A Computational Anthropic Principle: Where is the Hardest Problem in the Multiverse?

As Hilbert himself expressed some decades earlier it would be equally desirable to make some progress axiomatizing physics. In some ways this is both an easier and harder problem: easier because one has the real world to guide one’s choice of axioms, harder because, unlike mathematics, it’s not at all clear what physics is. While axiomatizing physics is tricky and barely defined, the lack of a question is rarely a sufficient blockade to physicists searching for answers. So, what should we require from a set of physical axioms? Well, at the very least, since we conventionally describe the physical world in the language of mathematics, we should insist on Hilbert’s criteria as a minimum. Unfortunately, we famously run into trouble here since thanks to Gödel [2], Church [3] and Turing [4], we know that whatever axioms we choose to use in mathematics, we can expect neither consistency, completeness nor decidability. Having said that, however, there is a real sense in which the above desiderata are still, well, desirable for physics. We want our axioms (and thus our physical theories) not to be mutually inconsistent; we would like, in principle, to be able to make as many predictions (engender as many explanations) about the world as we can; and we would further hope that predictions are also easy in practice, i.e. we should be able to compute as many consequences of our theory as possible. It is this last issue, computation, that we will focus on here. We have given the Entscheidungsproblem (the decision problem) a slightly different meaning when we apply it to physics. Rather than asking if the veracity of every statement can be established, instead we ask what the most complex question that can be asked and answered is in a given physical theory. Moreover, we consider theories to be more desirable they permit calculations of greater complexity. But why? The why is perhaps best motivated by asserting that good physical theories have a further desideratum: