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Von Neumann, too, was deeply concerned about the inability of humanity to keep up with its own inventions. “What we are creating now,” he said to his wife Klári in 1945, “is a monster whose influence is going to change history, provided there is any history left.” Moving to the subject of future computing machines he became even more agitated, foreseeing disaster if “people” could not “keep pace with what they create.”

Oppenheimer, Einstein, von Neumann and other Institute faculty channeled much of their effort toward what AI researchers today call the “alignment” problem: how to make sure our discoveries serve us instead of destroying us. Their approaches to this increasingly pressing problem remain instructive.

Von Neumann focused on applying the powers of mathematical logic, taking insights from games of strategy and applying them to economics and war planning. Today, descendants of his “game theory” running on von Neumann computing architecture are applied not only to our nuclear strategy, but also many parts of our political, economic and social lives. This is one approach to alignment: humanity survives technology through more technology, and it is the researcher’s role to maximize progress.

A panoply of devices and ideas are named after Archimedes. Besides the Archimedes screw, there is the Archimedes principle, the law of buoyancy that states the upward force on a submerged object equals the weight of the liquid displaced. There is the Archimedes claw, a weapon that most likely did exist, grabbing onto Roman ships and tipping them over. And there is the Archimedes sphere, a forerunner of the planetarium — a hand-held globe that showed the constellations as well as the locations of the sun and the planets in the sky.

Dr. Rorres said the singular genius of Archimedes was that he not only was able to solve abstract mathematics problems, but also used mathematics to solve physics problems, and he then engineered devices to take advantage of the physics. “He came up with fundamental laws of nature, proved them mathematically and then was able to apply them,” Dr. Rorres said.

Mr. Derman is perhaps a bit too harsh when he describes EMM—the so-called Efficient Market Model. EMM does not, as he claims, imply that prices are always correct and that price always equals value. Prices are always wrong. What EMM says is that we can never be sure if prices are too high or too low.

The Efficient Market Model does not suggest that any particular model of valuation—such as the Capital Asset Pricing Model—fully accounts for risk and uncertainty or that we should rely on it to predict security returns. EMM does not, as Mr. Derman says, "stubbornly assume that all uncertainty about the future is quantifiable." The basic lesson of EMM is that it is very difficult—well nigh impossible—to beat the market consistently.

Mr. Derman gives an eloquent description of James Clerk Maxwell's electromagnetic theory in a chapter titled "The Sublime." He writes: "The electromagnetic field is not like Maxwell's equations; it is Maxwell's equations."

economists have increasingly become the go-to experts on every manner of business and public policy issue facing society

The economics profession that Mr. Thaler entered in the early 1970s was deeply invested in proving that it was more than a mere social science

To achieve the same mathematical precision of hard sciences, economists made a radically simplifying assumption that people are “optimizers” whose behavior is as predictable as the speed of physical body falling through space.

the book is part memoir, part attack on a breed of economist who dominated the academy – particularly, the Chicago School that dominated economic theory at the University of Chicago – for the much of the latter part of the 20th century.

economists have increasingly become the go-to experts on every manner of business and public policy issue facing society.

rather than being a disgruntled former employee or otherwise easily marginalized whistle-blower, Mr. Thaler recently took the reins as president of the American Economic Association (and still teaches at Chicago’s graduate business program

We’ve all heard the argument that philosophy is isolated, an “ivory tower” discipline cut off from virtually every other progress-making pursuit of knowledge, including math and the sciences, as well as from the actual concerns of daily life. The reasons given for this are many. In a widely read essay in this series, “When Philosophy Lost Its Way,” Robert Frodeman and Adam Briggle claim that it was philosophy’s institutionalization in the university in the late 19th century that separated it from the study of humanity and nature, now the province of social and natural sciences.

This institutionalization, the authors claim, led it to betray its central aim of articulating the knowledge needed to live virtuous and rewarding lives. I have a different view: Philosophy isn’t separated from the social, natural or mathematical sciences, nor is it neglecting the study of goodness, justice and virtue, which was never its central aim.

identified philosophy with informal linguistic analysis. Fortunately, this narrow view didn’t stop them from contributing to the science of language and the study of law. Now long gone, neither movement defined the philosophy of its day and neither arose from locating it in universities.

