Big Number Fallacy

posted Dec 29, 2003

I couldn't find this on the Internet, so I present to you the "Big Number Fallacy":

Big Number Fallacy: All big numbers are infinite. In particular, they are all larger then each other, smaller then each other, and the same as each other, randomly.

In mathematics, depending on your choice of axioms, infinity is a "number" that can be greater then itself, less then itself, and equal to itself at the same time. Its ability to perform this feat is fundamentally why it's not a number (hence the scarequotes); it does not act like a number, therefore it isn't one.

When thinking about large numbers, people often internally promote them to these infinities, and then reason with them poorly. For instance, consider the following:

There's a chance that the Earth could be hit with a meteor the size of the mountain. Wow, that's big! That could knock the Earth farther away from the sun and we'd all freeze to death, even if we survived the impact!

The Earth is "really big", the asteroid is "really big", therefore, they are roughly equal in mass.

In reality, the number obtained by dividing the mass of the Earth by the mass of such an asteroid itself qualifies for "really big" status. Such an impact would be disasterous to us surface dwellers, but it's not even close to large enough to cause what we'd consider a change to Earth's orbit.

Of course it will have an impact, but for the climate to change we'd certainly need many thousands of miles of change, and to "freeze to death" would need many millions of miles of change. Any single impact large enough to do this would splatter the Earth. (If you want to change the orbit of a planet without killing everyone on it, you need to use gravity from other planet-sized objects, which of course is itself a problem... if you can move those planets that way why not use that technique to move Earth directly? There are some answers to that question (see "A World out of Time" by Niven), but I'm far enough off track.)

A quintillion is a million times larger then a trillion, and usually, factors of a million matter. Those are all Big Numbers, but don't fall prey to the Big Number fallacy; either do the math, or resist trying to guess how the Big Numbers will interact.

(Incidentally, pet peeve: Sci-fi movies showing the Earth blowing up. At planetary scales, everything is a liquid. "Blowing up the Earth" is more like blowing up a raindrop with a tiny explosive, not blowing up a rock with a tiny explosive. Someday, I hope to see a movie where they blow up a planet correctly. Be a great supercomputer simulation, the kind you run to get you on the news when you build The World's Greatest Super Computer. Of course, the amount of energy needed to blow up the Earth is itself a bona fide Big Number. (Give the linked site a chance; it has an unusual premise but it's high quality work.))