The Nature of Large Numbers

Or why we don't get them

There is nothing much more simple than whole numbers.  Add 1 over and over again, and you will (eventually) get any number you want.  Simple?  Well, so long as you are talking about small(ish) numbers.  Once the number gets big, our grasp on what it means weakens.

In the quaint and intelligent film "Local Hero", MacIntyre is trying to buy Ben's beach.  Ben is an eccentric beachcomber, MacIntyre represents an oil company, so you would expect the deal to be done.  MacIntyre certainly does. The only problem is that Ben doesn't want to sell.  So he takes a fistful of sand on the beach and says his price is one dollar for every grain of sand in his fist.  MacIntyre balks.  He has no idea, no grasp of how many grains of sand are in Ben's hand.  It could be millions. Billions even.  MacIntyre has no idea.  He understands millions of dollars, not fistfuls of sand, even when, as it turns out, there are at the most ten thousand grains of sand in Ben's fist.

So something is wrong with how we grasp large numbers.

A million dollars sounds a lot. As in a million one dollar bills or twenty thousand meals at an expensive restaurant.  But most of all, hard to get.  

A million litres of wine sounds like a large amount too.  Hard to store.  Hard to drink it all.  Hard to make.

A hundred years seems a long time, not to mention a thousand.  So the age of the oldest known galaxy, some thirteen billion years, seems so big it is almost meaningless, at least for a species whose average lifespan less than one hundred years.  And yet in other contexts, the national debt of countries (or banks, for that matter) the odd billion isn't all that much.

What emerges from these simple examples of 'large' numbers is that we think of them by means comparisons:  by comparison to a scale (eg, a human life span) or by comparing them to a sense of difficulty.  What we don't do is think of large numbers mathematically, or even logically.  We think of them psychologically: we are counting something specific, either years or pounds or litres of wine, we assign that count a psychological value.

If you think that trained physicists or economists are different you are probably wrong.  All they have done is applied a slightly larger scale.  A billion years, or dollars.  At which point, the age of the universe becomes a manageable quantity.

That leaves mathematicians.  But they are an odd bunch, so we can pretty safely ignore them.  That and the fact that they would probably have balked just as MacIntyre did when asked to estimate the number of grains of sand in his hand.

So let's look at the bare, naked numbers.  Because we want to look at large numbers, we need an easy notation to describe numbers, as otherwise it will become cumbersome to write the kind of numbers that get interesting.