## Posts Tagged ‘**Mathematics**’

## “I’m envious of people who can sleep as long as they want. I have the circadian rhythm of a farmer.”*…

After World War II, scientists began studying the internal clocks of animals in earnest. They discovered that mammals and other creatures are ruled by their own, internal body clock, what is commonly referred to today as a biological clock. The German physician and biologist Jürgen Aschoff wondered if this might also be true of humans. In the early 1960s, as head of a new department for biological timing at the Max Planck Institute for Behavioral Physiology, Aschoff and his research partner Rütger Wever designed an experiment to find out.

To study the inner workings of human biological clocks, Aschoff built a soundproof underground bunker in the foothills of a mountain deep in the Bavarian countryside, just up the road from the well-known beer-brewing monastery Kloster Andechs. Through a series of investigations that included 200 subjects and spanned two decades, Aschoff’s bunker experiments would become a pioneering study in the field of chronobiology, changing the way we think about time today…

What Is Chronobiology? Does it explain why we’re having so much trouble sleeping? Find out here.

* Moby

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**As we hit the hay,** we might spare a thought for Urbain Le Verrier; he died on this date in 1877. An astronomer and mathematician who specialized in celestial mechanics, he’s best remembered for predicting the existence and position of the planet Neptune using only mathematics. Le Verrier sent the coordinates to Johann Gottfried Galle at the New Berlin Observatory, asking him to verify. Galle found Neptune in the same night he received Le Verrier’s letter– this date in 1846. The planet was within 1° of the predicted position.

## “If a ‘religion’ is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one”*…

In 1931, the Austrian logician Kurt Gödel pulled off arguably one of the most stunning intellectual achievements in history.

Mathematicians of the era sought a solid foundation for mathematics: a set of basic mathematical facts, or axioms, that was both consistent — never leading to contradictions — and complete, serving as the building blocks of all mathematical truths.

But Gödel’s shocking incompleteness theorems, published when he was just 25, crushed that dream. He proved that any set of axioms you could posit as a possible foundation for math will inevitably be incomplete; there will always be true facts about numbers that cannot be proved by those axioms. He also showed that no candidate set of axioms can ever prove its own consistency.

His incompleteness theorems meant there can be no mathematical theory of everything, no unification of what’s provable and what’s true. What mathematicians can prove depends on their starting assumptions, not on any fundamental ground truth from which all answers spring.

In the 89 years since Gödel’s discovery, mathematicians have stumbled upon just the kinds of unanswerable questions his theorems foretold. For example, Gödel himself helped establish that the continuum hypothesis, which concerns the sizes of infinity, is undecidable, as is the halting problem, which asks whether a computer program fed with a random input will run forever or eventually halt. Undecidable questions have even arisen in physics, suggesting that Gödelian incompleteness afflicts not just math, but — in some ill-understood way — reality…

A (relatively) simple explanation of the incompleteness theorem– which destroyed the search for a mathematical theory of everything: “How Gödel’s Proof Works.”

* John D. Barrow, *The Artful Universe*

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**As we noodle on the unknowable,** we might spare a thought for Vilfredo Federico Damaso Pareto; he died on this date in 1923. An engineer, sociologist, economist, political scientist, and philosopher, he made several important contributions to economics, sociology, and mathematics.

He introduced the concept of Pareto efficiency and helped develop the field of microeconomics. He was also the first to discover that income follows a Pareto distribution, which is a power law probability distribution. The Pareto principle, named after him, generalized on his observations on wealth distribution to suggest that, in most systems/settings, 80% of the effects come from 20% of the causes– the “80-20 rule.” He was also responsible for popularizing the use of the term “elite” in social analysis.

As Benoit Mandelbrot and Richard L. Hudson observed, “His legacy as an economist was profound. Partly because of him, the field evolved from a branch of moral philosophy as practised by Adam Smith into a data intensive field of scientific research and mathematical equations.”

The future leader of Italian fascism Benito Mussolini, in 1904, when he was a young student, attended some of Pareto’s lectures at the University of Lausanne. It has been argued that Mussolini’s move away from socialism towards a form of “elitism” may be attributed to Pareto’s ideas.

