Mathematical psychology is an approach to psychological research that is based on

mathematical modeling A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, b ...

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 Memory is the faculty of the mind by which data or information is encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If past events could not be remembered ...

perception Perception () is the organization, identification, and interpretation of sensory information in order to represent and understand the presented information or environment. All perception involves signals that go through the nervous system ...

and psychophysics,

choice A choice is the range of different things from which a being can choose. The arrival at a choice may incorporate motivators and models. For example, a traveler might choose a route for a journey based on the preference of arriving at a give ...

and decision-making,

language Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of ...

and

thinking In their most common sense, the terms thought and thinking refer to conscious cognitive processes that can happen independently of sensory stimulation. Their most paradigmatic forms are judging, reasoning, concept formation, problem solving, an ...

, and measurement and scaling. Although psychology, as an independent subject of science, is a more recent discipline than

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

, the application of mathematics to psychology has been done in the hope of emulating the success of this approach in the

physical sciences Physical science is a branch of natural science that studies non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called the "physical sciences". Definition Phy ...

, which dates back to at least the

seventeenth century The 17th century lasted from January 1, 1601 ( MDCI), to December 31, 1700 ( MDCC). It falls into the early modern period of Europe and in that continent (whose impact on the world was increasing) was characterized by the Baroque cultural moveme ...

. 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 Psychometrics is a field of study within psychology concerned with the theory and technique of measurement. Psychometrics generally refers to specialized fields within psychology and education devoted to testing, measurement, assessment, and ...

. 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 Computational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is a branch of neuroscience which employs mathematical models, computer simulations, theoretical analysis and abstractions of the brain to u ...

and

econometrics Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...

, 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 Sensation (psychology) refers to the processing of the senses by the sensory system. Sensation or sensations may also refer to: In arts and entertainment In literature * Sensation (fiction), a fiction writing mode * Sensation novel, a Britis ...

and

, problem solving, decision-making, learning,

, and the quantitative analysis of behavior, and contribute to the work of other subareas of psychology such as clinical psychology,

social psychology Social psychology is the scientific study of how thoughts, feelings, and behaviors are influenced by the real or imagined presence of other people or by social norms. Social psychologists typically explain human behavior as a result of the ...

educational psychology Educational psychology is the branch of psychology concerned with the scientific study of human learning. The study of learning processes, from both cognitive and behavioral perspectives, allows researchers to understand individual differences in ...

, and

psychology of music ''Psychology of Music'' is a peer-reviewed academic journal that publishes papers in the field of music psychology. The editor-in-chief is Alexandra Lamont (Keele University). It was established in 1973 and is published by SAGE Publications on be ...

History

Mathematics and psychology before the 19th century

Choice and decision making theory are rooted in the development of probability theory. Blaise Pascal considered situations in gambling and further extended to Pascal's wager. In the 18th century, Nicolas Bernoulli proposed the

St. Petersburg Paradox The St. Petersburg paradox or St. Petersburg lottery is a paradox involving the game of flipping a coin where the expected payoff of the theoretical lottery game approaches infinity but nevertheless seems to be worth only a very small amount to t ...

in decision making, Daniel Bernoulli gave a solution and Laplace proposed a modification to the solution later on. In 1763, Bayes published the paper "

An Essay towards solving a Problem in the Doctrine of Chances ''An Essay towards solving a Problem in the Doctrine of Chances'' is a work on the mathematical theory of probability by Thomas Bayes, published in 1763, two years after its author's death, and containing multiple amendments and additions due to h ...

", which is the milestone of Bayesian statistics. Robert Hooke worked on modeling human memory, which is a precursor of the study of memory.

Mathematics and psychology in the 19th century

The research developments in German and England in the 19th century made psychology a new academic subject. Since German approach emphasized experiments in the investigation of the psychological processes that all human share and the England approach was the measurement of individual differences, the applications of mathematics are also different. In German, Wilhelm Wundt established the first experimental psychology laboratory. The math in German psychology is mainly applied in sensory and psychophysics. Ernst Weber (1795–1878) created the first mathematical law of the mind, Weber's law, based on a variety of experiments.

Gustav Fechner Gustav Theodor Fechner (; ; 19 April 1801 – 18 November 1887) was a German physicist, philosopher, and experimental psychologist. A pioneer in experimental psychology and founder of psychophysics (techniques for measuring the mind), he ins ...

