Mathematical psychology

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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. 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.

Psychology is the science of behavior and mind. Psychology includes the study of conscious and unconscious phenomena, as well as feeling and thought. It is an academic discipline of immense scope. Psychologists seek an understanding of the emergent properties of brains, and all the variety of phenomena linked to those emergent properties, joining this way the broader neuroscientific group of researchers. As a social science it aims to understand individuals and groups by establishing general principles and researching specific cases.


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.

Measurement Process of assigning numbers to objects or events

Measurement is the assignment of a number to a characteristic of an object or event, which can be compared with other objects or events. The scope and application of measurement are dependent on the context and discipline. In the natural sciences and engineering, measurements do not apply to nominal properties of objects or events, which is consistent with the guidelines of the International vocabulary of metrology published by the International Bureau of Weights and Measures. However, in other fields such as statistics as well as the social and behavioral sciences, measurements can have multiple levels, which would include nominal, ordinal, interval and ratio scales.

Psychometrics is a field of study concerned with the theory and technique of psychological measurement. As defined by the US National Council on Measurement in Education (NCME), psychometrics refers to psychological measurement. Generally, it refers to the field in psychology and education that is devoted to testing, measurement, assessment, and related activities.

Experimental psychology refers to work done by those who apply experimental methods to psychological study and the processes that underlie it. Experimental psychologists employ human participants and animal subjects to study a great many topics, including sensation & perception, memory, cognition, learning, motivation, emotion; developmental processes, social psychology, and the neural substrates of all of these.

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.

Psychophysics quantitatively investigates the relationship between physical stimuli and the sensations and perceptions they produce. Psychophysics has been described as "the scientific study of the relation between stimulus and sensation" or, more completely, as "the analysis of perceptual processes by studying the effect on a subject's experience or behaviour of systematically varying the properties of a stimulus along one or more physical dimensions".

Perception Organization, identification, and interpretation of sensory information in order to represent and understand the environment

Perception is the organization, identification, and interpretation of sensory information in order to represent and understand the presented information, or the environment.

Problem solving consists of using generic or ad hoc methods in an orderly manner to find solutions to problems. Some of the problem-solving techniques developed and used in philosophy, artificial intelligence, computer science, engineering, mathematics, or medicine are related to mental problem-solving techniques studied in psychology.


Ernst Heinrich Weber. Ernst Heinrich Weber.jpg
Ernst Heinrich Weber.
Gustav Fechner. Gustav Fechner.jpg
Gustav Fechner.

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.

Ernst Heinrich Weber German psychologist

Ernst Heinrich Weber was a German physician who is considered one of the founders of experimental psychology. He was an influential and important figure in the areas of physiology and psychology during his lifetime and beyond. His studies on sensation and touch, along with his emphasis on good experimental techniques gave way to new directions and areas of study for future psychologists, physiologists, and anatomists.

Gustav Fechner German philosopher, physicist and experimental psychologist

Gustav Theodor Fechner was a German philosopher, physicist and experimental psychologist. An early pioneer in experimental psychology and founder of psychophysics, he inspired many 20th-century scientists and philosophers. He is also credited with demonstrating the non-linear relationship between psychological sensation and the physical intensity of a stimulus via the formula: , which became known as the Weber–Fechner law.

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.

Astronomy Universe events since the Big Bang 13.8 billion years ago

Astronomy is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry to try and explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galaxies, and comets. Relevant phenomena include supernova explosions, gamma ray bursts, quasars, blazars, pulsars, and cosmic microwave background radiation. More generally, astronomy studies everything that originates outside Earth's atmosphere. Cosmology is a branch of astronomy. It studies the Universe as a whole.

Friedrich Bessel German astronomer and mathematician

Friedrich Wilhelm Bessel 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 of parallax. A special type of mathematical functions were named Bessel functions after Bessel's death, though they had originally been discovered by Daniel Bernoulli and then generalised by Bessel.

Hermann von Helmholtz physicist and physiologist

Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions in several scientific fields. The largest German association of research institutions, the Helmholtz Association, is named after him.

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]

Franciscus Donders Dutch ophthalmologist

Franciscus (Franz) Cornelius Donders FRS FRSE was a Dutch ophthalmologist. During his career, he was a professor of physiology in Utrecht, and was internationally regarded as an authority on eye diseases, directing the Netherlands Hospital for Eye Patients. Along with Graefe and Helmholtz, he was one of the primary founders of scientific ophthalmology.

Mental chronometry is the study of reaction time in perceptual-motor tasks to infer the content, duration, and temporal sequencing of mental operations. Mental chronometry is one of the core methodological paradigms of human experimental and cognitive psychology, but is also commonly analyzed in psychophysiology, cognitive neuroscience, and behavioral neuroscience to help elucidate the biological mechanisms underlying perception, attention, and decision-making across species.

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. Cattell 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 hearing).

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 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 [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 gravitated towards computational mechanisms and devices. Examples of the latter constitute so called cognitive architectures (e.g., production rule systems, ACT-R) as well as connectionist systems or neural networks.

