Posts Tagged ‘analysis’
“The control of large numbers is possible, and like unto that of small numbers, if we subdivide them”*…
It’s always been intuitively obvious that we handle small numbers more easily than large ones. But the discovery that the brain has different systems for representing small and large numbers provokes new questions about memory, attention, and mathematics…
More than 150 years ago, the economist and philosopher William Stanley Jevons discovered something curious about the number 4. While musing about how the mind conceives of numbers, he tossed a handful of black beans into a cardboard box. Then, after a fleeting glance, he guessed how many there were, before counting them to record the true value. After more than 1,000 trials, he saw a clear pattern. When there were four or fewer beans in the box, he always guessed the right number. But for five beans or more, his quick estimations were often incorrect.
Jevons’ description of his self-experiment, published in Nature in 1871, set the “foundation of how we think about numbers,” said Steven Piantadosi, a professor of psychology and neuroscience at the University of California, Berkeley. It sparked a long-lasting and ongoing debate about why there seems to be a limit on the number of items we can accurately judge to be present in a set.
Now, a new study in Nature Human Behaviour has edged closer to an answer by taking an unprecedented look at how human brain cells fire when presented with certain quantities. Its findings suggest that the brain uses a combination of two mechanisms to judge how many objects it sees. One estimates quantities. The second sharpens the accuracy of those estimates — but only for small numbers…
Although the new study does not end the debate, the findings start to untangle the biological basis for how the brain judges quantities, which could inform bigger questions about memory, attention and even mathematics…
One, two, three, four… and more: “Why the Human Brain Perceives Small Numbers Better,” from @QuantaMagazine.
* Sun Tzu
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As we stew over scale, we might spare a thought for a man untroubled by larger (and more complicated) numbers, Émile Picard; he died on this date in 1941. A mathematician whose theories did much to advance research into analysis, algebraic geometry, and mechanics, he made his most important contributions in the field of analysis and analytic geometry. He used methods of successive approximation to show the existence of solutions of ordinary differential equations. Picard also applied analysis to the study of elasticity, heat, and electricity. He and Henri Poincaré have been described as the most distinguished French mathematicians in their time.
Indeed, Picard was elected the fifteenth member to occupy seat 1 of the Académie française in 1924.
“People overestimate what they can do in one year and underestimate what they can do in 10 years”*…
This is especially true, argue Sam Butler-Sloss and Kingsmill Bond of the Rocky Mountain Institute, when it comes to assessing our progress in addressing the challenges of climate change with renewable energy solutions…
The renewable revolution is advancing at remarkable speed. In fact, the speed of the renewable revolution has defied many leading energy commentators who have continuously underestimated its true trajectory. They have suffered from what statisticians call a systematic bias, that is, an error that consistently skews in one direction. Noise, or a random error, is inherent to forecasting; bias, however, requires a deeper explanation.
So why do so many intelligent people undersell the pace and dynamism of the renewable revolution? Leaving aside the inherent bias of those seeking to prop up the fossil fuel system in order to enjoy the largesse of its annual $2 trillion in rents, we identify eight deadly sins of the energy transition.
Whether intentional or unwitting, these eight general errors of perspective are holding back understanding, wasting time and capital, and fueling unproductive climate pessimism…
The renewable revolution is plainly gaining speed and impact. Read on to learn why are so many analysts so wrong about the pace and scale of innovation: “The Eight Deadly Sins of Analyzing the Energy Transition,” from @SamButl3r and @KingsmillBond at @RockyMtnInst. (TotH to friend MZ)
See also: “When Idiot Savants Do Climate Economics.”
* Bill Gates
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As we contemplate compounding, we might recall that it was on this date in 1896 that Nikola Tesla and Westinghouse Electric achieved the first long-distance transmission of hydroelectricity: from the Niagara Falls Power Company to Buffalo, N.Y., 26 miles away.
“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.
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