(1801–1887) contributed math theories in sensations and perceptions and one of them is the Fechner's law, which modifies Weber's law. Mathematical modeling has a long history in psychology starting in the 19th century with Ernst Weber (1795–1878) and

(1801–1887) being among the first to apply successful mathematical technique of

functional equations In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...

from physics to psychological processes. They thereby established the fields of experimental psychology in general, and that of psychophysics in particular. Researchers in

astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...

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 Friedrich Wilhelm Bessel (; 22 July 1784 – 17 March 1846) was a German astronomer, mathematician, physicist, and geodesist. He was the first astronomer who determined reliable values for the distance from the sun to another star by the method ...

(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 Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associatio ...

measured reaction times to determine nerve conduction speed, developed resonance theory of hearing and the Young-Helmholtz theory of color vision. 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 Mental chronometry is the scientific study of processing speed or reaction time on cognitive tasks to infer the content, duration, and temporal sequencing of mental operations. Reaction time (RT; sometimes referred to as "response time") is meas ...

'' to scientifically infer the elements of complex cognitive activity by measurement of ''simple reaction time'' 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. The origin of English psychology can be traced to the theory of evolution by Darwin. But the emergence of English psychology is because of Francis Galton, who interested in individual differences between humans on psychological variables. The math in English psychology is mainly statistics and the work and methods of Galton is the foundation of

. Galton introduced bivariate normal distribution in modeling the traits of the same individual, he also investigated measurement error and built his own model, and he also developed a stochastic branching process to examine the extinction of family names. There is also a tradition of the interest in studying intelligence in English psychology started from Galton. James McKeen Cattell and

Alfred Binet Alfred Binet (; 8 July 1857 – 18 October 1911), born Alfredo Binetti, was a French psychologist who invented the first practical IQ test, the Binet–Simon test. In 1904, the French Ministry of Education asked psychologist Alfred Binet to ...

developed tests of intelligence. 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 Introspection is the examination of one's own conscious thoughts and feelings. In psychology, the process of introspection relies on the observation of one's mental state, while in a spiritual context it may refer to the examination of one's sou ...

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. Cattell soon adopted the methods of Galton and helped laying the foundation of psychometrics.

20th century

Many statistical methods were developed even before the 20th century:

Charles Spearman Charles Edward Spearman, FRS (10 September 1863 – 17 September 1945) was an English psychologist known for work in statistics, as a pioneer of factor analysis, and for Spearman's rank correlation coefficient. He also did seminal work on mod ...

invented

factor analysis Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. For example, it is possible that variations in six observed ...

which studies individual differences by the variance and covariance. German psychology and English psychology have been combined and taken over by the United States. The statistical methods dominated the field during the beginning of the century. There are two important statistical developments: Structural Equation Modeling (SEM) and

analysis of variance Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician ...

(ANOVA). Since factor analysis unable to make causal inferences, the method of structural equation modeling was developed by

Sewall Wright Sewall Green Wright FRS(For) Honorary FRSE (December 21, 1889March 3, 1988) was an American geneticist known for his influential work on evolutionary theory and also for his work on path analysis. He was a founder of population genetics alongsi ...

to correlational data to infer causality, which is still a major research area today. Those statistical methods formed psychometrics. The Psychometric Society was established in 1935 and the journal ''Psychometrika'' was published since 1936. 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. In Europe introspection survived in

Gestalt psychology Gestalt-psychology, gestaltism, or configurationism is a school of psychology that emerged in the early twentieth century in Austria and Germany as a theory of perception that was a rejection of basic principles of Wilhelm Wundt's and Edward ...

. Behaviorism dominated American psychology until the end of the

Second World War World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposi ...

, and largely refrained from inference on mental processes. Formal theories were mostly absent (except for

vision Vision, Visions, or The Vision may refer to: Perception Optical perception * Visual perception, the sense of sight * Visual system, the physical mechanism of eyesight * Computer vision, a field dealing with how computers can be made to gain und ...

and

hearing Hearing, or auditory perception, is the ability to perceive sounds through an organ, such as an ear, by detecting vibrations as periodic changes in the pressure of a surrounding medium. The academic field concerned with hearing is audit ...

). During the war, developments in

engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

and computability theory,

computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...

