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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. In practice "quantifiable behavior" is 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.
Contents 1 History 1.1 20th century 2 Influential mathematical psychologists 3 Important theories and models8 3.1 Sensation Perception and Psychophysics 3.2 Simple detection 3.3 Stimulus identification 3.4 Simple decision 3.5 Memory scanning visual search 3.6 Error response times 3.7 Sequential effects 3.8 Learning 3.9 Measurement Theory 4 Journals

and organizations 5 References 6 External links // History Ernst Heinrich Weber. Gustav Fechner.
Mathematical modeling has a long history in psychology starting in the 1800s with Heinrich Weber (17951878) and Gustav Fechner (18011887) 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 1800s 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 am

ount 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 time1

The first psychological laboratory was established in Germany by 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. Part of the problems was due to the 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 introspection

ism 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 psychologist 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.12
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 Mosteller3 instigated the linear operator approach to learning and a paper by Estes4 that started the stimulus sampling tradition in psychological theorizing. 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 choi

ce 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 hand full by more than a tenfold not counting psychometricians. Most of these were concentrated at the University of Indiana Michigan Pennsylvania and Stanford.25 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 readings6 and three handbooks7. 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 r

ule 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. Influential mathematical psychologists Richard C. Atkinson C. H. Coombs Robyn Dawes William Estes B. F. Green Daniel Kahneman D. H. Krantz D. R. J. Laming R. Duncan Luce David Marr James L. McClelland Louis Narens Allen Newell David E. Rumelhart Herbert A. Simon Stanley S. Stevens Saul Sternberg Patrick Suppes John A. Swets Louis L. Thurstone Amos Tversky Thomas D. Wickens Important theories and models8 Sensation Perception and Psychophysics Stevens' Power Law Weber-Fechner Law Simple detection Signal detection theory Stimulus identification Accumulator models Diffusion models Neural network/connectionist models Race models Random walk models Renewal models Simple decision Casca

de model Level and change race model Recruitment model SPRT Decision field theory 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 Measurement Theory Theory of conjoint measurement Journals and organizations
Central journals are the Journal of Mathematical Psychology and the British Journal of Mathematical and Statistical Psychology. There are two annual conferences in the field the annual meeting of the Society for Mathematical Psychology in the U.S and the annual European Mathematical Psychology Group (EMPG) meeting. References a b c Leahey T. H. (1987). A History of Psychology. Englewood Cliffs NJ: Prentice Hall. a b c Batchelder W. H. (2002). Mathematical Psychology. In A. E. Kazdin (Ed.) Encyclopedia of Psychology Washington/New York: APA/Oxford University Press. Bush R. R. & Mosteller F. (1951). A mathematical model for simple learning. Psychological Review 58:313-323. Estes W. K. (1950). Towards a statistical theory of learning. Psychological Review 57:94-107. a b Estes W. K. (2002). History of the Society Luce R. D. Bush R. R. & Galanter E. (Eds.) (1963). Readings in mathematical psychology. Volumes I & II. New York: Wiley. Luce R. D. Bush R. R. & Galanter E. (Eds.) (1963). Handbook of mathematical psychology. Volumes I-III. New York: Wiley. Vo