephen Hawking, traveling salesman, Turing machine, Turing test, Watson beat the top human players on Jeopardy!, William of Occam
Suppose you have actually found a solution to the P versus NP problem. How do you get your $1 million check from the Clay Mathematics Institute? Slow down. You almost surely don’t have a proof. Realize why your proof doesn’t work, and you will have obtained enlightenment. Let me mention a few of the common mistakes people make when thinking they have a proof. Perhaps the first bad P ≠ NP proof goes back to 1550 and the writings of Gerolamo Cardano, an Italian mathematician considered one of the founders of the field of probability. Cardano, in creating a new cryptographic system, argued for the security of his system because there were too many keys to check them all. But his system was easily broken. You don’t have to check all the keys when doing cryptanalysis on secret messages. Variations on Cardano’s basic error recur in many modern attempts at proving P ≠ NP.
Whiplash: How to Survive Our Faster Future by Joi Ito, Jeff Howe
3D printing, Albert Michelson, Amazon Web Services, artificial general intelligence, basic income, Bernie Sanders, bitcoin, Black Swan, blockchain, Burning Man, buy low sell high, Claude Shannon: information theory, cloud computing, Computer Numeric Control, conceptual framework, crowdsourcing, cryptocurrency, data acquisition, disruptive innovation, Donald Trump, double helix, Edward Snowden, Elon Musk, Ferguson, Missouri, fiat currency, financial innovation, Flash crash, frictionless, game design, Gerolamo Cardano, informal economy, interchangeable parts, Internet Archive, Internet of things, Isaac Newton, Jeff Bezos, John Harrison: Longitude, Joi Ito, Khan Academy, Kickstarter, Mark Zuckerberg, microbiome, Nate Silver, Network effects, neurotypical, Oculus Rift, pattern recognition, peer-to-peer, pirate software, pre–internet, prisoner's dilemma, Productivity paradox, race to the bottom, RAND corporation, random walk, Ray Kurzweil, Ronald Coase, Ross Ulbricht, Satoshi Nakamoto, self-driving car, SETI@home, side project, Silicon Valley, Silicon Valley startup, Simon Singh, Singularitarianism, Skype, slashdot, smart contracts, Steve Ballmer, Steve Jobs, Steven Levy, Stewart Brand, Stuxnet, supply-chain management, technological singularity, technoutopianism, The Nature of the Firm, the scientific method, The Signal and the Noise by Nate Silver, There's no reason for any individual to have a computer in his home - Ken Olsen, Thomas Kuhn: the structure of scientific revolutions, universal basic income, unpaid internship, uranium enrichment, urban planning, WikiLeaks
Alberti was far from the only Renaissance scholar fascinated by cryptography—the growing sophistication of European mathematics; the search for hidden patterns in nature, which might illuminate religious mysteries or reveal secret knowledge; the unprecedented spread of information enabled by the printing press; and the tangled diplomatic environment of Renaissance Europe all provided fertile ground for the development of ever more complex methods of cryptography and cryptanalysis. In the sixteenth century, Johannes Trithemius and Giovan Battista Bellasso created their own polyalphabetic ciphers, while Gerolamo Cardano and Blaise de Vigenère pioneered autokey ciphers, in which the message itself is incorporated into the key.29 All of these cryptographic innovations were matched by innovations in cryptanalysis—a Renaissance version of the same escalation that drives advances in both cybersecurity and cyber attacks today. The relatively primitive mechanical devices of early cryptography, like the cipher disk Alberti used to track his shifting alphabets, grew increasingly complex, culminating in advanced cryptographic machines like the German Enigma machine of World War II, whose theoretically unbreakable ciphers were betrayed by a simple design flaw—no letter encoded by an Enigma would ever be encoded as itself.
But What if We're Wrong? Thinking About the Present as if It Were the Past by Chuck Klosterman
a long time ago in a galaxy far, far away, Affordable Care Act / Obamacare, British Empire, citizen journalism, cosmological constant, dark matter, Edward Snowden, Elon Musk, Francis Fukuyama: the end of history, Frank Gehry, George Santayana, Gerolamo Cardano, ghettoisation, Howard Zinn, Isaac Newton, Joan Didion, non-fiction novel, obamacare, pre–internet, Ralph Nader, Ray Kurzweil, Ronald Reagan, Silicon Valley, Stephen Hawking, the medium is the message, the scientific method, Thomas Kuhn: the structure of scientific revolutions, too big to fail, Y2K
He knows a great deal about probability theory,35 so I asked him if our contemporary understanding of probability is still evolving and if the way people understood probability three hundred years ago has any relationship to how we will gauge probability three hundred years from today. His response: “What we think about probability in 2016 is what we thought in 1716, for sure . . . probably in 1616, for the most part . . . and probably what [Renaissance mathematician and degenerate gambler Gerolamo] Cardano thought in 1564. I know this sounds arrogant, but what we’ve believed about probability since 1785 is still what we’ll believe about probability in 2516.” If we base any line of reasoning around consistent numeric values, there is no way to be wrong, unless we are (somehow) wrong about the very nature of the numbers themselves. And that possibility is a non-math conversation. I mean, can 6 literally turn out to be 9?
The Perfect Bet: How Science and Math Are Taking the Luck Out of Gambling by Adam Kucharski
Ada Lovelace, Albert Einstein, Antoine Gombaud: Chevalier de Méré, beat the dealer, Benoit Mandelbrot, butterfly effect, call centre, Chance favours the prepared mind, Claude Shannon: information theory, collateralized debt obligation, correlation does not imply causation, diversification, Edward Lorenz: Chaos theory, Edward Thorp, Everything should be made as simple as possible, Flash crash, Gerolamo Cardano, Henri Poincaré, Hibernia Atlantic: Project Express, if you build it, they will come, invention of the telegraph, Isaac Newton, Johannes Kepler, John Nash: game theory, John von Neumann, locking in a profit, Louis Pasteur, Nash equilibrium, Norbert Wiener, p-value, performance metric, Pierre-Simon Laplace, probability theory / Blaise Pascal / Pierre de Fermat, quantitative trading / quantitative ﬁnance, random walk, Richard Feynman, Ronald Reagan, Rubik’s Cube, statistical model, The Design of Experiments, Watson beat the top human players on Jeopardy!, zero-sum game
Write down the amount you’ve bet and the amount you could win, and you’ll always come out on top. The calculations have a flaw only when they meet reality. On paper, the martingale seems to work fine; in practical terms, it’s hopeless. When it comes to gambling, understanding the theory behind a game can make all the difference. But what if that theory hasn’t been invented yet? During the Renaissance, Gerolamo Cardano was an avid gambler. Having frittered away his inheritance, he decided to make his fortune by betting. For Cardano, this meant measuring how likely random events were. Probability as we know it did not exist in Cardano’s era. There were no laws about chance events, no rules about how likely something was. If someone rolled two sixes while playing dice, it was simply good luck. For many games, nobody knew precisely what a “fair” wager should be.
The Doomsday Calculation: How an Equation That Predicts the Future Is Transforming Everything We Know About Life and the Universe by William Poundstone
Albert Einstein, anthropic principle, Any sufficiently advanced technology is indistinguishable from magic, Arthur Eddington, Bayesian statistics, Benoit Mandelbrot, Berlin Wall, bitcoin, Black Swan, conceptual framework, cosmic microwave background, cosmological constant, cosmological principle, cuban missile crisis, dark matter, digital map, discounted cash flows, Donald Trump, Doomsday Clock, double helix, Elon Musk, Gerolamo Cardano, index fund, Isaac Newton, Jaron Lanier, Jeff Bezos, John Markoff, John von Neumann, mandelbrot fractal, Mark Zuckerberg, Mars Rover, Peter Thiel, Pierre-Simon Laplace, probability theory / Blaise Pascal / Pierre de Fermat, RAND corporation, random walk, Richard Feynman, ride hailing / ride sharing, Rodney Brooks, Ronald Reagan, Ronald Reagan: Tear down this wall, Sam Altman, Schrödinger's Cat, Search for Extraterrestrial Intelligence, self-driving car, Silicon Valley, Skype, Stanislav Petrov, Stephen Hawking, strong AI, Thomas Bayes, Thomas Malthus, time value of money, Turing test
It is conjectured that Bayes’s work on probability was motivated by Hume’s debunking of miracles. But there is no mention of Hume or miracles in Bayes’s one influential work, the one describing his theorem: An Essay Towards Solving a Problem in the Doctrine of Chances. Nor do we know for certain when the Essay was written. Richard Price discovered it after Bayes’s death, filed among papers of the late 1740s. Bayes’s Theorem The theory of probability began at the gambling table. Gerolamo Cardano was the ultimate Renaissance man—a philosopher, mathematician, physicist, astronomer, astrologer, inventor, chemist, biologist, and fashionable physician. He was also a compulsive gambler who by his admission bet daily for twenty-five years. Cardano’s short treatise on probability was an attempt to understand how so much money had slipped through his fingers. Gamblers already knew how cards, dice, and roulette wheels worked.