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 Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...

, information theory, linear systems and

filter theory Filter design is the process of designing a signal processing filter that satisfies a set of requirements, some of which may be conflicting. The purpose is to find a realization of the filter that meets each of the requirements to a sufficient ...

, game theory, stochastic processes and

gained a large influence on psychological thinking. 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, and a paper by Estes that started the stimulus sampling tradition in psychological theorizing. These two papers presented the first detailed formal accounts of data from learning experiments. Mathematical modeling of learning process were greatly developed in the 1950s as the behavioral learning theory was flourishing. One development is the stimulus sampling theory by Williams K. Estes, the other is linear operator models by Robert R. Bush, and

Frederick Mosteller Charles Frederick Mosteller (December 24, 1916 – July 23, 2006) was an American mathematician, considered one of the most eminent statisticians of the 20th century. He was the founding chairman of Harvard's statistics department from 19 ...

. Signal processing and detection theory are broadly used in perception, psychophysics and nonsensory area of cognition.

Von Neumann Von Neumann may refer to: * John von Neumann (1903–1957), a Hungarian American mathematician * Von Neumann family * Von Neumann (surname), a German surname * Von Neumann (crater), a lunar impact crater See also * Von Neumann algebra * Von Ne ...

's book '' The Theory of Games and Economic Behavior'' establish the importance of game theory and decision making. R. Duncan Luce and

Howard Raiffa Howard Raiffa (; January 24, 1924 – July 8, 2016) was an American academic who was the Frank P. Ramsey Professor (Emeritus) of Managerial Economics, a joint chair held by the Business School and Harvard Kennedy School at Harvard University. He w ...

contributed to the choice and decision making area. The area of language and thinking comes into the spotlight with the development of computer science and linguistics, especially information theory and computation theory. Chomsky proposed the model of linguistics and computational hierarchy theory. Allen Newell and Herbert Simon proposed the model of human solving problems. The development in artificial intelligence and human computer interface are active areas in both computer science and psychology. Before the 1950s, psychometricians emphasized the structure of measurement error and the development of high-power statistical methods to the measurement of psychological quantities but little of the psychometric work concerned the structure of the psychological quantities being measured or the cognitive factors behind the response data. Scott and Suppes studied relationship between the structure of data and the structure of numerical systems that represent the data. Coombs constructed formal cognitive models of the respondent in a measurement situation rather than statistical data processing algorithms, for example the unfolding model. Another breakthrough is the development of a new form of the psychophysical scaling function along with new methods of collecting psychophysical data, like Stevens' power law. 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 Detection theory or signal detection theory is a means to measure the ability to differentiate between information-bearing patterns (called stimulus in living organisms, signal in machines) and random patterns that distract from the information (ca ...

for human stimulus detection, and Miller's approach to information processing. 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.Estes, W. K. (2002). History of the Society
/ref> 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 the 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 and three handbooks. 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 The ''Journal of Mathematical Psychology'' is a peer-reviewed scientific journal established in 1964. It covers all areas of mathematical and theoretical psychology, including sensation and perception, psychophysics, learning and memory, problem ...

'' in January 1964. Under the influence of developments in computer science, logic, and language theory, in the 1960s modeling gravitated towards computational mechanisms and devices. Examples of the latter constitute so called

cognitive architectures A cognitive architecture refers to both a theory about the structure of the human mind and to a computational instantiation of such a theory used in the fields of artificial intelligence (AI) and computational cognitive science. The formalized mode ...

(e.g., production rule systems,

ACT-R ACT-R (pronounced /ˌækt ˈɑr/; short for "Adaptive Control of Thought—Rational") is a cognitive architecture mainly developed by John Robert Anderson and Christian Lebiere at Carnegie Mellon University. Like any cognitive architecture, ACT-R ...

) as well as

connectionist Connectionism refers to both an approach in the field of cognitive science that hopes to explain mental phenomena using artificial neural networks (ANN) and to a wide range of techniques and algorithms using ANNs in the context of artificial int ...

systems or neural networks. Important mathematical expressions for relations between physical characteristics of stimuli and subjective perception are

Weber–Fechner law The Weber–Fechner laws are two related hypotheses in the field of psychophysics, known as Weber's law and Fechner's law. Both laws relate to human perception, more specifically the relation between the actual change in a physical stimulus an ...

