In my last post, I argued that intrinsic uncertainty between existing and non-existing objects should be incorporated into today’s mathematics, and I wondered about the best place to do so. Being a fundamental concept, intrinsic uncertainty should be part of the foundation of mathematics. It should be among the most basic mathematical concepts we have. And here comes the surprise: Intrinsic uncertainty is already there! We do not need to add intrinsic uncertainty. It has been present all along, but has largely gone unnoticed.
When we look at the root of mathematics, where it all begins, we find sets and numbers. The entire mathematical edifice, no matter how complex it gets, builds upon these elementary concepts. For instance, natural numbers allow us to do basic things like counting and arithmetic. They also play an important part in theoretical computer science, where they can represent computer programs or theorems. Mathematicians spent a great deal of time investigating natural numbers, and yet, I think natural numbers are still not fully understood.
The Peano axioms, named after the Italian mathematician Giuseppe Peano (1858-1932), are a typical axiomatization of natural numbers. They allow us to construct the natural numbers and derive theorems. Here are a few examples of these axioms:
- There exists a natural number 0 (The first Peano axiom).
- Every natural number n has a successor S(n) that is a natural number.
- The natural number 0 is not the successor of any natural number.
- Distinct natural numbers have distinct successors.
The first Peano axiom is special in the sense that it guarantees the existence of at least one natural number. This number serves as the root number from which all other numbers can be derived via a successor function. It is exactly here, where we encounter intrinsic uncertainty. With no way of knowing whether the root number really exists, the assertion made by the first Peano axiom is intrinsically uncertain. In fact, the first Peano axiom subsumes the uncertainty about existence and non-existence into one single axiom. Unfortunately, we cannot get rid of this uncertainty. Removing the first Peano axiom from the definition of natural numbers merely shifts the uncertainty from the root number to other numbers.
I suspect that Peano and his colleagues had hoped to get rid of intrinsic uncertainty once and for all when they introduced the first Peano axiom. What they overlooked is that the first Peano axiom only subsumes the uncertainty; it does not help us avoid uncertainty. In my recent paper “Computational Complexity on Signed Numbers,”, I exploit this subtlety to tackle the P/NP problem. If any of the readers knows about papers dealing with that intrinsic uncertainty aspect of natural numbers, I would appreciate hearing from you.