Today is your lucky day. Actually, this week is your lucky week!
Until Friday, September 10, you can download Math for Grownups for free — yep, $0 0¢ — on your eReader or computer. That’s how much I and my publisher (Adams Media) love you.
I was surprised to learn that the majority work in geometry, supposedly the most ‘visual’ discipline, and fascinated to learn that they generally believe the experience of sight puts people at a disadvantage because it locks us into a perception-led view of space.
The Mandelbrot set in a certain sense is a **** of a dream I had and an uncle of mine had since I was about 20. I was a student of mathematics, but not happy with mathematics that I was taught in France. Therefore, looking for other topics, an uncle of mine, who was a very well-known pure mathematician, wanted me to study a certain theory which was then many years old, 30 years old or something, but had in a way stopped developing. When he was young he had tried to get this theory out of a rut and he didn’t succeed, nobody succeeded. So, there was a case of two men, Julia, a teacher of mine, and Fatu, who had died, had a very good year in 1910 and then nothing was happening. My uncle was telling me, if you look at that, if you find something new, it would be a wonderful thing because I couldn’t – nobody could.
I looked at it and found it too difficult. I just could see nothing I could do. Then over the years, I put that a bit in the back of my mind until one day I read an obituary. It is an interesting story that I was motivated by an obituary, an obituary of a great man named Poincaré, and in that obituary this question was raised again. At that time, I had a computer and I had become quite an expert in the use of the computer for mathematics, for physics, and for many sciences. So, I decided, perhaps the time has come to please my uncle; 35 years later, or something. To please my uncle and do what my uncle had been pushing me to do so strongly.
But I approached this topic in a very different fashion than my uncle. My uncle was trying to think of something, a new idea, a new problem, a new way of developing the theory of Fatu and Julia. I did something else. I went to the computer and tried to experiment. I introduced a very high level of experiment in very pure mathematics. I was at IBM, I had the run of computers which then were called very big and powerful, but in fact were less powerful than a handheld machine today. But I had them and I could make the experiments. The conditions were very, very difficult, but I knew how to look at pictures. In fact, the reason I did not go into pure mathematics earlier was that I was dominated by visual. I tried to combine the visual beauty and the mathematics.
Fortunately there are a few Web sites that provide daily links to the best that is thought and said. Arts and Letters Daily is the center of high-toned linkage on the Web. The Browser is a trans-Atlantic site with a superb eye for the interesting and the profound. Book Forum has a more academic feel, but it is also worth a daily read.
Among Dyson’s gifts is interpretive clarity, a penetrating ability to grasp the method and significance of what many kinds of scientists do. His thoughts about how science works appear in a series of lucid, elegant books for nonspecialists that have made him a trusted arbiter of ideas ranging far beyond physics. Dyson has written more than a dozen books, including “Origins of Life” (1999), which synthesizes recent discoveries by biologists and geologists into an evaluation of the double-origin hypothesis, the possibility that life began twice; “Disturbing the Universe” (1979) tries among other things to reconcile science and humanity. “Weapons and Hope” (1984) is his meditation on the meaning and danger of nuclear weapons that won a National Book Critics Circle Award. Dyson’s books display such masterly control of complex matters that smart young people read him and want to be scientists; older citizens finish his books and feel smart.
Yet even while probing and sifting, Dyson is always whimsically gazing into the beyond. As a boy he sketched plans for English rocket ships that could explore the stars, and then, in midlife, he helped design an American spacecraft to be powered by exploding atomic bombs — a secret Air Force project known as Orion. Dyson remains an armchair astronaut who speculates with glee about the coming of cheap space travel, when families can leave an overcrowded earth to homestead on asteroids and comets, swooping around the universe via solar sail craft. Dyson is convinced that our current “age of computers” will soon give way to “the age of domesticated biotechnology.” Bio-tech, he writes in his book, “Infinite in All Directions” (1988), “offers us the chance to imitate nature’s speed and flexibility,” and he imagines the furniture and art that people will “grow” for themselves, the pet dinosaurs they will “grow” for their children, along with an idiosyncratic menagerie of genetically engineered cousins of the carbon-eating tree: termites to consume derelict automobiles, a potato capable of flourishing on the dry red surfaces of Mars, a collision-avoiding car.
