Normativity in Logic

Incompleteness — the absence of alternative natural numbers — can be ascribed to a ready-made normativity, inducing a rigid departure syntax/semantics. Geometry of Interaction, set in the noncommutative universe of von Neumann algebras, makes normative assumptions explicit, thus rending possible their internalisation, a possible way out from the semantic aporia. As an illustration, we define an alternative A model B: logspace integers. Foundational questions are cognitive: A What can we know? B, A How do we know? B, A What are our preconceptions? B. Thus, the open problems in algorithmic complexity, which address the efficiency of computation, are foundational, although far from the stereotyped problem of consistency. Foundations can — and must — question everything including questions, the only limit to this interrogation being efficiency. And the rigid departure between syntax and semantics — which is only appropriate in A usual B situations — is the first dogma that foundations should put into question. Consider natural numbers: several systems, yet only one model, the standard N. This extreme poverty (incompleteness) of the semantic universe is a by-product of normativity. Indeed, the question A what is standard? B is booby-trapped: it induces a meaningless dichotomy between standard, normal, integers and non-standard, abnormal ones. There is no way to escape this aporia while sticking to the rigid distinction between syntax and semantics, where subject and object — both clearly individuated — relate according to a fixed protocol. This normativity is ready-made, i.e., hidden and external: it goes without saying, moreover it proceeds from the sky. By