# Mathematical psychology

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Mathematical psychology is an approach to psychological research that is based on mathematical modeling of perceptual, cognitive and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus characteristics with quantifiable behavior. The mathematical approach is used with the goal of deriving hypotheses that are more exact and thus yield stricter empirical validations. Quantifiable behavior is in practice often constituted by task performance.

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/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; limited vs. unlimited processing capacity, serial vs. parallel processing, etc., 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, and language, collectively known as cognitive psychology, and the quantitative analysis of behavior but also, e.g., in clinical psychology, social psychology, and psychology of music.

## History

Mathematical modeling has a long history in psychology starting in the 19th century with Ernst Weber (1795–1878) and Gustav Fechner (1801–1887) being among the first to apply successful mathematical technique of functional equations from physics to psychological processes. They thereby established the fields of experimental psychology in general, and that of psychophysics in particular.

Researchers in astronomy in the 19th century were mapping distances between stars by denoting the exact time of a star's passing of a cross-hair on a telescope. For lack of the automatic registration instruments of the modern era, these time measurements relied entirely on human response speed. It had been noted that there were small systematic differences in the times measured by different astronomers, and these were first systematically studied by German astronomer Friedrich Bessel (1782–1846). Bessel constructed personal equations from measurements of basic response speed that would cancel out individual differences from the astronomical calculations. Independently, physicist Hermann von Helmholtz measured reaction times to determine nerve conduction speed.

These two lines of work came together in the research of Dutch physiologist F. C. Donders and his student J. J. de Jaager, who recognized the potential of reaction times for more or less objectively quantifying the amount of time elementary mental operations required. Donders envisioned the employment of his mental chronometry to scientifically infer the elements of complex cognitive activity by measurement of simple reaction time[1]

The first psychological laboratory was established in Germany by Wilhelm Wundt, who amply used Donders' ideas. However, findings that came from the laboratory were hard to replicate and this was soon attributed to the method of introspection that Wundt introduced. Some of the problems resulted from individual differences in response speed found by astronomers. Although Wundt did not seem to take interest in these individual variations and kept his focus on the study of the general human mind, Wundt's U.S. student James McKeen Cattell was fascinated by these differences and started to work on them during his stay in England.

The failure of Wundt's method of introspection led to the rise of different schools of thought. 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. Catell soon adopted the methods of Galton and helped laying the foundation of psychometrics.

### 20th century

In the United States, behaviorism arose in opposition to introspectionism and associated reaction-time research, and turned the focus of psychological research entirely to learning theory.[1] In Europe introspection survived in Gestalt psychology. Behaviorism dominated American psychology until the end of the Second World War, and largely refrained from inference on mental processes. Formal theories were mostly absent (except for vision and audition).

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. Out of this mix of different disciplines mathematical psychology arose. Especially the developments in signal processing, information theory, linear systems and filter theory, game theory, stochastic processes and mathematical logic gained a large influence on psychological thinking.[1][2]

Two seminal papers on learning theory in Psychological Review helped to establish the field in a world that was still dominated by behaviorists: A paper by Bush and Mosteller instigated the linear operator approach to learning,[3] and a paper by Estes that started the stimulus sampling tradition in psychological theorizing.[4] These two papers presented the first detailed formal accounts of data from learning experiments.

The 1950s saw a surge in mathematical theories of psychological processes, including Luce's theory of choice, Tanner and Swets' introduction of signal detection theory for human stimulus detection, and Miller's approach to information processing.[2] By the end of the 1950s, the number of mathematical psychologists had increased from a handful by more than a tenfold, not counting psychometricians. Most of these were concentrated at the Indiana University, Michigan, Pennsylvania, and Stanford.[2][5] Some of these were regularly invited by the U.S. Social Science Research Counsel to teach in summer workshops in mathematics for social scientists at Stanford University, promoting collaboration.

To better define field of mathematical psychology, the mathematical models of the 1950s were brought together in sequence of volumes edited by Luce, Bush, and Galanter: Two readings[6] and three handbooks.[7] This series of volumes turned out to be helpful in the development of the field. 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.[5]

Under the influence of developments in computer science, logic, and language theory, in the 1960s modeling became more in terms of computational mechanisms and devices. Examples of the latter constitute so called cognitive architectures (e.g., production rule systems, ACT-R) as well connectionist systems or neural networks.

Important mathematical expressions for relations between physical characteristics of stimuli and subjective perception are Weber's law (which is now sometimes called Weber-Fechner Law), Ekman's Law, Stevens' Power Law, Thurstone's Law of Comparative Judgment, the Theory of Signal Detection (borrowed from radar engineering), the Matching Law, and Rescorla-Wagner rule for classical conditioning. While the first three laws are all deterministic in nature, later established relations are more fundamentally stochastic. This has been a general theme in the evolution in mathematical modeling of psychological processes: from deterministic relations as found in classical physics to inherently stochastic models.

## Important theories and models[8]

### Stimulus identification

• Accumulator models
• Diffusion models
• Neural network/connectionist models
• Race models
• Random walk models
• Renewal models

### Memory scanning, visual search

• Push-down stack
• Serial exhaustive search (SES) model

### Error response times

• Fast guess model

### Sequential effects

• Linear operator model

### Learning

• Linear operator model
• Stochastic learning theory

## Journals and organizations

Central journals are the Journal of Mathematical Psychology and the British Journal of Mathematical and Statistical Psychology. There are three annual conferences in the field, the annual meeting of the Society for Mathematical Psychology in the U.S, the annual European Mathematical Psychology Group meeting in Europe, and the Australasian Mathematical Psychology conference.