Mathematical psychology is an approach to psychological research that is based on mathematical modeling of perceptual, thought, cognitive and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus characteristics with quantifiable behavior (in practice often constituted by task performance). The mathematical approach is used with the goal of deriving hypotheses that are more exact and thus yield stricter empirical validations. There are five major research areas in mathematical psychology: learning and memory, perception and psychophysics, choice and decision making, language and thinking, and measurement and scaling.
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Although psychology, as an independent subject of science, is a more recent discipline than physics, the application of mathematics to psychology has been done in the hope of emulating the success of this approach in the physical sciences, which dates back to at least the seventeenth century. Mathematics in psychology is used extensively roughly in two areas: one is the mathematical modeling of psychological theories and experimental phenomena, which leads to mathematical psychology, the other is the statistical approach of quantitative measurement practices in psychology, which leads to psychometrics.
As quantification of behavior is fundamental in this endeavor, the theory of measurement is a central topic in mathematical psychology. Mathematical psychology is therefore closely related to psychometrics. However, where psychometrics is concerned with individual differences (or population structure) in mostly static variables, mathematical psychology focuses on process models of perceptual, cognitive and motor processes as inferred from the 'average individual'. Furthermore, where psychometrics investigates the stochastic dependence structure between variables as observed in the population, mathematical psychology almost exclusively focuses on the modeling of data obtained from experimental paradigms and is therefore even more closely related to experimental psychology, cognitive psychology, and psychonomics. Like computational neuroscience and econometrics, mathematical psychology theory often uses statistical optimality as a guiding principle, assuming that the human brain has evolved to solve problems in an optimized way. Central themes from cognitive psychology (e.g., limited vs. unlimited processing capacity, serial vs. parallel processing) and their implications are central in rigorous analysis in mathematical psychology.
Mathematical psychologists are active in many fields of psychology, especially in psychophysics, sensation and perception, problem solving, decision-making, learning, memory, language, and the quantitative analysis of behavior, and contribute to the work of other subareas of psychology such as clinical psychology, social psychology, educational psychology, and psychology of music.