Is God a Mathematician? by Mario Livio
Albert Einstein, Antoine Gombaud: Chevalier de Méré, Brownian motion, cellular automata, correlation coefficient, correlation does not imply causation, cosmological constant, Dava Sobel, double helix, Edmond Halley, Eratosthenes, Georg Cantor, Gerolamo Cardano, Gödel, Escher, Bach, Henri Poincaré, Isaac Newton, Johannes Kepler, John von Neumann, music of the spheres, Myron Scholes, probability theory / Blaise Pascal / Pierre de Fermat, Russell's paradox, Thales of Miletus, The Design of Experiments, the scientific method, traveling salesman
To be sure, mathematicians did realize that mathematics dealt only with rather abstract Platonic forms, but those were regarded as reasonable idealizations of the actual physical elements. In fact, the feeling that the book of nature was written in the language of mathematics was so deeply rooted that many mathematicians absolutely refused even to consider mathematical concepts and structures that were not directly related to the physical world. This was the case, for instance, with the colorful Gerolamo Cardano (1501–76). Cardano was an accomplished mathematician, renowned physician, and compulsive gambler. In 1545 he published one of the most influential books in the history of algebra—the Ars Magna (The Great Art). In this comprehensive treatise Cardano explored in great detail solutions to algebraic equations, from the simple quadratic equation (in which the unknown appears to the second power: x2) to pioneering solutions to the cubic (involving x3), and quartic (involving x4) equations.
Adapt: Why Success Always Starts With Failure by Tim Harford
Andrew Wiles, banking crisis, Basel III, Berlin Wall, Bernie Madoff, Black Swan, car-free, carbon footprint, Cass Sunstein, charter city, Clayton Christensen, clean water, cloud computing, cognitive dissonance, complexity theory, corporate governance, correlation does not imply causation, creative destruction, credit crunch, Credit Default Swap, crowdsourcing, cuban missile crisis, Daniel Kahneman / Amos Tversky, Dava Sobel, Deep Water Horizon, Deng Xiaoping, disruptive innovation, double entry bookkeeping, Edmond Halley, en.wikipedia.org, Erik Brynjolfsson, experimental subject, Fall of the Berlin Wall, Fermat's Last Theorem, Firefox, food miles, Gerolamo Cardano, global supply chain, Intergovernmental Panel on Climate Change (IPCC), Isaac Newton, Jane Jacobs, Jarndyce and Jarndyce, Jarndyce and Jarndyce, John Harrison: Longitude, knowledge worker, loose coupling, Martin Wolf, mass immigration, Menlo Park, Mikhail Gorbachev, mutually assured destruction, Netflix Prize, New Urbanism, Nick Leeson, PageRank, Piper Alpha, profit motive, Richard Florida, Richard Thaler, rolodex, Shenzhen was a fishing village, Silicon Valley, Silicon Valley startup, South China Sea, special economic zone, spectrum auction, Steve Jobs, supply-chain management, the market place, The Wisdom of Crowds, too big to fail, trade route, Tyler Cowen: Great Stagnation, web application, X Prize, zero-sum game
It is impossible to estimate a percentage return on blue-sky research, and it is delusional even to try. Most new technologies fail completely. Most original ideas turn out either to be not original after all, or original for the very good reason that they are useless. And when an original idea does work, the returns can be too high to be sensibly measured. The Spitfire is one of countless examples of these unlikely ideas, which range from the sublime (the mathematician and gambler Gerolamo Cardano first explored the idea of ‘imaginary numbers’ in 1545; these apparently useless curiosities later turned out to be essential for developing radio, television and computing) to the ridiculous (in 1928, Alexander Fleming didn’t keep his laboratory clean, and ended up discovering the world’s first antibiotic in a contaminated Petri dish). We might be tempted to think of such projects as lottery tickets, because they pay off rarely and spectacularly.
