A Physical Quantum Model in a Smooth Topos

Abstract

We strengthen the case that the new logical perspective afforded by topos theory is suitable to the task of describing the physical world around us. In exploring some of the aspects of construction of a simple quantum-mechanical system in a mathematical universe different from that represented by set theory, we show that more thought and a better appreciation of the assumptions going into any mathematical model of the physical world are needed. We reflect on some of the mathematical consequences of this wider perspective of physics, explaining one interpretation of probabilistic values and numerical calculations in two mathematical universes governed by the less restrictive intuitionistic logic, and known to support the theory of synthetic differential geometry.