Posts Tagged ‘Probability’
“Oops, I did it again”*…
(Roughly) Daily has contemplated game theory a number of times (e.g., here). The Generalist Academy offers a particularly poignant example…
Football [or, as it’s called in the U.S., soccer] has a lot of strange rules – like Ted Lasso, I still don’t understand exactly how the offside rule works. But the basic game is pretty simple: get the ball into your opponent’s goal, and prevent them from getting the ball into your goal. Scoring a goal against your own side is a rare and accidental embarrassment. Usually.
The qualification round for the 1994 Caribbean Cup had some unusual rules. No match could end in a draw; if the teams were tied at the end of regular time, they would go into sudden death extra time. But! Any goal scored in extra time would count as two goals. This was presumably done because this tournament, like many, used goal difference to break ties in the qualifying groups. (Goal difference = total number of goals they’ve scored minus the number of goals they’ve conceded.) So that extra time “golden goal” would give a team an edge in the overall competition. Little did the organisers know that it would also lead to one of the strangest football games ever seen…
A truly remarkable match: “Own-Goal Football,” from @GeneralistAcad.
* Britney Spears (Songwriters: Martin Max / Rami Yacoub)
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As we work backwards, we might send carefully-calculated birthday greetings to Félix Édouard Justin Émile Borel; he was born on this date in 1871. A mathematician (and politician, who served as French Minister of the Navy), he is remembered for his foundational work in measure theory and probability. He published a number of research papers on game theory and was the first to define games of strategy.
But Borel may be best remembered for a thought experiment he introduced in one of his books, proposing that a monkey hitting keys at random on a typewriter keyboard will – with absolute certainty – eventually type every book in France’s Bibliothèque Nationale de France. This is now popularly known as the infinite monkey theorem.
“Why, sometimes I’ve believed as many as six impossible things before breakfast”*…
Imaginary numbers were long dismissed as mathematical “bookkeeping.” But now, as Karmela Padavic-Callaghan explains, physicists are proving that they describe the hidden shape of nature…
Many science students may imagine a ball rolling down a hill or a car skidding because of friction as prototypical examples of the systems physicists care about. But much of modern physics consists of searching for objects and phenomena that are virtually invisible: the tiny electrons of quantum physics and the particles hidden within strange metals of materials science along with their highly energetic counterparts that only exist briefly within giant particle colliders.
In their quest to grasp these hidden building blocks of reality scientists have looked to mathematical theories and formalism. Ideally, an unexpected experimental observation leads a physicist to a new mathematical theory, and then mathematical work on said theory leads them to new experiments and new observations. Some part of this process inevitably happens in the physicist’s mind, where symbols and numbers help make invisible theoretical ideas visible in the tangible, measurable physical world.
Sometimes, however, as in the case of imaginary numbers – that is, numbers with negative square values – mathematics manages to stay ahead of experiments for a long time. Though imaginary numbers have been integral to quantum theory since its very beginnings in the 1920s, scientists have only recently been able to find their physical signatures in experiments and empirically prove their necessity…
Learn more at “Imaginary numbers are real,” from @Ironmely in @aeonmag.
* The Red Queen, in Lewis Carroll’s Through the Looking Glass
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As we get real, we might spare a thought for two great mathematicians…
Georg Friedrich Bernhard Riemann died on this date in 1866. A mathematician who made contributions to analysis, number theory, and differential geometry, he is remembered (among other things) for his 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, regarded as one of the most influential papers in analytic number theory.
Andrey (Andrei) Andreyevich Markov died on this date in 1922. A Russian mathematician, he helped to develop the theory of stochastic processes, especially those now called Markov chains: sequences of random variables in which the future variable is determined by the present variable but is independent of the way in which the present state arose from its predecessors. (For example, the probability of winning at the game of Monopoly can be determined using Markov chains.) His work on the study of the probability of mutually-dependent events has been developed and widely applied to the biological, physical, and social sciences, and is widely used in Monte Carlo simulations and Bayesian analyses.
