### Numbers big and small, and the anthropic principle

I’d like to share a couple of spooky things about numbers which a friend told me about years ago.

In our laws and formulae in science, we need some constants to make them work. Things like the speed of light, the mass of an electron, Avogadro’s number, and so on. These are some of the numbers that a professor puts on a test’s formula sheet so that students do not have to memorize them.

There would be no value in memorizing them as they are all over the place, like they had been pulled out of a hat at random: as examples, the speed of light is 2.998 × 10^{8} m/s, the mass of an electron is 9.11 × 10^{-31} kg, and Avogadro’s number is 6.022 × 10^{23}.

You can see what I mean from just these three numbers, and there are a huge number of other physical constants: who would want to memorize these numbers, and what value would that have?

Here is the spooky thing, or rather part one of the spooky thing. Take all the leading digits of all the physical constants, not just the three I mentioned (where the leading digits are 2, 9, and 6, respectively), but all of them from a book of constants. Now, see how many are 1s, how many are 2s, and so on. The natural thing to expect, as I said, is that these numbers are all over the place, so one might think the leading digits to be equally distributed: 11% are 1s, 11% are 2s, 11% are 9s. Instead they are mostly 1s, 2s, and 3s (about 60 per cent), with a small number being 7s, 8s, and 9s (about 16 per cent). This is called Benford’s law.

Here is why. The speed of light is 2.998 × 10^{8} m/s, using the metric system. In the old English system it is 6.706 × 10^{8} mph. So, depending on the units, the leading digit will change from 2 to 6. If we say the distribution of the numbers cannot depend on the units—a more rigorous way to require the numbers are all over the place, and are pulled out of a hat at random—one can show all the numbers sit on a scale invariant distribution, and we recover Benford’s law. I won’t do the algebra here, but it is easy to show.

This is a handy result, which is true for a great deal of statistical data, not just physical constants. As such, it is used to check statistics that are suspect for some reason: if the statistics do not satisfy Benford’s law, it suggests someone may have created and inserted data by hand, fine-tuning that data to force some desired result, rather than collected real data from a real experiment. This was one of the ways used to look at questionable data from Enron.

Which, finally, brings me to the second half of the spooky thing, the anthropic principle. The anthropic principle argues that physical constants are fine-tuned so that the universe is just right for life—or, in the words of Goldilocks, not too hot, and not too cold. For example, if the speed of light was a little faster or a little slower, or if the mass of an electron was a little bit larger or smaller, it is argued that galaxies, stars, planets—and you and me—might not be here.

The anthropic principle is controversial. We held a public science debate on the anthropic principle a couple of years ago at McGill, sponsored by Lorne Trottier. Every year we have a public debate sponsored by Lorne. It is a great way to bring real science to the public in a way that was common in the 19^{th} century but is relatively uncommon now, as public science has become more and more the domain of popularizers.

The debate on the anthropic principle was very well attended, and very stimulating. I encourage you to check out the links, including a **webcast of the debate**.

Also, remember what I said about Benford’s law, and how handy it is for checking if someone has fine-tuned their data! Whatever value the anthropic principle may have, the distribution of physical constants shows no evidence of fine tuning, just the opposite: they are consistent, in a not-very-obvious-way, with being pulled out of a hat at random.

PS: the fifth annual Lorne Trottier Public Science Symposium takes place on November 19, 2009, and this year’s topic is “Avoiding dangerous climate change: Geo-engineering or mitigation?” I hope to see you there! Find out more on the **Trottier Symposium website.**

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– Victor Chisholm, Moderator