The Real and the Mathematical in Quantum Modeling: From Principles to Models and from Models to Principles

Abstract

The history of mathematical modeling outside physics has been dominated by the use of classical mathematical models, C-models, primarily those of a probabilistic or statistical nature. More recently, however, quantum mathematical models, Q-models, based in the mathematical formalism of quantum theory have become more prominent in psychology, economics, and decision science. The use of Q-models in these fields remains controversial, in part because it is not entirely clear whether Q-models are necessary for dealing with the phenomena in question or whether C-models would still suffice. My aim, however, is not to assess the necessity of Q-models in these fields, but instead to reflect on what the possible applicability of Q-models may tell us about the corresponding phenomena there, vis-à-vis quantum phenomena in physics. In order to do so, I shall first discuss the key reasons for the use of Q-models in physics. In particular, I shall examine the fundamental principles that led to the development of quantum mechanics. Then I shall consider a possible role of similar principles in using Q-models outside physics. Psychology, economics, and decision science borrow already available Q-models from quantum theory, rather than derive them from their own internal principles, while quantum mechanics was derived from such principles, because there was no readily available mathematical model to handle quantum phenomena, although the mathematics ultimately used in quantum did in fact exist then. I shall argue, however, that the principle perspective on mathematical modeling outside physics might help us to understand better the role of Q-models in these fields and possibly to envision new models, conceptually analogous to but mathematically different from those of quantum theory, that may be helpful or even necessary there or in physics itself. I shall, in closing, suggest one possible type of such models, singularized probabilistic models, SP-models, some of which are time-dependent, TDSP-models. The necessity of using such models may change the nature of mathematical modeling in science and, thus, the nature of science, as it happened in the case of Q-models, which not only led to a revolutionary transformation of physics but also opened new possibilities for scientific thinking and mathematical modeling beyond physics.

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“…There is an additional possibility, which leads to a different type of models or theories, conforming to the QI, no-cloning, QD, and QP (but not QS) postulates, and RWR views, which I would like to suggest in closing, extending and revising the argument offered in Plotnitsky (2017) . I shall call such models or theories singularized probabilistic (SP) models or theories, keeping in mind their RWR-type character (Realist SP models are possible, too, but are put aside here).…”

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“…might generate a negative reaction)" [34] (p. 9). Here, in closing, I will risk a more sustained appeal to Freud, in part following [52] (pp. [11][12], even if without fully subscribing to Freud's views either, in part given their ultimately realist nature, vs. the RWR/IWI view assumed here.…”