Square root of the population

27 years ago I wrote,

There are some 1010 people in the world. Roughly speaking, I am among the top 105 most successful. Accounting for this improbably high position in the hierarchy is a troubling problem for me. A tantalizing idea suggests itself: can one argue somehow that in a random population of n people, the expected position to find an introspective person of my sort is on the order of √n from the top?

Well, here’s an argument that gets exactly that answer. The person in position #1 will be pretty interested in his or her good fortune. Suppose we imagine that person #2 is half as interested, person #3 is 1/3 as interested, and so on: essentially, each person’s interest in their luck is proportional to their luck. Then the total amount of human interest in positional luck, integrated over the whole of humanity, is about log n.

Now pick a human being at random, weighted by their interest in their luck. The middle position is ½log n, which is the logarithm of √n. So there you have it. If I am person number 105, then half the population’s interest in their luck is to be found in people higher up than I am, and half in people lower down, making me quite entirely typical. Throwing in a few trillion dogs, squirrels, and mosquitoes, with their lower levels of introspection, won’t change the result.

[12 April 2011]

Exponentials give us randomness, and certainty

Here is a curious symmetry. To achieve randomness in science or technology, our best strategy is exponentials. You can toss a coin, but the outcome isn’t so random because it is sensitive only algebraically to the details of the throw. For truer randomness you need a chaotic system with exponential sensitivities, like a pinball machine or the Lorenz equations. Run such a system for a moment and your randomness might be 99%. If that’s not enough, run it a little longer to get 99.99%. The point is that with each new step, your knowledge about the system shrinks by a constant factor, soon reaching zero for practical purposes.

And to achieve certainty, our best strategy is exponentials again! At the level of fundamental physics, anything can happen because of quantum tunnelling. But some things “never” happen in practice, such as the radioactive decay of an iron-56 atom. Why? Because the frequency of quantum events shrinks exponentially with the width of a potential barrier. Thickening up that barrier in a physics experiment is like adding another level of error correction in an electronic circuit or taking another step of a random algorithm or making another compressed sensing measurement. With each new step, your uncertainty about the system shrinks by a constant factor, soon reaching zero for practical purposes.

[1 September 2011]

Protecting one’s woman

Men being bigger and stronger than women, I was musing how in the old days, a man used his strength to protect his woman. However evolved you may be, it’s hard not to feel a tinge of pride in being associated with this classic role.

These fine feelings were dimmed a little by the thought that, realistically speaking, probably about 80% of that protection was against other men.

[15 June 2018]