“Nothing in life is certain except death, taxes and the second law of thermodynamics”*…
The second law of thermodynamics– asserting that the entropy of a system increases with time– is among the most sacred in all of science, but it has always rested on 19th century arguments about probability. As Philip Ball reports, new thinking traces its true source to the flows of quantum information…
In all of physical law, there’s arguably no principle more sacrosanct than the second law of thermodynamics — the notion that entropy, a measure of disorder, will always stay the same or increase. “If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations — then so much the worse for Maxwell’s equations,” wrote the British astrophysicist Arthur Eddington in his 1928 book The Nature of the Physical World. “If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.” No violation of this law has ever been observed, nor is any expected.
But something about the second law troubles physicists. Some are not convinced that we understand it properly or that its foundations are firm. Although it’s called a law, it’s usually regarded as merely probabilistic: It stipulates that the outcome of any process will be the most probable one (which effectively means the outcome is inevitable given the numbers involved).
Yet physicists don’t just want descriptions of what will probably happen. “We like laws of physics to be exact,” said the physicist Chiara Marletto of the University of Oxford. Can the second law be tightened up into more than just a statement of likelihoods?
A number of independent groups appear to have done just that. They may have woven the second law out of the fundamental principles of quantum mechanics — which, some suspect, have directionality and irreversibility built into them at the deepest level. According to this view, the second law comes about not because of classical probabilities but because of quantum effects such as entanglement. It arises from the ways in which quantum systems share information, and from cornerstone quantum principles that decree what is allowed to happen and what is not. In this telling, an increase in entropy is not just the most likely outcome of change. It is a logical consequence of the most fundamental resource that we know of — the quantum resource of information…
Is that most sacrosanct natural laws, second law of thermodynamics, a quantum phenomenon? “Physicists Rewrite the Fundamental Law That Leads to Disorder,” from @philipcball in @QuantaMagazine.
* “Nothing in life is certain except death, taxes and the second law of thermodynamics. All three are processes in which useful or accessible forms of some quantity, such as energy or money, are transformed into useless, inaccessible forms of the same quantity. That is not to say that these three processes don’t have fringe benefits: taxes pay for roads and schools; the second law of thermodynamics drives cars, computers and metabolism; and death, at the very least, opens up tenured faculty positions.” — Seth Lloyd
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As we get down with disorder, we might spare a thought for Francois-Marie Arouet, better known as Voltaire; he died on this date in 1778. The Father of the Age of Reason, he produced works in almost every literary form: plays, poems, novels, essays, and historical and scientific works– more than 2,000 books and pamphlets (and more than 20,000 letters). He popularized Isaac Newton’s work in France by arranging a translation of Principia Mathematica to which he added his own commentary.
A social reformer, Voltaire used satire to criticize the intolerance, religious dogma, and oligopolistic privilege of his day, perhaps nowhere more sardonically than in Candide.
“We do not inherit the earth from our ancestors, we borrow it from our children”*…
… and the interest rate on that loan is rising.
There’s much discussion of what’s causing the sudden-feeling spike in prices that we’re experiencing: pandemic disruptions, nativist and protectionist policies, the over-taxing of over-optimized supply chains, and others. But Robinson Meyer argues that there’s another issue, an underlying cause, that’s not getting the attention it deserves… one that will likely be even harder to address…
Over the past year, U.S. consumer prices have risen 7 percent, their fastest rate in nearly four decades, frustrating households and tanking President Joe Biden’s approval rating. And no wonder. High inflation corrodes the basic machinery of the economy, unsettling consumers, troubling companies, and preventing everyone from making sturdy plans for the future…
For years, scientists and economists have warned that climate change could cause massive shortages of major commodities, such as wine, chocolate, and cereals. Financial regulators have cautioned against a “disorderly transition,” in which the world commits only haphazardly to leaving fossil fuels, so it does not invest enough in their zero-carbon replacements. In an economy as prosperous and powerful as America’s, those problems are likely to show up—at least at first—not as empty grocery shelves or bankrupt gas stations but as price increases.
That phenomenon, long hypothesized, may be starting to actually arrive. Over the past year, unprecedented weather disasters have caused the price of key commodities to spike, and a volatile oil-and-gas market has allowed Russia and Saudi Arabia to exert geopolitical force.
“This climate-change risk to the supply chain—it’s actually real. It is happening now,” Mohamed Kande, the U.S. and global advisory leader at the accounting firm PwC, told me.