, Ekman's law,

Stevens's power law Stevens' power law is an empirical relationship in psychophysics between an increased intensity or strength in a physical stimulus and the perceived magnitude increase in the sensation created by the stimulus. It is often considered to supersede ...

, Thurstone's

law of comparative judgment The law of comparative judgment was conceived by L. L. Thurstone. In modern-day terminology, it is more aptly described as a model that is used to obtain measurements from any process of pairwise comparison. Examples of such processes are the compa ...

, the theory of signal detection (borrowed from radar engineering), the

matching law In operant conditioning, the matching law is a quantitative relationship that holds between the relative rates of response and the relative rates of reinforcement in concurrent schedules of reinforcement. For example, if two response alternative ...

, 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

* John Anderson * Richard C. Atkinson * William H. Batchelder * Jerome Busemeyer * Hans Colonius * C. H. Coombs * Robyn Dawes * Adele Diederich * Ehtibar Dzhafarov *

William Kaye Estes William Kaye Estes (June 17, 1919 – August 17, 2011) was an American psychologist. A ''Review of General Psychology'' survey, published in 2002, ranked Estes as the 77th most cited psychologist of the 20th century. In order to develop a statist ...

* Jean-Claude Falmagne * B. F. Green * Daniel Kahneman * Roger E. Kirk * D. H. Krantz * D. R. J. Laming * Michael D. Lee * Philip Marcus Levy * R. Duncan Luce * David Marr * James L. McClelland * Jeff Miller * Jay Myung * Louis Narens * Allen Newell * Robert M. Nosofsky * Roger Ratcliff * David E. Rumelhart *

Herbert A. Simon Herbert Alexander Simon (June 15, 1916 – February 9, 2001) was an American political scientist, with a Ph.D. in political science, whose work also influenced the fields of computer science, economics, and cognitive psychology. His primary ...

Roger Shepard Roger Newland Shepard (January 30, 1929 – May 30, 2022) was an American cognitive scientist and author of the "universal law of generalization" (1987). He was considered a father of research on spatial relations. He studied mental rotation, an ...

Richard Shiffrin Richard Shiffrin (born March 13, 1942) is an American psychologist, professor of cognitive science in the Department of Psychological and Brain Sciences at Indiana University, Indiana University, Bloomington. Shiffrin has contributed a number of t ...

* Philip L. Smith * Stanley S. Stevens *

George Sperling George Sperling (born 1934) is an American cognitive psychologist, researcher, and educator. Sperling documented the existence of iconic memory (one of the sensory memory subtypes). Through several experiments, he showed support for his hypothe ...

* Saul Sternberg *

Patrick Suppes Patrick Colonel Suppes (; March 17, 1922 – November 17, 2014) was an American philosopher who made significant contributions to philosophy of science, the theory of measurement, the foundations of quantum mechanics, decision theory, psychology ...

* John A. Swets *

Joshua Tenenbaum Joshua Brett Tenenbaum (Josh Tenenbaum) is Professor of Computational Cognitive Science at the Massachusetts Institute of Technology. He is known for contributions to mathematical psychology and Bayesian cognitive science. According to the MacA ...

* James T. Townsend * Louis L. Thurstone * Amos Tversky * Rolf Ulrich * Dirk Vorberg * Eric-Jan Wagenmakers * Elke U. Weber * Thomas D. Wickens

Important theories and models

Sensation, perception, and psychophysics

Stevens' power law Stevens' power law is an empirical relationship in psychophysics between an increased intensity or strength in a physical stimulus and the perceived magnitude increase in the sensation created by the stimulus. It is often considered to supersede ...

Stimulus detection and discrimination

Signal detection theory Detection theory or signal detection theory is a means to measure the ability to differentiate between information-bearing patterns (called stimulus in living organisms, signal in machines) and random patterns that distract from the information (ca ...

Stimulus identification

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

Simple decision

* Cascade 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 The theory of conjoint measurement (also known as conjoint measurement or additive conjoint measurement) is a general, formal theory of continuous quantity. It was independently discovered by the French economist Gérard Debreu (1960) and by the Am ...

Journals and organizations

Central journals are the ''

'' 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.

References

External links

''British Journal of Mathematical and Statistical Psychology''

European Mathematical Psychology Group

''Journal of Mathematical Psychology''