It is worth noting that it is not possible within algorithmic information theory to demonstrate that a particular object is incompressible so history could not have happened another way. No matter how many times you run history, the number theorists would have obtained the prime number theorem first both for this reason and the fact that any civilisation capable of developing computers would have probably developed number theory first. However, it is the number theorists that would have to wait for the development of theoretical computer science in order to obtain the correct interpretation of the prime number theorem. That is, the prime number theorem is a statement that prime numbers have a maximum entropy distribution and that prime encodings are algorithmically random.
Moreover, since it can’t be proven within algorithmic information theory the best we can do is perform rigorous numerical experiments. I think this presents a strong argument for experimental mathematics in general. There is however, one more twist and turn in this story.
The experimental verification of the Monte Carlo hypothesis has important consequences for the existence of computable prime formulas as well as the existence of proofs of the Riemann hypothesis if it is true.
- Aidan Rocke. https://github.com/AidanRocke/Monte_Carlo_hypothesis 2021.
- Aidan Rocke. The Monte Carlo Hypothesis, a scientific investigation into the origins of randomness in the natural world. 2021.
- Aidan Rocke. Determining the Monte Carlo constant. 2021.