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How to respond to these problems? The U.S. government has one tool to slow down the great chase of inflation: Leash up its dollars. By raising the rate at which the federal government lends money to banks, the Federal Reserve makes it more expensive for businesses or consumers to take out loans themselves. This brings demand in the economy more in line with supply. It is like the king in our thought experiment deciding to buy back some of his gold coins.
But wait—is it always appropriate to focus on dollars? What if the problem was caused by too few goods? Worse, what if the economy lost the ability to produce goods over time, throwing off the dollars-to-goods ratio? Then what was once an adequate number of dollars will, through no fault of its own, become too many...
… if the climate scars on supply continue to grow, does the Federal Reserve have the right tools to manage? Stinson Dean, the lumber trader, is doubtful. “Raising interest rates will blunt demand for housing—no doubt. But if you blunt demand enough to bring lumber prices down, you’re destroying the economy,” Dean told me. “For us to have lower lumber prices, we can only build a million homes a year. Do you really want to do that?
“Raising rates,” he said, “doesn’t grow more trees.” Nor does it grow more coffee, end a drought, or bring certainty to the energy transition. And if our new era of climate-driven inflation takes hold, America will need more than higher interest rates to bring balance to supply and demand.
A provocative look at the tangled roots of our inflation, suggesting that “The World Isn’t Ready for Climate-Change-Driven Inflation,” from @yayitsrob in @TheAtlantic. Eminently worth reading in full. Via @sentiers.
* Native American proverb
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As we dig deeper, we might send carefully calculated birthday greetings to Frank Plumpton Ramsey; he was born on this date in 1903. A philosopher, mathematician, and economist, he made major contributions to all three fields before his death (at the age of 26) on this date in 1930.
While he is probably best remembered as a mathematician and logician and as Wittgenstein’s friend and translator, he wrote three paper in economics: on subjective probability and utility (a response to Keynes, 1926), on optimal taxation (1927, described by Joseph E. Stiglitz as “a landmark in the economics of public finance”), and optimal economic growth (1928; hailed by Keynes as “”one of the most remarkable contributions to mathematical economics ever made”). The economist Paul Samuelson described them in 1970 as “three great legacies – legacies that were for the most part mere by-products of his major interest in the foundations of mathematics and knowledge.”
For more on Ramsey and his thought, see “One of the Great Intellects of His Time,” “The Man Who Thought Too Fast,” and Ramsey’s entry in the Stanford Encyclopedia of Philosophy.
“No structure, even an artificial one, enjoys the process of entropy. It is the ultimate fate of everything, and everything resists it.”*…
A 19th-century thought experiment that motivates physicists– and information scientists– still…
The universe bets on disorder. Imagine, for example, dropping a thimbleful of red dye into a swimming pool. All of those dye molecules are going to slowly spread throughout the water.
Physicists quantify this tendency to spread by counting the number of possible ways the dye molecules can be arranged. There’s one possible state where the molecules are crowded into the thimble. There’s another where, say, the molecules settle in a tidy clump at the pool’s bottom. But there are uncountable billions of permutations where the molecules spread out in different ways throughout the water. If the universe chooses from all the possible states at random, you can bet that it’s going to end up with one of the vast set of disordered possibilities.
Seen in this way, the inexorable rise in entropy, or disorder, as quantified by the second law of thermodynamics, takes on an almost mathematical certainty. So of course physicists are constantly trying to break it.
One almost did. A thought experiment devised by the Scottish physicist James Clerk Maxwell in 1867 stumped scientists for 115 years. And even after a solution was found, physicists have continued to use “Maxwell’s demon” to push the laws of the universe to their limits…
A thorny thought experiment has been turned into a real experiment—one that physicists use to probe the physics of information: “How Maxwell’s Demon Continues to Startle Scientists,” from Jonathan O’Callaghan (@Astro_Jonny)
* Philip K. Dick
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As we reconsider the random, we might send carefully-calculated birthday greetings to Félix Édouard Justin Émile Borel; he was born on this date in 1871. A mathematician (and politician, who served as French Minister of the Navy), he is remembered for his foundational work in measure theory and probability. He published a number of research papers on game theory and was the first to define games of strategy.
But Borel may be best remembered for a thought experiment he introduced in one of his books, proposing that a monkey hitting keys at random on a typewriter keyboard will – with absolute certainty – eventually type every book in France’s Bibliothèque Nationale de France. This is now popularly known as the infinite monkey theorem.
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