1. Consider the activity of coin-flipping.
2. In the beginning of a sequence of coin-flips, relatively long runs of heads and tails are possible, but if the coin is flipped numerous times the ratio of heads to tails converges on 1/2 with an error of a certain square-root size.
3. Why should this be? Why should the error have the size that it does?
4. In the beginning scene of Tom Stoppard’s play “Rosencrantz and Guildenstern are Dead” in which Rosencrantz bets on heads and wins 92 times in a row, which prompts Guildenstern to say that they may be “within un, sub or supernatural forces”.
5. The answer starts with consciousness. Something about consciousness that is not mysterious, is the need for a distinction between consciousness and its content, between the observer and the observed.
6. If the separation between the two were to reach certain extremes, then any sense of pattern would disappear. We find for example that if we stand very close to, or very far from a landscape painting, we will be unable to discern the landscape. If there is zero distance between observer and observed, there is a disappearance of any pattern whatever.
7. This thought gives rise to the notion of randomness as a loss of arithmetic continuity, and there are two directions in which such a loss can occur – toward more separation and dis-uniformity or towards less.
8. In 1974, H L Montogomery discovered that the spacings of the imaginary parts of the non – trivial zeros of the Riemann zeta function are identical to those of phenomena having a distribution governed by the Gaussian Unitary Ensemble or GUE).
9. The size of the raindrops in a steady rainfall are happy either to be heaped up upon each other or wildly separated, but the energy levels of the nuclei of atoms and the imaginary parts of the non-trivial zeros of the zeta function dislike these extremes.
10. The the size of raindrops have a “Poisson” distribution, while the energy-levels and the zeros have a distribution have as “Gaussian”‘ distribution, the one is toward greater randomness, the other is towards less.
11. The eigenvalues of a matrix represent points at which one unit gives way to another, and the spacing between the eigenvalues of matrices governing phenomena as diverse as the energy levels of the nuclei of atoms and the waiting times between buses share this similarity, as if the uniformity required to count these things is underlain by a particular form of dis-uniformity.
12. Since the non-trivial zeros of the zeta function can be used to predict the locations of the prime numbers in the number line, this striking similarity -known as the Montogomery-Odlyzco Law- can be seen as evidence of a coming new physics, a physics having to do with the connection between the distribution of the primes and the natural world.
13. The central mathematical principle behind this new physics is something known as the Riemann Hypothesis, according to which the behaviour of the non-trivial zeros of the zeta function -and therefore the distribution of the primes- are strictly inter-related to ensure that classical phenomena do not depart in either a positive or the negative direction from the randomness of the coin-flip.
14. By subjecting the Mertens function to the re-arrangements depicted, we see that the Riemann Hypothesis says that error between the count of the primes and the primes is no more nor less random than a series of coin-flips.
15. Seeing that arithmetic progressions are associated to wave-forms, and that if and only if the associated wave-form has the randomness of the wave-form associated to a series of coin-flips does the progression involve a potentially infinite number of primes, it follows that the falsity of the hypothesis implies that there are a strictly number of primes.
16. Since the Riemann Hypothesis imposes the necessary and sufficient conditions of arithmetic continuity, and therefore on consciousness, it must be the governing principle of the physical universe generally.
17. It is therefore the principle that sets the masses of particles, and gives all the parameters in nature that have been set only from experiment.
18. Behind this universe -its means of proof tells us- are worlds that are not arithmetically continuous, and in which consciousness in the purely quantificational sense championed by scientists and mathematicians doesn’t exist.
19. In the same way that the painter creates the illusion of a representational image out of abstract brushstrokes, a supremely intelligent force behind the universe has created the illusion of arithmetic continuity out of proto-arithmetic chaos.