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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


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.

Urbain Le Verrier


Written by LW

September 23, 2020 at 1:01 am

“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


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]

220px-Vilfredo_Pareto_1870s2 source



“If you are not completely confused by quantum mechanics, you do not understand it”*…




If we can harness it, quantum technology promises fantastic new possibilities. But first, scientists need to coax quantum systems to stay yoked for longer than a few millionths of a second.

A team of scientists at the University of Chicago’s Pritzker School of Molecular Engineering announced the discovery of a simple modification that allows quantum systems to stay operational—or “coherent”—10,000 times longer than before. Though the scientists tested their technique on a particular class of quantum systems called solid-state qubits, they think it should be applicable to many other kinds of quantum systems and could thus revolutionize quantum communication, computing and sensing…

Down at the level of atoms, the world operates according to the rules of quantum mechanics—very different from what we see around us in our daily lives. These different rules could translate into technology like virtually unhackable networks or extremely powerful computers; the U.S. Department of Energy released a blueprint for the future quantum internet in an event at UChicago on July 23. But fundamental engineering challenges remain: Quantum states need an extremely quiet, stable space to operate, as they are easily disturbed by background noise coming from vibrations, temperature changes or stray electromagnetic fields.

Thus, scientists try to find ways to keep the system coherent as long as possible…

“This breakthrough lays the groundwork for exciting new avenues of research in quantum science,” said study lead author David Awschalom, the Liew Family Professor in Molecular Engineering, senior scientist at Argonne National Laboratory and director of the Chicago Quantum Exchange. “The broad applicability of this discovery, coupled with a remarkably simple implementation, allows this robust coherence to impact many aspects of quantum engineering. It enables new research opportunities previously thought impractical.”…

Very big news at a very small scale: “Scientists discover way to make quantum states last 10,000 times longer.”

*John Wheeler


As we strive for stability, we might send calculated birthday greetings to Brook Taylor; he was born on this date in 1685.  A mathematician, he is best known for his work in describing and understanding oscillation.  In 1708, Taylor produced a solution to the problem of the center of oscillation.  His Methodus incrementorum directa et inversa (“Direct and Indirect Methods of Incrementation,” 1715) introduced what is now called the calculus of finite differences.  Using this, he was the first to express mathematically the movement of a vibrating string on the basis of mechanical principles.  Methodus also contained Taylor’s theorem, later recognized by Joseph Lagrange as the basis of differential calculus.

A gifted artist, Taylor also wrote on the basic principles of perspective, including the first general treatment of the principle of vanishing points.

220px-BTaylor source



“We can judge our progress by the courage of our questions and the depth of our answers, our willingness to embrace what is true rather than what feels good”*…




If one takes Donald Trump and his administration to embody modern conservatism, it is easy to see in their response to the coronavirus pandemic the right’s final divorce from science and expertise. There was the case of Rick Bright, the Health and Human Services scientist who claims that the Trump administration retaliated against him when he objected to the administration’s rapid push to distribute anti-malaria drugs that were largely untested for treating coronavirus patients. There are reports that the president for months ignored his own intelligence experts’ warnings that the virus threatened our shores. There was the ongoing drama over whether Trump would fire Anthony Fauci, who has headed the National Institute of Allergy and Infectious Diseases since 1984. And there was the president’s daily passion play—the White House press briefings where he’d stand next to scientists who grimaced as he speculated that the death toll was exaggerated and that sunlight inside the body might kill the virus.

The White House’s sorry Covid-19 track record has sparked a chorus of dissent recently distilled by New York Times columnist Michelle Goldberg, who argues that the crisis displays conservatives’ long-standing “antipathy to science,” owing to “populist distrust of experts, religious rejection of information that undermines biblical literalism and efforts by giant corporations to evade regulation.” But this narrative is too pat. While something is plainly amiss in the relationship of the Trumpian right to science, it is hardly as principled as the religious objections of, say, creationists opposing evolutionary theory. Neither is it straightforwardly hostile.

What’s more curious about the response by the president and his allies to the virus is rather their embrace of scientific expertise of a sort…

The story of the crisis is not quite that of scientists who knew the answers and one political party that just wouldn’t listen to them. Rather, it is a story of fracture—of conflict and confusion, of experts earning mistrust, of each side cultivating its own class of experts to own the other’s. It is also a perverse story of how a group of self-styled truth-telling outsiders turned science’s mythology against its institutions, warping it from a tool to fight the virus into a tool to attack the establishment.

How did we get here?…

Ari Schulman (@AriSchulman) explains how a new class of outsider experts is exploiting institutional failures and destabilizing knowledge: “The Coronavirus and the Right’s Scientific Counterrevolution.”

TotH to Byrne Hobart, who notes (in his nifty newsletter, The Diff):

… this essay obviously takes a side, but it tries to be fair to the side it disagrees with. Which means there are two Straussian readings: maybe it’s an essay about how science is on one side in an American political context, and the other side only makes vague gestures towards empiricism. Alternatively, it could be an essay on how science never answers political questions, but politics corrupts science. (Why doesn’t science answer political questions? Because you can’t build a coalition out of stating the obvious, but you can build one from denying it—if your beliefs are crazy, you can spot members of the ingroup. So most scientific questions are irrelevant to politics, and when they’re relevant, politics wins by default in the short term, even if it loses long-term. To build a coherent and healthy ingroup, you need beliefs that are crazy but don’t lead to bad decisions.)

Pair with another of Hobart’s suggestions: “On Cultures That Build” (and the reasons why, the author argues. the U.S. is not one).

* Carl Sagan


As we commit to learning, we might note that today is the birthday of not one but two extraordinary mathematicians:  Gottfried Wilhelm Leibniz (1646; variants on his date of birth are due to calendar changes), the German  philosopher, scientist, mathematician, diplomat, librarian, lawyer, co-inventor, with Newton, of The Calculus, and “hero” (well, one hero) of Neal Stephenson’s Baroque Trilogy…  and  Alan Turing (1912), British mathematician, computer science pioneer (inventor of the Turing Machine, creator of “the Turing Test” and inspiration for “The Turing Prize”), and cryptographer (leading member of the team that cracked the Enigma code during WWII).

Go figure…

Turing (source: Univ. of Birmingham)

Giambattista Vico was also born on this date in 1668.  A political philosopher, rhetorician, historian, and jurist, Vico was one of the greatest Enlightenment thinkers.  Best known for the Scienza Nuova (1725, often published in English as New Science), he famously criticized the expansion and development of modern rationalism and was an apologist for classical antiquity.

He was an important precursor of systemic and complexity thinking (as opposed to Cartesian analysis and other kinds of reductionism); and he can be credited with the first exposition of the fundamental aspects of social science, though his views did not necessarily influence the first social scientists.  Vico is often claimed to have fathered modern philosophy of history (although the term is not found in his text; Vico speaks of a “history of philosophy narrated philosophically”). While he was not strictly speaking a historicist, interest in him has been driven by historicists (like Isaiah Berlin).



Written by LW

June 23, 2020 at 1:01 am

“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 Succession and Showtime’s Billions. 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 must be 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 the Harvard 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


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.

220px-Alan_Turing_Aged_16 source


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