The Science of Discworld IV Judgement Da - By Terry Pratchett, Ian Stewart Page 0,127

is then argued that the probability of all thirty parameters being in the right range is 1/10 raised to the power 30. This is 10-30, one part in a nonillion (ten billion billion billion). It is so ridiculously small that there is absolutely no serious prospect of it happening by chance. This calculation is the origin of the ‘knife edge’ image.

It is also complete nonsense.

It’s like starting at Centrepoint, in the middle of London, and going a few metres westwards along New Oxford Street, a few metres northwards up Tottenham Court Road, and imagining you’ve covered the whole of London. You haven’t even explored a few metres in a north-westerly direction, let alone anything further away. Mathematically, what is being explored by each change to a single parameter is a tiny interval along an axis in parameter space. When you multiply the associated probabilities together, you are exploring a tiny box whose sides correspond to the changes made to individual parameters – without considering changing any of the others. The car example shows how silly this type of calculation is.

Even using the constants for this universe, we can’t deduce the structure of something as apparently simple as a helium atom from the laws of physics, let alone a bacterium or a human being. Our understanding of everything more complex than hydrogen relies on clever approximations, refined by comparison with actual observations. But when we start thinking about other universes, we don’t have any observations to compare with; we must rely on the mathematical consequences of our equations. For anything interesting, even helium, we can’t do the sums. So we take short cuts, and rule out particular structures, such as stars or atoms, on various debatable grounds.

However, what such calculations actually rule out (even when they’re correct) are stars just like those in this universe and atoms just like those in this universe. Which isn’t quite the point when we’re discussing a different universe. What other structures could exist? Could they be complex enough to constitute a form of life? The mathematics of complex systems shows that simple rules can lead to astonishingly complex behaviour. Such systems typically behave in many different interesting ways, but not in just one interesting way. They don’t just sit there being dull and boring, except for one special ‘finely tuned’ set of constants where all hell breaks loose.

Stenger gives an instructive example of the fallacy of varying parameters one at a time. He works with just two: nuclear efficiency and the fine structure constant.

Nuclear efficiency is the fraction of the mass of a helium atom that is greater than the combined masses of two protons and two neutrons. This is important because the helium nucleus consists of just that combination. Add two electrons, and you’re done. In our universe, this parameter has the value 0.007. It can be interpreted as how sticky the glue that holds the nucleus together is, so its value affects whether helium (and other small atoms like hydrogen and deuterium) can exist. Without any of these atoms, stars could not be powered by nuclear fusion, so this is a vital parameter for life. Calculations that vary only this parameter, keeping all others fixed, show that it has to lie between 0.006 and 0.008 for fusion-powered stars to be feasible. If it is less than 0.006, deuterium’s two positively charged protons can push each other apart despite the glue. If it is more than 0.008, protons stick together, so there would be no free protons. Since a free proton is the nucleus of hydrogen, that means no hydrogen.

The fine structure constant determines the strength of electromagnetic forces. Its value in our universe is 0.007. Similar calculations show that it has to lie in the range from 0.006 to 0.008. (It seems to be coincidence that these values are essentially the same as those for nuclear efficiency. They’re not exactly equal.)

Does this mean that in any universe with fusion-powered stars, both the nuclear efficiency and the fine structure constant must lie in the range from 0.006 to 0.008? Not at all. Changes to the fine structure constant can compensate for the changes to the nuclear efficiency. If their ratio is approximately 1, that is, if they have similar values, then the required atoms can exist and are stable. We can make the nuclear efficiency much larger, well outside the tiny range from 0.006 to 0.008, provided we also make the fine structure constant larger. The same goes if we make one of them