Turtles All the Way Down // 155

Graph showing in how many ways (vertical axis) a given even

number (horizontal axis) can be expressed as a sum of two

primes. The lowest points in the graph move upwards as we go

from left to right, indicating that there are many ways to achieve

this. However, for all we know an occasional point might fall on

the horizontal axis. Just one such point would disprove the

Goldbach Conjecture.

In 1923 Godfrey Hardy and John Littlewood obtained a heuristic formula ­ one that they could not prove rigorously, but looked plausible ­ for the number of different ways to write a given even integer as a sum of two primes. This formula, which agrees with numerical evidence, indicates that when the number gets large, there are many ways to write it as a sum of two primes. Therefore we may expect the smallest of the two primes to be

¨ relatively tiny. In 2001 Jorg Richstein observed that for numbers up to 1014, the smaller prime is at most 5,569, and this occurs for

389;965;026;819;938 ¼ 5;569 þ 389;965;814;369

........................................... Turtles All the Way Down Infinity is a slippery idea. People talk fairly casually of `eternity' ­ an infinite period of time. According to the Big Bang theory, the universe came into being about 13 billion years ago. Not only 156 // Turtles All the Way Down

was there no universe before then ­ there was no `before' before then.* Some people worry about that, and most of them seem much happier with the idea that the universe `has always existed'. That is, its past has already been infinitely long.

This alternative seems to solve the difficult question of the origin of the universe, by denying that it ever had an origin. If something has always been here, it's silly to ask why it's here now. Isn't it?

Probably. But that still doesn't explain why it's always been here.

This can be a difficult point to grasp. To bring it into perspective, let me compare it with a rather different proposal. There is an amusing (and very likely true) tale that a famous scientist ­ Stephen Hawking is often mentioned because he told the story in A Brief History of Time ­ was giving a lecture about the universe, and a lady in the audience pointed out that the Earth floats in space because it rests on the back of four elephants, which in turn rest on the back of a turtle.

`Ah, but what supports the turtle?' the scientist asked.

`Don't be silly,' she said. `It's turtles all the way down!'

Turtles all the

way down.

* Some cosmologists now think that there could have been

something before the Big Bang after all ­ our universe may be

part of a `multiverse' in which individual universes could come

into existence or fade away again. The theory is nice, but it's

difficult to find any way to test it.

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