Does resolving PvNP require a paradigm shift?

Abstract

I shall argue that a resolution of the PvNP problem requires building an iff bridge between the domain of provability and that of computability. The former concerns how a human intelligence decides the truth of number-theoretic relations, and is formalised by the first-order Peano Arithmetic PA following Dededekind's axiomatisation of Peano's Postulates. The latter concerns how a human intelligence computes the values of number-theoretic functions, and is formalised by the operations of a Turing Machine following Turing's analysis of computable functions. I shall show that such a bridge requires objective definitions of both an `algorithmic' interpretation of PA, and an `instantiational' interpretation of PA. I shall show that both interpretations are implicit in the definition of the subjectively defined `standard' interpretation of PA. However the existence of, and distinction between, the two objectively definable interpretations---and the fact that the former is sound whilst the latter is not---is obscured by the extraneous presumption under the `standard' interpretation of PA that Aristotle's particularisation must hold over the structure N of the natural numbers. I shall argue that recognising the falseness of this belief awaits a paradigm shift in our perception of the application of Tarski's analysis (of the concept of truth in the languages of the deductive sciences) to the `standard' interpretation of PA. I shall then show that an arithmetical formula [F] is PA-provable if, and only if, [F] interprets as true under an algorithmic interpretation of PA. I shall finally show how it then follows from Goedel's construction of a formally `undecidable' arithmetical proposition that there is a Halting-type PA formula which---by Tarski's definitions---is algorithmically verifiable as true, but not algorithmically computable as true, under a sound interpretation of PA.