These ideas attract derision similar to Dyson’s essays on climate change, but he is an undeterred octogenarian futurist. “I don’t think of myself predicting things,” he says. “I’m expressing possibilities. Things that could happen. To a large extent it’s a question of how badly people want them to. The purpose of thinking about the future is not to predict it but to raise people’s hopes.” Formed in a heretical and broad-thinking tradition of British public intellectuals, Dyson left behind a brooding England still stricken by two bloody world wars to become an optimistic American immigrant with tremendous faith in the creative imagination’s ability to invent technologies that would overcome any predicament. And according to the physicist and former Caltech president Marvin Goldberger, Dyson is himself the living embodiment of that kind of ingenuity. “You point Freeman at a problem and he’ll solve it,” Goldberger says. “He’s extraordinarily powerful.” Dyson seems to see the world as an interdisciplinary set of problems out there for him to evaluate. Climate change is the big scientific issue of our time, so naturally he finds it irresistible. But to Dyson this is really only one more charged conundrum attracting his interest just as nuclear weapons and rural poverty have. That is to say, he is a great problem-solver who is not convinced that climate change is a great problem.
Dyson is well aware that “most consider me wrong about global warming.” That educated Americans tend to agree with the conclusion about global warming reached earlier this month at the International Scientific Conference on Climate Change in Copenhagen (“inaction is inexcusable”) only increases Dyson’s resistance. Dyson may be an Obama-loving, Bush-loathing liberal who has spent his life opposing American wars and fighting for the protection of natural resources, but he brooks no ideology and has a withering aversion to scientific consensus. The Nobel physics laureate Steven Weinberg admires Dyson’s physics — he says he thinks the Nobel committee fleeced him by not awarding his work on quantum electrodynamics with the prize — but Weinberg parts ways with his sensibility: “I have the sense that when consensus is forming like ice hardening on a lake, Dyson will do his best to chip at the ice.”
Dyson says he doesn’t want his legacy to be defined by climate change, but his dissension from the orthodoxy of global warming is significant because of his stature and his devotion to the integrity of science. Dyson has said he believes that the truths of science are so profoundly concealed that the only thing we can really be sure of is that much of what we expect to happen won’t come to pass. In “Infinite in All Directions,” he writes that nature’s laws “make the universe as interesting as possible.” This also happens to be a fine description of Dyson’s own relationship to science. In the words of Avishai Margalit, a philosopher at the Institute for Advanced Study, “He’s a consistent reminder of another possibility.” When Dyson joins the public conversation about climate change by expressing concern about the “enormous gaps in our knowledge, the sparseness of our observations and the superficiality of our theories,” these reservations come from a place of experience. Whatever else he is, Dyson is the good scientist; he asks the hard questions. He could also be a lonely prophet. Or, as he acknowledges, he could be dead wrong.
The New York University political science professor has developed a computerized game theory model that predicts the future of many business and political negotiations and also figures out ways to influence the outcome. Two independent evaluations, one by academics and one by the U.S. Central Intelligence Agency, have both shown that about 90 percent of his predictions have been accurate. Most recently, he has used his mathematical tools to offer approaches for handling the growing nuclear crisis with Iran...
Former CIA analyst Stanley Feder says that he has used Bueno de Mesquita's model well over a thousand times since the early 1980s to make predictions about specific policies. Like others, he has found it to be more than 90 percent accurate. In situations where predictions of the model differed from experts' predictions, the model always turned out to be correct.
"I'm always stunned that it works so well," Bueno de Mesquita says. "This 90 percent is not my assessment."