The Physics of Wall Street: A Brief History of Predicting the Unpredictable by James Owen Weatherall
Albert Einstein, algorithmic trading, Antoine Gombaud: Chevalier de Méré, Asian financial crisis, bank run, beat the dealer, Benoit Mandelbrot, Black Swan, Black-Scholes formula, Bonfire of the Vanities, Bretton Woods, Brownian motion, business cycle, butterfly effect, buy and hold, capital asset pricing model, Carmen Reinhart, Claude Shannon: information theory, collateralized debt obligation, collective bargaining, dark matter, Edward Lorenz: Chaos theory, Edward Thorp, Emanuel Derman, Eugene Fama: efficient market hypothesis, financial innovation, fixed income, George Akerlof, Gerolamo Cardano, Henri Poincaré, invisible hand, Isaac Newton, iterative process, John Nash: game theory, Kenneth Rogoff, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, martingale, Myron Scholes, new economy, Paul Lévy, Paul Samuelson, prediction markets, probability theory / Blaise Pascal / Pierre de Fermat, quantitative trading / quantitative ﬁnance, random walk, Renaissance Technologies, risk-adjusted returns, Robert Gordon, Robert Shiller, Robert Shiller, Ronald Coase, Sharpe ratio, short selling, Silicon Valley, South Sea Bubble, statistical arbitrage, statistical model, stochastic process, The Chicago School, The Myth of the Rational Market, tulip mania, Vilfredo Pareto, volatility smile
MIT’s mathematics library, despite its enormous holdings, did not have a copy of the obscure 1914 textbook. But Samuelson did find something else by Bachelier that piqued his interest: Bachelier’s dissertation, published under the title A Theory of Speculation. He checked it out of the library and brought it back to his office. Bachelier was not, of course, the first person to take a mathematical interest in games of chance. That distinction goes to the Italian Renaissance man Gerolamo Cardano. Born in Milan around the turn of the sixteenth century, Cardano was the most accomplished physician of his day, with popes and kings clamoring for his medical advice. He authored hundreds of essays on topics ranging from medicine to mathematics to mysticism. But his real passion was gambling. He gambled constantly, on dice, cards, and chess — indeed, in his autobiography he admitted to passing years in which he gambled every day.
