mental-math   40

Introduction to Scaling Laws
http://galileo.phys.virginia.edu/classes/304/scaling.pdf

Galileo’s Discovery of Scaling Laws: https://www.mtholyoke.edu/~mpeterso/classes/galileo/scaling8.pdf
Days 1 and 2 of Two New Sciences

An example of such an insight is “the surface of a small solid is comparatively greater than that of a large one” because the surface goes like the square of a linear dimension, but the volume goes like the cube.5 Thus as one scales down macroscopic objects, forces on their surfaces like viscous drag become relatively more important, and bulk forces like weight become relatively less important. Galileo uses this idea on the First Day in the context of resistance in free fall, as an explanation for why similar objects of different size do not fall exactly together, but the smaller one lags behind.
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august 2017 by nhaliday
Kelly criterion - Wikipedia
In probability theory and intertemporal portfolio choice, the Kelly criterion, Kelly strategy, Kelly formula, or Kelly bet, is a formula used to determine the optimal size of a series of bets. In most gambling scenarios, and some investing scenarios under some simplifying assumptions, the Kelly strategy will do better than any essentially different strategy in the long run (that is, over a span of time in which the observed fraction of bets that are successful equals the probability that any given bet will be successful). It was described by J. L. Kelly, Jr, a researcher at Bell Labs, in 1956.[1] The practical use of the formula has been demonstrated.[2][3][4]

The Kelly Criterion is to bet a predetermined fraction of assets and can be counterintuitive. In one study,[5][6] each participant was given \$25 and asked to bet on a coin that would land heads 60% of the time. Participants had 30 minutes to play, so could place about 300 bets, and the prizes were capped at \$250. Behavior was far from optimal. "Remarkably, 28% of the participants went bust, and the average payout was just \$91. Only 21% of the participants reached the maximum. 18 of the 61 participants bet everything on one toss, while two-thirds gambled on tails at some stage in the experiment." Using the Kelly criterion and based on the odds in the experiment, the right approach would be to bet 20% of the pot on each throw (see first example in Statement below). If losing, the size of the bet gets cut; if winning, the stake increases.
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august 2017 by nhaliday
The Flynn effect for verbal and visuospatial short-term and working memory: A cross-temporal meta-analysis
Specifically, the Flynn effect was found for forward digit span (r = 0.12, p < 0.01) and forward Corsi block span (r = 0.10, p < 0.01). Moreover, an anti-Flynn effect was found for backward digit span (r = − 0.06, p < 0.01) and for backward Corsi block span (r = − 0.17, p < 0.01). Overall, the results support co-occurrence theories that predict simultaneous secular gains in specialized abilities and declines in g. The causes of the differential trajectories are further discussed.

http://www.unz.com/jthompson/working-memory-bombshell/
https://www.newscientist.com/article/2146752-we-seem-to-be-getting-stupider-and-population-ageing-may-be-why/
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august 2017 by nhaliday
A sense of where you are | West Hunter
Nobody at the Times noticed it at first. I don’t know that they ever did notice it by themselves- likely some reader brought it to their attention. But this happens all the time, because very few people have a picture of the world in their head that includes any numbers. Mostly they don’t even have a rough idea of relative size.

In much the same way, back in the 1980s,lots of writers were talking about 90,000 women a year dying of anorexia nervosa, another impossible number. Then there was the great scare about 1,000,000 kids being kidnapped in the US each year – also impossible and wrong. Practically all the talking classes bought into it.

You might think that the people at the top are different – but with a few exceptions, they’re just as clueless.
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may 2017 by nhaliday
Learning mathematics in a visuospatial format: A randomized, controlled trial of mental abacus instruction
We asked whether MA improves students’ mathematical abilities, and whether expertise – which requires sustained practice of mental imagery – is driven by changes to basic cognitive capacities like working memory. MA students improved on arithmetic tasks relative to controls, but training was not associated with changes to basic cognitive abilities. Instead, differences in spatial working memory at the beginning of the study mediated MA learning. We conclude that MA expertise can be achieved by many children in standard classrooms and results from efficient use of pre-existing abilities.

