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 “model”: logspace integers. Foundational questions are cognitive: “What can we know?”, “How do we know?”, “What are our preconceptions?”. 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 “usual” 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 “what is standard?” 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 making normative requirements explicit, by internalising them as parts of the “semantic”, objective, universe, we produce new “models” for natural numbers. In some sense, normativity appears as a mobile curtain separating the object from the subject; by the way, this would not be the only mobile curtain of logic, think of the departure between sets and proper classes.