Mathematical psychology
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.
[2] As quantification of behavior is fundamental in this endeavor, the theory of measurement is a central topic in mathematical psychology.
In 1763, Bayes published the paper "An Essay Towards Solving a Problem in the Doctrine of Chances", which is the milestone of Bayesian statistics.
For lack of the automatic registration instruments of the modern era, these time measurements relied entirely on human response speed.
Bessel constructed personal equations from measurements of basic response speed that would cancel out individual differences from the astronomical calculations.
Donders envisioned the employment of his mental chronometry to scientifically infer the elements of complex cognitive activity by measurement of simple reaction time[5] Although there are developments in sensation and perception, Johann Herbart developed a system of mathematical theories in cognitive area to understand the mental process of consciousness.
Wundt's laboratory was directed towards conscious human experience, in line with the work of Fechner and Weber on the intensity of stimuli.
In the United Kingdom, under the influence of the anthropometric developments led by Francis Galton, interest focussed on individual differences between humans on psychological variables, in line with the work of Bessel.
Many statistical methods were developed even before the 20th century: Charles Spearman invented factor analysis which studies individual differences by the variance and covariance.
There are two important statistical developments: Structural Equation Modeling (SEM) and analysis of variance (ANOVA).
Behaviorism dominated American psychology until the end of the Second World War, and largely refrained from inference on mental processes.
During the war, developments in engineering, mathematical logic and computability theory, computer science and mathematics, and the military need to understand human performance and limitations, brought together experimental psychologists, mathematicians, engineers, physicists, and economists.
One development is the stimulus sampling theory by Williams K. Estes, the other is linear operator models by Robert R. Bush, and Frederick Mosteller.
Signal processing and detection theory are broadly used in perception, psychophysics and nonsensory area of cognition.
In the summer of 1963 the need was felt for a journal for theoretical and mathematical studies in all areas in psychology, excluding work that was mainly factor analytical.
An initiative led by R. C. Atkinson, R. R. Bush, W. K. Estes, R. D. Luce, and P. Suppes resulted in the appearance of the first issue of the Journal of Mathematical Psychology in January 1964.
Mathematical modeling is useful in developmental psychology for implementing theory in a precise and easy-to-study manner, allowing generation, explanation, integration, and prediction of diverse phenomena.