Cohen’s d = .60 (95% CI: .30 - .89) for arithmetic, .24 (-.05 - .52) for WJ-III, and .28 (.00 - .57) for place value

This finding suggests that the development of MA expertise is mediated by children’s pre-existing cognitive abilities, and thus that MA may not be suitable for all K-12 classroom environments, especially in groups of children who have low spatial working memory or attentional capacities (which may have been the case in our study). Critically, this does not mean that MA expertise depends on unusually strong cognitive abilities. Perhaps because we studied children from relatively disadvantaged backgrounds, few Mental Abacus 21 children in our sample had SWM capacities comparable to those seen among typical children in the United States.
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march 2017 by nhaliday
Sir Ronald Aylmer Fisher | West Hunter
In 1930 he published The Genetical Theory of Natural Selection, which completed the fusion of Darwinian natural selection with Mendelian inheritance. James Crow said that it was ‘arguably the deepest and most influential book on evolution since Darwin’. In it, Fisher analyzed sexual selection, mimicry, and sex ratios, where he made some of the first arguments using game theory. The book touches on many other topics. As was the case with his other works, The Genetical Theory is a dense book, not easy for most people to understand. Fisher’s tendency to leave out mathematical steps that he deemed obvious (a leftover from his early training in mental mathematics) frustrates many readers.

The Genetical Theory is of particular interest to us because Fisher there lays out his ideas on how population size can speed up evolution. As we explain elsewhere, more individuals mean there will be more mutations, including favorable mutations, and so Fisher expected more rapid evolution in larger populations. This idea was originally suggested, in a nonmathematical way, in Darwin’s Origin of Species.

Although Fisher was fiercely loyal to friends and could be very charming, he had a quick temper and was a fine hater. The same uncompromising spirit that fostered his originality led to constant conflict with authority. He had a long conflict with Karl Pearson, who had also played an important part in the development of mathematical statistics. In this case, Pearson was more at fault, resisting the advent of a more talented competitor, as well as being an eminently hateable person in general. Over time Fisher also became increasing angry at Sewall Wright (another one of the founders of population genetics) due to scientific disagreements – and this was just wrong, because Wright was a sweetheart.

Fisher’s personality decreased his potential influence. He was not a school-builder, and was impatient with administrators. He expected to find some form of war-work in the Second World War, but his characteristics had alienated too many people, and thus his team dispersed to other jobs during the war. He returned to Rothamsted for the duration. This was a difficult time for him: his marriage disintegrated and his oldest son, an RAF pilot, was killed in the war.

...

Fisher’s ideas in genetics have taken an odd path. The Genetical Theory was not widely read, sold few copies, and has never been translated. Only gradually did its ideas find an audience. Of course, that audience included people like Bill Hamilton, the greatest mathematical biologist of the last half of the 20th century, who was strongly influenced by Fisher’s work. Hamilton said “By the time of my ultimate graduation,will I have understood all that is true in this book and will I get a First? I doubt it. In some ways some of us have overtaken Fisher; in many, however, this brilliant, daring man is still far in front.“

In fact, over the past generation, much of Fisher’s work has been neglected – in the sense that interest in population genetics has decreased (particularly interest in selection) and fewer students are exposed to his work in genetics in any way. Ernst Mayr didn’t even mention Fisher in his 1991 book One Long Argument: Charles Darwin and the Genesis of Modern Evolutionary Thought, while Stephen Jay Gould, in The Structure of Evolutionary Theory, gave Fisher 6 pages out of 1433. Of course Mayr and Gould were both complete chuckleheads.

Fisher’s work affords continuing insight, including important implications concerning human evolution that have emerged more than 50 years after his death. We strongly discourage other professionals from learning anything about his ideas.
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january 2017 by nhaliday
Shut Up And Guess - Less Wrong
At what confidence level do you guess? At what confidence level do you answer "don't know"?

I took several of these tests last month, and the first thing I did was some quick mental calculations. If I have zero knowledge of a question, my expected gain from answering is 50% probability of earning one point and 50% probability of losing one half point. Therefore, my expected gain from answering a question is .5(1)-.5(.5)= +.25 points. Compare this to an expected gain of zero from not answering the question at all. Therefore, I ought to guess on every question, even if I have zero knowledge. If I have some inkling, well, that's even better.

You look disappointed. This isn't a very exciting application of arcane Less Wrong knowledge. Anyone with basic math skills should be able to calculate that out, right?

I attend a pretty good university, and I'm in a postgraduate class where most of us have at least a bachelor's degree in a hard science, and a few have master's degrees. And yet, talking to my classmates in the cafeteria after the first test was finished, I started to realize I was the only person in the class who hadn't answered "don't know" to any questions.

even more interesting stories in the comments
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september 2016 by nhaliday

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