Rage Inside the Machine: The Prejudice of Algorithms, and How to Stop the Internet Making Bigots of Us All by Robert Elliott Smith
Ada Lovelace, affirmative action, AI winter, Alfred Russel Wallace, Amazon Mechanical Turk, animal electricity, autonomous vehicles, Black Swan, British Empire, cellular automata, citizen journalism, Claude Shannon: information theory, combinatorial explosion, corporate personhood, correlation coefficient, crowdsourcing, Daniel Kahneman / Amos Tversky, desegregation, discovery of DNA, Douglas Hofstadter, Elon Musk, Fellow of the Royal Society, feminist movement, Filter Bubble, Flash crash, Gerolamo Cardano, gig economy, Gödel, Escher, Bach, invention of the wheel, invisible hand, Jacquard loom, Jacques de Vaucanson, John Harrison: Longitude, John von Neumann, Kenneth Arrow, low skilled workers, Mark Zuckerberg, mass immigration, meta analysis, meta-analysis, mutually assured destruction, natural language processing, new economy, On the Economy of Machinery and Manufactures, p-value, pattern recognition, Paul Samuelson, performance metric, Pierre-Simon Laplace, precariat, profit maximization, profit motive, Silicon Valley, social intelligence, statistical model, Stephen Hawking, stochastic process, telemarketer, The Bell Curve by Richard Herrnstein and Charles Murray, The Future of Employment, the scientific method, The Wealth of Nations by Adam Smith, The Wisdom of Crowds, theory of mind, Thomas Bayes, Thomas Malthus, traveling salesman, Turing machine, Turing test, twin studies, Vilfredo Pareto, Von Neumann architecture, women in the workforce
As mysticism gave way to emerging scientific enquiry in the early days of the Renaissance, the mysteries of the world’s workings and man’s role within it became the subject of intense speculation. Although people still played the astragalus in the streets of Milan, Pavia and Bologna, Italy’s big university towns, new games of chance and business opportunities were tempting Renaissance men to try their luck at taming the goddess Fortuna. One such man was mathematician, scientist, philosopher and physician Gerolamo Cardano,6 who was born in Pavia in 1501. Gerolamo was one of the greatest intellectuals of Renaissance Italy, but from the outset the Fates dealt him a difficult hand, and despite his many talents and achievements his life was beset by unfortunate circumstances beyond his control. He was the bastard son to a mathematician friend of Leonardo da Vinci, Fazio Cardano, who held a chair of mathematics at Pavia University, one of the oldest institutions of higher learning in the world.
Surfaces and Essences by Douglas Hofstadter, Emmanuel Sander
affirmative action, Albert Einstein, Arthur Eddington, Benoit Mandelbrot, Brownian motion, cognitive dissonance, computer age, computer vision, dematerialisation, Donald Trump, Douglas Hofstadter, Ernest Rutherford, experimental subject, Flynn Effect, Georg Cantor, Gerolamo Cardano, Golden Gate Park, haute couture, haute cuisine, Henri Poincaré, Isaac Newton, l'esprit de l'escalier, Louis Pasteur, Mahatma Gandhi, mandelbrot fractal, Menlo Park, Norbert Wiener, place-making, Sapir-Whorf hypothesis, Silicon Valley, statistical model, Steve Jobs, Steve Wozniak, theory of mind, upwardly mobile, urban sprawl, yellow journalism, zero-sum game
This little guess, sliding a couple of times from two-ness to three-ness, and also once from three-ness to four-ness (which in itself comes from a mini-analogy: “4 is to 3 as 3 is to 2”) seems like an utter triviality, but without very simple-seeming conceptual slippages of this sort, which crop up absolutely everywhere in mathematics, it would be impossible to make any kind of progress at all. Let’s return to the story of the solution of “the” cubic equation (the reason for the quote marks will emerge shortly). It all took place in Italy — first in Bologna (Scipione del Ferro), and a bit later in Brescia (Niccolò Tartaglia) and Milan (Gerolamo Cardano). Del Ferro found a partial solution first but didn’t publish it; some twenty years later, Tartaglia found essentially the same partial solution; finally, Cardano generalized their findings and published them in a famous book called Ars Magna (“The Great Art”). The odd thing is that, as things were coming into focus, in order to list all the “different” solutions of the cubic equation, Cardano had to use thirteen chapters!
One can imagine that at this stage mathematicians might well have joyfully concluded that they had at last arrived at the end of the number trail: that there were no more numbers left to be discovered. But the predilection of the human mind to make analogies left and right was far too strong for that to be the case. The discovery of the solution of the cubic by the Italians in the sixteenth century inspired European mathematicians to seek analogous solutions to equations having higher degrees than 3. In fact, Gerolamo Cardano himself, aided by Lodovico Ferrari, solved the quartic — the fourth-degree equation. Even though there was no geometric interpretation for an expression like “x4 ”, the purely formal analogy between the equation ax3 +bx2 + cx + d = 0 and its longer cousin ax4 + bx3 + cx2 + dx + e = 0 was so alluring to Cardano that he could not resist tackling the challenge. And in short order Cardano and Ferrari, using methods analogous to those that had turned the trick for the cubic, came up with the solution.