Modelling the measurement ("active observation") process makes it possible to express in strict terms the degree to which the logic of the observer determines what he "sees", and formalize the difficult concept of rational behaviour. Presented here is a rigorous formulation of several implicit assumptions of standard physics which leads to a first-order theory shown to possess a real-world model: if an observer's logic is Boolean, he is bound to perceive his spacetime as a four-dimensional pseudo-Riemannian manifold of signature 2, with an ideal big bang geometry. The connections between the type of an observer's logic and large-scale structure of the observable universe yield a testable prediction, existence of positive cosmological constant and suggest a non-standard integration-over-spacetime technique. They strongly favour non-local reality and deliver an operational explanation of the number of particle generations. The result casts some doubts (arising also from the necessity of renormalization procedures) that classical mathematics (i.e. the mathematics of the topos of sets) is the "natural" mathematics of our world, and offers a new candidate for this role, that differs from its classical counterpart. In general, the scheme outlines a formal way to unify the logical, physical and, possibly, psychological templates of perception, which can be briefly expressed as "physics is an exponent-image of psychology".