The main reason that the model generates more reliable predictions than experts do is that "the computer doesn't get bored, it doesn't get tired, and it doesn't forget," he says. In the analysis of nuclear technology development in Iran, for example, experts identified 80 relevant players. Because no individual can keep track of all the possible interactions between so many players, human analysts focus on five or six key players. The lesser players may not have a lot of power, Buena de Mesquita says, but they tend to be knowledgeable enough to influence how key decision-makers understand the issues. His model can keep track of those influences when a human can't.
"Given expert input of data for the variables for such a model, it would not surprise me in the least to see that it would perform well," says Branislav L. Slantchev, a political scientist and game theorist at the University of California at San Diego. Predictions based on game theory can fail in a context where people don't act rationally, but in Buena de Mesquita's work, Slantchev says, rational action mostly means that the players are promoting their own perceived interests as best they can, something humans tend to do.
Heads or tails? It’s a simple question with a far from simple answer. One that takes us into the strange and complex world of probability.
Probability is the field of maths relating to random events and, although commonplace now, the idea that you can pluck a piece of maths from the tumbling of dice, the shuffling of cards or the odds in the local lottery is a relatively recent and powerful one. It may start with the toss of a coin but probability reaches into every area of the modern world, from the analysis of society to the decay of an atom.
WSJ: If you can pick an index-outperforming stock 51% of the time, how many picks do you need to make to have better than a 99% chance of outperforming the index? (We’ll assume your picks are uncorrelated and that the magnitude of any outperformance or underperformance is the same.)
Mr. Mlodinow: Consider a stock analyst versus an index fund in a kind of stock-picking World Series. The law of large numbers says if you play a best-of-X series you can be confident that the best team will win — if X is large enough. But for small X, say, a best-of-seven series, there is a surprisingly large chance that the lesser team will win. So in sports just because one team is superior doesn’t mean it will win the series.
The same uncertainty applies to the market. For example, suppose the stock picker has a 51/49 edge over the index fund, meaning he or she will outperform it, in the long run, in 51% of the years in which they compete. How long is the long run in this case? The mathematics shows that in order to justify 99% confidence that the stock picker will outperform the index fund more often than it underperforms it, the contest would have to go on for about 13,700 years...
WSJ: After a particular drug is on the market, it will cause a particularly serious adverse effect to happen to one of every 3,000 patients in an epidemiological, i. e., post-hoc analysis. In retrospect, how many patients must be tested in the randomized, double-blinded, placebo-controlled, clinical trial to achieve 95% confidence that the side effect will show up?
Mr. Mlodinow: You need roughly 14,000 patients. Here is how you get that: The process is governed by the binomial distribution, which can be approximated by the normal distribution. The chance of an adverse reaction in any one patient is one in 3,000. Since you want a 95% confidence interval for one (or more) reactions, you want enough patients so that 1.00 is 1.64 standard deviations (or more) below the mean. With 14,000 patients the mean number of adverse reactions will be about 4.6 and the standard deviation is about 2.2, which gives you what you require. (I have rounded my answer to the nearest 1,000).
Correction: To achieve 95% confidence that the side effect will show up, you need 8,985 patients receiving the drug. This blog post misstated the number as 14,000. See the comments of this post for more details.
WSJ: Might we need to proceed irrationally in our lives to succeed? In other words, if we really believed that so much of success was the result of luck, wouldn’t a lot of us just give up trying?
Mr. Mlodinow: Some theorize that this is the evolutionary reason that we like to assume we are in control, even when we clearly aren’t. That may be so, but I don’t mourn the role of luck, I celebrate it. All else equal, it is a lot more fun not knowing how your book will do, or how your life will turn out, than it would be if everything could be determined by a logical calculation. Moreover, the fact that luck matters means you can help yourself by being persistent. A failure doesn’t mean you are unworthy, nor does it preclude success on the next try. As Thomas J. Watson, the highly successful IBM pioneer, said, “If you want to succeed, double your failure rate.”