Our standard framework for thinking about engineering failures—reflected, for instance, in regulations for medical devices—was developed shortly after World War II, before the advent of software, for electromechanical systems. The idea was that you make something reliable by making its parts reliable (say, you build your engine to withstand 40,000 takeoff-and-landing cycles) and by planning for the breakdown of those parts (you have two engines). But software doesn’t break. Intrado’s faulty threshold is not like the faulty rivet that leads to the crash of an airliner. The software did exactly what it was told to do. In fact it did it perfectly. The reason it failed is that it was told to do the wrong thing.

Software failures are failures of understanding, and of imagination. Intrado actually had a backup router, which, had it been switched to automatically, would have restored 911 service almost immediately. But, as described in a report to the FCC, “the situation occurred at a point in the application logic that was not designed to perform any automated corrective actions.”

The introduction of programming languages like Fortran and C, which resemble English, and tools, known as “integrated development environments,” or IDEs, that help correct simple mistakes (like Microsoft Word’s grammar checker but for code), obscured, though did little to actually change, this basic alienation—the fact that the programmer didn’t work on a problem directly, but rather spent their days writing out instructions for a machine.

haped abstractions called vectors. Pondering Hilbert space makes me feel like a lump of dumb, decrepit flesh trapped in a squalid, 3-D prison. Far from exploring Hilbert space, I can’t even find a window through which to peer into it. I envision it as an immaterial paradise where luminescent cognoscenti glide to and fro, telepathically swapping witticisms about adjoint operators.

Reality, great sages have assured us, is essentially mathematical. Plato held that we and other things of this world are mere shadows of the sublime geometric forms that constitute reality. Galileo declared that “the great book of nature is written in mathematics.” We’re part of nature, aren’t we? So why does mathematics, once we get past natural numbers and basic arithmetic, feel so alien to most of us?

Physicists’ theories work. They predict the arc of planets and the flutter of electrons, and they have spawned smartphones, H-bombs and—well, what more do we need? But scientists, and especially physicists, aren’t just seeking practical advances. They’re after Truth. They want to believe that their theories are correct—exclusively correct—representations of nature. Physicists share this craving with religious folk, who need to believe that their path to salvation is the One True Path.

The sages had placed the Earth at the centre of the universe for nearly two thousand years until Copernicus arrived on the scene and let it spin like a top around the Sun, as we know it today.

Nicholas Copernicus (1473-1543) was not the first to explain that everything revolves around the Sun, but he did it so thoroughly, in that book, that he initiated a scientific revolution against the universal order established by the greatest scholar ever known, the Greek philosopher Aristotle.

Aristotle said in the fourth century BC that a mystical force moved the Sun and the planets in perfect circles around the Earth. Although this was much to the taste of the Church, in order to fit this idea with the strange movements of the planets seen in the sky, astronomers had to resort to the mathematical juggling that another Greek, Ptolemy, invented in the second century AD.

Aristotelian thought had dominated mathematics and astronomy for centuries, until revolutionaries like Nicolaus Copernicus and Galileo Galilei challenged those views.

The mathematization of physics was a crucial step in the advancement of science. It was realized that the mathematical tools we had at the time weren’t strong enough.

By trial and error, Kepler worked and worked until finally, he hit upon the shape that worked—elliptical orbits with the Sun at one focus. It turned out to perfectly fit the known observations.

I wanted to be a scientist. So why did I find the actual work of science so boring? In college science courses, I had occasional bursts of mind-expanding insight. For the most part, though, I was tortured by drudgery.

I’d found that science was two-faced: simultaneously thrilling and tedious, all-encompassing and narrow. And yet this was clearly an asset, not a flaw. Something about that combination had changed the world completely.

“Science is an alien thought form,” he writes; that’s why so many civilizations rose and fell before it was invented. In his view, we downplay its weirdness, perhaps because its success is so fundamental to our continued existence.

Various studies point to the conclusion that subjecting the mind to formal discipline — as when studying geometry or Latin — does not, in general, engender a broad transfer of learning. There is no sweeping increase of a general capacity for tasks like writing a speech or balancing a checkbook.

Many reasons have been advanced for this poor showing, including the lack of relevance of such an abstract exercise to people’s daily lives.

Most people reflexively eliminate the cards not explicitly specified in the rule (the F and the 2) and then continue with slower, more analytic processing only for the E and the 5. In this, they rely on an initial snap judgment about superficial similarity, a tendency that some scholars speculate evolved in humans because in most real-world contexts, quickly detecting such similarities is a good strategy for survival.

