A CLASSICAL-MODAL INTERPRETATION OF SMOOTH INFINITESIMAL ANALYSIS

Abstract Smooth Infinitesimal Analysis (SIA) is a remarkable late twentieth-century theory of analysis. It is based on nilsquare infinitesimals, and does not rely on limits. SIA poses a challenge of motivating its use of intuitionistic logic beyond merely avoiding inconsistency. The classical-modal account(s) provided here attempt to do just that. The key is to treat the identity of an arbitrary nilsquare, e , in relation to 0 or any other nilsquare, as objectually vague or indeterminate—pace a famous argument of Evans [10]. Thus, we interpret the necessity operator of classical modal logic as “determinateness” in truth-value, naturally understood to satisfy the modal system, S4 (the accessibility relation on worlds being reflexive and transitive). Then, appealing to the translation due to Gödel et al., and its proof-theoretic faithfulness (“mirroring theorem”), we obtain a core classical-modal interpretation of SIA. Next we observe a close connection with Kripke semantics for intuitionistic logic. However, to avoid contradicting SIA’s non-classical treatment of identity relating nilsquares, we translate “=” with a non-logical surrogate, ‘ E ,’ with requisite properties. We then take up the interesting challenge of adding new axioms to the core CM interpretation. Two mutually incompatible ones are considered: one being the positive stability of identity and the other being a kind of necessity of indeterminate identity (among nilsquares). Consistency of the former is immediate, but the proof of consistency of the latter is a new result. Finally, we consider moving from CM to a three-valued, semi-classical framework, SCM, based on the strong Kleene axioms. This provides a way of expressing “indeterminacy” in the semantics of the logic, arguably improving on our CM. SCM is also proof-theoretically faithful, and the extensions by either of the new axioms are consistent.