This short chapter contains basics of the mathematical formalism for thequantum measurement theory. In this book we proceed mainly withthe von Neumann measurement theory in which observables are given byHermitian operators and the state update by projections. However, we alsomention the measurement formalism based on quantum instruments, sinceit gives the general framework for quantum measurements. This formalismis used only in Chapters 10 and 18. The latter chapter is devoted to quantum-likemodeling – the applications of the mathematical formalism and methodologyof quantum mechanics (QM) to cognition, psychology, and decision making.Surprisingly, in such applications even the simplest effects can’t be described bythe von Neumann theory. One should use quantum instruments (compare withquantum physics where the main body of theory can be presented solelywithin the von Neumann measurement theory).

This paper concentrates on Kant’s precritical prize essay, Inquiry Concerning the Distinctness of the Principles of Natural Theology and Morality, (1763; published 1764). In the prize essay Kant first emphasizes the essential use of symbols in mathematics as opposed to philosophy. Dunlop argues that in his account of the mathematical use of symbols, Kant has the materials to explain those symbols’ extramental reference. Hence symbolization at least partially fulfills the function assigned to sensible intuition in Kant’s critical philosophy. By comparing Kant’s views in the prize essay with those of Moses Mendelssohn, taken as a representative of the still-dominant Wolffian approach, this paper shows that Kant’s account of the use of symbols makes use of resources already available in the Wolffian tradition. Their familiarity raises the question of what is novel in Kant’s prize essay, and the correspondence between the essay and the first Critique invites us to ask what purpose is served by introducing a faculty of intuition. Dunlop claims that Kant’s view of pure intuition as a condition on empirical intuition is the most important development in the critical account of mathematical discourse’s extramental reference.