This is a Epicurean (341–270 BC) belief that the gods are too busy to deal with the day-to-day running of the universe but they set it in motion using mathematics.

The same is true for language, art, music and other “Third World” constructs—these are incrementally evolving systems and form one of Karl Popper’s ontological tools (Carr, 1977).

Another similar type of psychology, psychometrics, differs in that it measures the behavior of populations while mathematics paired with psychology in this arena looks at the behavior of the “average individual.”

The American philosopher Wilfrid Sellars was challenging this assumption when he spoke of “material inferences.” Sellars was interested in inferences that we can only recognize as valid if we possess certain bits of factual knowledge.

That the use of standard algorithms isn’t merely mechanical is not by itself a reason to teach them. It is important to teach them because, as we already noted, they are also the most elegant and powerful methods for specific operations. This means that they are our best representations of connections among mathematical concepts. Math instruction that does not teach both that these algorithms work and why they do is denying students insight into the very discipline it is supposed to be about.

according to Wittgenstein, is why it is wrong to understand algorithm-based calculations as expressions of nothing more than “mental mechanisms.” Far from being genuinely mechanical, such calculations involve a distinctive kind of thought.

I am a deluded throwback to carefree days, and in my attempt to raise a conscious, creative and socially and environmentally responsible child while lacking the means to also finance her conscious, creative and environmentally and socially responsible lifestyle forever, I’d accidentally gone and raised a hothouse serf. Oops.

Reich’s thesis is that some inequality is inevitable, even necessary, in a free-market system. But what makes an economy stable and prosperous is a strong, vibrant, growing middle class. In the three decades after World War II, a period that Reich calls “the great prosperity,” the G.I. Bill, the expansion of public universities and the rise of labor unions helped create the biggest, best-educated middle class in the world. Reich describes this as an example of a “virtuous circle” in which productivity grows, wages increase, workers buy more, companies hire more, tax revenues increase, government invests more, workers are better educated. On the flip side, when the middle class doesn’t share in the economic gains, it results over time in a downward vicious cycle: Wages stagnate, workers buy less, companies downsize, tax revenues decrease, government cuts programs, workers are less educated, unemployment rises, deficits grow. Since the crash that followed the deregulation of the financial markets, we have struggled to emerge from such a cycle.

What if the kid got it in her head that it was a good idea to go into public service, the helping professions, craftsmanship, scholarship or — God help her — the arts? Wouldn’t a greedier, more back-stabby style of early education be more valuable to the children of the shrinking middle class ­ — one suited to the world they are actually living in?

Every living thing is a pulse. We quicken, then we fade. There is a deep beauty in this, but deeper down, inside every plant, every leaf, inside every living thing (us included) sits a secret.

Everything alive will eventually die, we know that, but now we can read the pattern and see death coming. We have recently learned its logic, which "You can put into mathematics," says physicist Geoffrey West. It shows up with "extraordinary regularity," not just in plants, but in all animals, from slugs to giraffes. Death, it seems, is intimately related to size.

Life is short for small creatures, longer in big ones.

The largest prime number yet has been discovered — and it's 17,425,170 digits long. The new prime number crushes the last one discovered in 2008, which was a paltry 12,978,189 digits long.

The number — 2 raised to the 57,885,161 power minus 1 — was discovered by University of Central Missouri mathematician Curtis Cooper as part of a giant network of volunteer computers

"It's analogous to climbing Mt. Everest," said George Woltman, the retired, Orlando, Fla.-based computer scientist who created GIMPS. "People enjoy it for the challenge of the discovery of finding something that's never been known before."

If you take a look at the progress of science, the sciences are kind of a continuum, but they're broken up into fields. The greatest progress is in the sciences that study the simplest systems. So take, say physics -- greatest progress there. But one of the reasons is that the physicists have an advantage that no other branch of sciences has. If something gets too complicated, they hand it to someone else.

If a molecule is too big, you give it to the chemists. The chemists, for them, if the molecule is too big or the system gets too big, you give it to the biologists. And if it gets too big for them, they give it to the psychologists, and finally it ends up in the hands of the literary critic, and so on.

neuroscience for the last couple hundred years has been on the wrong track. There's a fairly recent book by a very good cognitive neuroscientist, Randy Gallistel and King, arguing -- in my view, plausibly -- that neuroscience developed kind of enthralled to associationism and related views of the way humans and animals work. And as a result they've been looking for things that have the properties of associationist psychology.