Mandelbrot summarized Pareto’s notions as follows:

At the bottom of the Wealth curve, he wrote, Men and Women starve and children die young. In the broad middle of the curve all is turmoil and motion: people rising and falling, climbing by talent or luck and falling by alcoholism, tuberculosis and other kinds of unfitness. At the very top sit the elite of the elite, who control wealth and power for a time – until they are unseated through revolution or upheaval by a new aristocratic class. There is no progress in human history. Democracy is a fraud. Human nature is primitive, emotional, unyielding. The smarter, abler, stronger, and shrewder take the lion’s share. The weak starve, lest society become degenerate: One can, Pareto wrote, ‘compare the social body to the human body, which will promptly perish if prevented from eliminating toxins.’ Inflammatory stuff – and it burned Pareto’s reputation… [

source]

## “Neoliberalization has meant, in short, the financialization of everything”*…

Investing and deal-making occupy an outsized role in popular depictions of “business” like HBO’s

Successionand Showtime’sBillions. They also occupy an outsized share of our elite: Over the last five years, the nation’s top business schools have sent nearly thirty percent of their graduating classes into finance.But the buying and selling of companies, the mergers and divestments, the hedging and leveraging, are not themselves valuable activity. They invent, create, build, and provide nothing. Their claim to value is purely derivative—by improving the allocation of capital and configuration of assets, they are supposed to make everyone operating in the real economy more productive. The practitioners are rewarded richly for their effort.

Does this work, or are the efforts largely wasted? One might default to the assumption that an industry attracting so much talent and generating so much profit

mustbe creating enormous value. But the elaborate financial engineering of the 2000s, which attempted an alchemy-like conversion of high-risk loans into rock-solid assets, and then placed highly leveraged bets against their performance, led to the collapse of some established Wall Street institutions, massive bailouts for others, and a global economic meltdown. Mergers and acquisitions, meanwhile, appear largely to be exercises in wheel-spinning: “M&A is a mug’s game,” explains Roger Martin in theHarvard Business Review, “in which typically 70%–90% of acquisitions are abysmal failures.”…Hedge funds and venture capital funds appear to badly underperform simple public market indexes, while buyout funds have performed roughly at par over the past decade. Of course, some funds deliver outsized returns in a given timeframe; even a random distribution has a right tail. And there are managers whose strong and consistent track records suggest the creation of real value.

In other words, most fund managers are generating the results that one might expect from an elaborate game of chance—placing bets in the market with odds similar to a coin flip. With enough people playing, some will always find themselves on winning streaks and claim the Midas touch, at least until the coin’s next flip. Except under these rules of “heads I win, tails you lose,” they collect their fees regardless…

In the U.S., finance, insurance and real estate (FIRE) sector now accounts for 20 percent of GDP– compared with only 10 percent in 1947. The thorough and thoughtful analysis– and critique– of the frothier components of that sector excerpted above is noteworthy, beyond its quality, for it’s origin; it is an early product of a new conservative think tank, American Compass.

Read it in full: “Coin-Flip Capitalism: A Primer.”

Pair with “What Kind of Country Do We Want?“, a resonant essay from the amazing Marilynne Robinson.

(image above: *source*)

* “Neoliberalization has meant, in short, the financialization of everything. There was unquestionably a power shift away from production to the world of finance… Neoliberalization has not been very effective in revitalizing global capital accumulation, but it has succeeded remarkably well in restoring, or in some instances (as in Russia and China) creating, the power of an economic elite. The theoretical utopianism of neoliberal argument has, I conclude, primarily worked as a system of justification and legitimation for whatever needed to be done to achieve this goal.” — David Harvey, * A Brief History of Neoliberalism *

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**As we look beyond price to value,** we might recall that it was on this date in 1936 that Alan Turing submitted his paper, “On Computable Numbers” for publication; its full title was “On Computable Numbers, with an Application to the Entscheidungsproblem.” In answer to Hibert’s and Ackermann’s 1928 challenge, Turing demonstrated that some purely mathematical yes-no questions can never be answered by computation; more technically, that some decision problems are “undecidable” in the sense that there is no single algorithm that infallibly gives a correct “yes” or “no” answer to each instance of the problem. In Turing’s own words: “…what I shall prove is quite different from the well-known results of Gödel … I shall now show that there is no general method which tells whether a given formula **U** is provable in **K**.”

Turing followed this proof with two others, both of which rely on the first. And all rely on his development of type-writer-like “computing machines” that obey a simple set of rules and his subsequent development of a “universal computing machine”– the “Turing Machine,” a key inspiration (to von Neumann and others) for the development of the digital computer.