A science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life. The theory of prime numbers satisfies no such criteria. Those who pursue it will, if they are wise, make no attempt to justify their interest in a subject so trivial and so remote, and will console themselves with the thought that the greatest mathematicians of all ages have found in it a mysterious attraction impossible to resist.-Hardy
If I were to awaken after having slept a thousand years, my first question would be: has the Riemann Hypothesis been proven?-Hilbert
Would human civilisation be possible without the integers? It is worth considering that they have been indispensable in the areas where they have found important applications. These applications range from accounting and finance to Quantum Mechanics and advanced encryption methods for secure communications. However, more advanced applications will require a much deeper understanding of their atomic constituents, the prime numbers, which we still understand very poorly. In particular, this will require a complete understanding of the laws governing the distribution of primes.
At present, it is essential to be honest about the current state of scientific knowledge. If we marshalled the world’s best mathematicians with the task of inventing the distribution of primes they would be incapable of doing such a thing. There is quite a big difference between the ability to use the integers for basic financial applications and the ability to invent them. In fact, as the integers provide the foundation for all of math and physics the laws governing the distribution of primes actually precede and govern the laws of physics. The pair-correlation conjecture formulated by Montgomery and Dyson is not an accident. It is for this reason that I suspect that creating the integers will require nothing less than building a computer capable of simulating all of physics. Perhaps we could call this Plato’s computer as it would be capable of simulating all Platonic forms that all material things implicitly refer to.
Incidentally, my own personal interest in the distribution of primes began around mid-February earlier this year a couple days after considering the notion of pre-established Harmony developed by Leibniz. In my attempt to reconcile the perceived randomness in the distribution of primes with the Theodicy of Leibniz, the Monte Carlo Hypothesis occurred to me. This Hypothesis was partly motivated as a scientific investigation into the origins of randomness in the natural world as well as an analysis of mathematical signatures of the Simulation Hypothesis.
I’d like to share one more insight that recently occurred to me. I suspect that a partial explanation fo why many excellent mathematicians have an irresistible attraction to the distribution of primes lies partly in information-theoretic approaches to human cognition. If science refers to an objective reality then this analysis will tell us more about humans than the distribution of primes. But, if the Universe is observer-dependent as some theories of Quantum measurement suggest then such an analysis would provide us with new insights into the distribution of primes.
Might there be a higher purpose in life than to attain a complete understanding of the distribution of primes? What I can say for certain is that the distribution of prime numbers contains the answer to questions which we are not yet able to ask.