Friday, August 5, 2016

A few fascinating laws and paradoxes

I recently spent an evening discussing the time-reversibility in Newtonian mechanics through the medium of 140 character tweets, after being introduced to one of my favourite things: a new paradox (hat tip to @MikeBenchCapon). This reminded me that there can be no excuse to be bored in this day and age when you can spend happy hours perusing the lists of eponymous laws and of paradoxes on Wikipedia. Here are a few of my favourites eponymous laws from those lists and elsewhere:

Benford's law: on the power-law distribution of specific digits in naturally occurring statistics.  The most commonly quoted part of the law is that about 30% of all statistics will start with the digit 1, compared to a naive expectation of around 11%. This law was used to show that Iran had been fabricating data relating to its nuclear program, since the digits in the data did not follow Benford's law. My favourite aspect of the law is that it can be derived from the assumption that if there does exist a distribution for the digits, it must be independent of the numerical basis used to represent the statistics.

Baumol's cost disease: why the cost of doctors, teachers and other service professionals increases over time. The efficiency of manufacturing has historically progressed faster than service sector occupations such as health care and education, through mechanisation. Instead of raising the salaries of manufacturing workers faster than service workers, all salaries tend to grow at roughly the same rate. As a result, labour-intensive industries become more costly over time relative to the price of manufactured goods. Expect tuition fees to carry on rising.

Goodhart's law: why you can't measure how well an intelligent system performs if you reward it for that performance (see also Campbell's law). Academics will be familiar with the gaming of league tables and the UK Research Excellence Framework by their institutions. When a body such as the government decides on metrics as a proxy to measure performance, and then rewards those who perform well by these measures, individuals choose to target the measures rather than genuinely improving overall performance. Hence we get teachers teaching-to-test, universities gaming the REF, scientists prioritising citations over true advances, and hospitals playing games with patient waiting times.

And of course Stigler's law of eponymy, which states that these laws were probably not named after the people who discovered them first. Stigler was, of course, not the first to propose this.

Here are a few of my favourites paradoxes, along with a rating for how genuinely paradoxical they seem to me:

Berkson's paradox: why the best-looking people you date have the worst personalities. While beautiful people may be no more or less pleasant in the population as a whole, you will let a bad personality slide for a beautiful mate, or date someone below your usual standards of physical beauty if they have a sparkling wit. As a result, in the group of people you date there will be an inverse correlation between beauty and personality. Paradox rating 1/10

The friendship paradox: why your friends probably are more successful and have more friends than you do. It is a simple result of networkm theory that you are most likely to be friends with people who have lots of friends, since they have more friendship links available. This means that a typical person is connected to people who have more friends than they do (while a few individuals are connected to lots of people with fewer friends). A simple corrolary is that if more successful people have more friends, then your friends will, on average, be more successful than you. In science, this selection effect is why everyone you know seems to be doing better than you are - the better they are doing, the more likely you are to be aware of them. Paradox rating 3/10

The envelope paradox: how a simple game tests the bounds of probability theory. A game show host offers you two envelopes and tells you that one contains twice as much money as the other. You open one envelope and find it contains £10. The other must contain either £5 or £20, with an average of £12.5. When the host offers to let you switch it seems that you should. But that choice would have been the same if you had never opened the envelope. The next time you don't even bother to open the envelope before switching, but now the same logic applies to the new envelope, making you switch back and forth forever. What has gone wrong? Paradox rating 7/10

Norton's dome: Theoretical departure from causality in Newtonian physics. A point mass sits atop a radially-symmetric, frictionless dome, with no force acting on it. After some arbitrary amount of time it begins to move spontaneously and rolls down the side of the dome. Its motion nonetheless obeys Newton's laws at all times, despite there being no way to predict, or even place probabilities on, the time elapsed before it starts to roll. Paradox rating 9/10

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