Wednesday, June 02, 2004

The Strong Law of Truly Misquoted Mathematics

On the one hand it is nice when technical terms from the world of math/CS sneak into common usage. On the other hand it can be rather grating. If I had a dime for every time someone used "exponential growth" to mean "something big", I could probably buy a single share of Google stock...

This excerpt from a Slate article on cell phone usage is another example:

...the cell-phone growth projections are based on premises that are more redolent of 1990s-era techno-enthusiasm than the current decade's putative realism. Some savvy observers of the telecom world detect the beginnings of a bubble. "At some point, we're going to run into the law of large numbers and it's going to be a nasty setup," said Cody Willard, general partner at CL Willard Capital and a columnist for TheStreet.com. "I just don't think we're quite there yet."

I have to ask: exactly what law is Mr. Willard scared of running into ? Let's consider the options:

The Weak Law of Large numbers:
The probability that the sum of i.i.d variables over the same distribution differs from the mean is arbitrarily small.

or even:
The Strong Law of Large Numbers
Not only is the probability of deviation small, the actual deviation of the sum from the mean is arbitrarily small.

Somehow I don't see how either of these fit into an analysis of the global cell phone market. But wait!, there are some more intriguing possibilities.

The Frivolous Theorem Of Arithmetic
Almost all natural numbers are really large

Possible, but Mr. Willard did after all quote a law of large numbers. We should grant him at least some degree of precision in his statements.

Strong Law of Small Numbers
There aren't enough small numbers to meet the many demands made of them..

This seems somewhat more plausible: after all, soon we might run out of small numbers to describe cell phone usage, at which point the market might collapse.

But I think the real answer lies in a deeper analysis of his prognostication. Note if you will the use of the phrase 'nasty setup'. This clearly indicates his invocation of:

The Law of Truly Large Numbers
With a large enough sample, any outrageous thing is likely to happen

This indeed settles it. As cell phone usage increases, something bad is going to happen. Now we know why they get paid the big bucks in Wall Street.