  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.
