The basic concepts of classical probability, negative probability, fuzzy sets are considered with appropriate analogies of the quantum mechanical approach to create mathematical models for describing social phenomena. It also briefly describes the features of the application of the theory of intentionality, which uses the concept of institutionality for social phenomena. At the same time, institutionality is understood as a property of consciousness directed at objects and states of affairs in the real world. The concept of collective institutionality is singled out, which forms social facts arising from the collective impact of members of society. The concept of status is given, which denotes a set of characteristic values of a subject or object that determine its position in the system. The features and scope for fuzzy models are described, which are used for the mathematical description of linguistic terms and intentionality in estimating the probabilities of events in fuzzy set theory. In addition, the concept of negative probability and the possibility of using it to describe social events are described. The following is an example illustrating the concept of a complex-valued function and its use to measure the characteristics of socio-economic processes. In conclusion, the concept of status functions is introduced for comparison with membership functions and wave functions.
The basic concepts of classical probability, negative probability, fuzzy sets are considered with appropriate analogies of the quantum mechanical approach to create mathematical models for describing social phenomena. It also briefly describes the features of the application of the theory of intentionality, which uses the concept of institutionality for social phenomena. At the same time, institutionality is understood as a property of consciousness directed at objects and states of affairs in the real world. The concept of collective institutionality is singled out, which forms social facts arising from the collective impact of members of society. The concept of status is given, which denotes a set of characteristic values of a subject or object that determine its position in the system. The features and scope for fuzzy models are described, which are used for the mathematical description of linguistic terms and intentionality in estimating the probabilities of events in fuzzy set theory. In addition, the concept of negative probability and the possibility of using it to describe social events are described. The following is an example illustrating the concept of a complex-valued function and its use to measure the characteristics of socio-economic processes. In conclusion, the concept of status functions is introduced for comparison with membership functions and wave functions.
It is proposed to use quantum-like models with the use of status functions for mathematical modeling and subsequent analysis of complex socio-economic systems. The limitations of the methods of classical probability theory and mathematical statistics, as well as the theory of fuzzy sets, algorithms Mamdani, Suzuki and others for solving similar problems are described. A description of the main assumptions that are used in the mathematical modeling of socio-economic objects based on status functions is given. Examples are considered that describe the features of transition paths through intermediate states. An operator for the transition of a socio-economic system to various states, similar to the Hamiltonian, is presented. A spectrum of possible virtual trajectories is introduced to describe transitions to different states. A mathematical model based on status functions is proposed to describe the transition of the system to a measurable state. In the proposed Hamiltonian, the first term represents a subsystem of indica-tors, the second is an analog of the energy of indicators in the information environment. At the same time, terms are distinguished that are analogues of the energies of the system of two controlled indica-tors: interaction, kinetic and potential. The description of the results of mathematical modeling and the analysis of the interaction of two hypothetical indicators of the socio-economic system are given. The indicators are taken from the statistics of innovation indicators of one of the regions of the Russian Federation.
The goal of this study is to extend quantum theory to the field of fuzzy phenomena. Using the idea and method of fuzzy mathematics, the quantum probability theory of fuzzy event is proposed, with the form of fuzzy event route integral, membership degree amplitude, fuzzy field function, Green function, physical quantity and fuzzy diagram. This theory changes quantum mechanics, making it a special instance of the former. Under some conditions, this theory breaches unitarity, gauge invariance, probability conservation, and information conservation, rendering these concepts approximate. This new theory, which does not require renormalization and can naturally produce meaningful results that are consistent with the experiments, is the proper theory for describing microscopic high-speed phenomena, whereas quantum mechanics is only a proper theory for describing microscopic low-speed phenomena. Because this theory is not divergent under the conditions that there is no renormalization and an infinite number of offsetting terms, it can constitute the theoretical framework required for gravity quantization.
This article examines some experimental conditions that invite and possibly require recourse to quantum-mechanical-like mathematical models (QMLMs), models based on the key mathematical features of quantum mechanics, in scientific fields outside physics, such as biology, cognitive psychology, or economics. In particular, I consider whether the following two correlative features of quantum phenomena that were decisive for establishing the mathematical formalism of quantum mechanics play similarly important roles in QMLMs elsewhere. The first is the individuality and discreteness of quantum phenomena, and the second is the irreducibly probabilistic nature of our predictions concerning them, coupled to the particular character of the probabilities involved, as different from the character of probabilities found in classical physics. I also argue that these features could be interpreted in terms of a particular form of epistemology that suspends and even precludes a causal and, in the first place, realist description of quantum objects and processes. This epistemology limits the descriptive capacity of quantum theory to the description, classical in nature, of the observed quantum phenomena manifested in measuring instruments. Quantum mechanics itself only provides descriptions, probabilistic in nature, concerning numerical data pertaining to such phenomena, without offering a physical description of quantum objects and processes. While QMLMs share their use of the quantum-mechanical or analogous mathematical formalism, they may differ by the roles, if any, the two features in question play in them and by different ways of interpreting the phenomena they considered and this formalism itself. This article will address those differences as well.
New scientific theories have been and will always be required for the progress of science. Two examples of new theories that were advanced in the 20th century are quantum mechanics and fuzzy set theory. Both deal with the concept of uncertainty and both use mathematical tools that are essentially different from those used in classical probability theory. In both, the introduction of new concepts (theoretical terms) was necessary because phenomena and processes observed in the real world could not be explained with existing mathematical concepts. This paper examines the possibilities of (re)interpreting uncertainty in quantum mechanics with the aid of fuzzy sets and systems
The aim of the paper is to seek and identify mathematical models designated to understand better the quantum scale phenomena, models which could permit to link them with the real world phenomena. This shall be made by analyzing the common features exhibited by both, normal scale and quantum scale phenomena and then, by creating mathematical models that are describing both, the quantum and the real world. One main idea of the paper, on which the involved models are built is that observed physical quantities such as position and momentum of a quantum particle, but also the laws of their transformation (of that physical quantities) from one state into another such as for example the probability wave function described by the Schrödinger’s equation, are signals transmitted out of the quantum world into the real physical world by the a certain real, not yet detected wave function, wave function, underlying the visible measurable reality and which is describing this more complex reality underlying the observable one, and unavailable not only to our limited senses but also not available, even though using the new discovered technologies. The two observed physical quantities, position and momentum, are describing a quantum system, that is to say the quantum system can be represented either as a wave function of the position of the particle or as a wave function of the momentum of the particle, this two different functions being linked one to another by means of the Fourier Transform. The paper is furthermore describing the way in which the theory of Signal Processing, a powerful and useful tool, is to be used in order to model both, quantum and large scale phenomena.