Important mathematical expressions for relations between physical characteristics of stimuli and subjective perception are Weber–Fechner law, Ekman's law, Stevens's 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

Important theories and models [8]

Sensation, perception, and psychophysics

Stimulus detection and discrimination

Stimulus identification

Simple decision

Error response times

Sequential effects


Measurement 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.

See also

Related Research Articles

Cognition is "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses many aspects of intellectual functions and processes such as attention, the formation of knowledge, memory and working memory, judgment and evaluation, reasoning and "computation", problem solving and decision making, comprehension and production of language. Cognitive processes use existing knowledge and generate new knowledge.

Introspection is the examination of one's own conscious thoughts and feelings. In psychology, the process of introspection relies exclusively on observation of one's mental state, while in a spiritual context it may refer to the examination of one's soul. Introspection is closely related to human self-reflection and is contrasted with external observation.

George Armitage Miller American cognitive psychologist

George Armitage Miller was an American psychologist who was one of the founders of the cognitive psychology, and more broadly, of cognitive science. He also contributed to the birth of psycholinguistics. Miller wrote several books and directed the development of WordNet, an online word-linkage database usable by computer programs. He authored the paper, "The Magical Number Seven, Plus or Minus Two," in which he observed that many different experimental findings considered together reveal the presence of an average limit of seven for human short-term memory capacity. This paper is frequently cited by psychologists and in the wider culture. Miller won numerous awards, including the National Medal of Science.

Stevens's 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 the Weber–Fechner law, based on a logarithmic relationship between stimulus and sensation, because the power law describes a wider range of sensory comparisons.

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 comparison of perceived intensity of physical stimuli, such as the weights of objects, and comparisons of the extremity of an attitude expressed within statements, such as statements about capital punishment. The measurements represent how we perceive objects, rather than being measurements of actual physical properties. This kind of measurement is the focus of psychometrics and psychophysics.

Quantitative psychology is a field of scientific study that focuses on the mathematical modeling, research design and methodology, and statistical analysis of human or animal psychological processes. It includes tests and other devices for measuring human abilities. Quantitative psychologists develop and analyze a wide variety of research methods, including those of psychometrics, a field concerned with the theory and technique of psychological measurement.

William Kaye Estes 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 statistical explanation for the learning phenomena, William Kaye Estes developed the Stimulus Sampling Theory in 1950 which suggested that a stimulus-response association is learned on a single trial; however, the learning process is continuous and consists of the accumulation of distinct stimulus-response pairings.

Functional psychology psychological philosophy that considers mental life and behaviour in terms of active adaptation to the persons environment

Functional psychology or functionalism refers to a psychological school of thought that was a direct outgrowth of Darwinian thinking which focuses attention on the utility and purpose of behavior that has been modified over years of human existence. Edward L. Thorndike, best known for his experiments with trial-and-error learning, came to be known as the leader of the loosely defined movement. This movement arose in the U.S. in the late 19th century in direct contrast to Edward Titchener's Structuralism which focused on the contents of consciousness rather than the motives and ideals of human behavior. Functionalism denies the principle of introspection which tends to investigate the inner workings of human thinking rather than understanding the biological processes of the human consciousness.

Eugene Galanter was one of the modern founders of cognitive psychology. He was an academic in the field of experimental psychology and an author. Dr. Galanter was Professor Emeritus of Psychology end Quondam Director of the Psychophysics Laboratory at Columbia University. He was also the co-founder, Chairman of the Board of Directors and Chief Scientific Officer of Children’s Progress, an award-winning New York City-based company that specializes in the use of computer technology in early education. The company's assessments and reports have been used in 40 states and 9 countries.

The theory of conjoint measurement is a general, formal theory of continuous quantity. It was independently discovered by the French economist Gérard Debreu (1960) and by the American mathematical psychologist R. Duncan Luce and statistician John Tukey.

Some of the research that is conducted in the field of psychology is more "fundamental" than the research conducted in the applied psychological disciplines, and does not necessarily have a direct application. The subdisciplines within psychology that can be thought to reflect a basic-science orientation include biological psychology, cognitive psychology, neuropsychology, and so on. Research in these subdisciplines is characterized by methodological rigor. The concern of psychology as a basic science is in understanding the laws and processes that underlie behavior, cognition, and emotion. Psychology as a basic science provides a foundation for applied psychology. Applied psychology, by contrast, involves the application of psychological principles and theories yielded up by the basic psychological sciences; these applications are aimed at overcoming problems or promoting well-being in areas such as mental and physical health and education.

Structuralism in psychology is a theory of consciousness developed by Wilhelm Wundt and his student Edward Bradford Titchener. This theory was challenged in the 20th century. It is debated who deserves the credit for finding this field of psychology, but it is widely accepted that Wundt created the foundation on which Titchener expanded. Structuralism as a school of psychology seeks to analyze the adult mind in terms of the simplest definable components and then to find how these components fit together to form more complex experiences as well as how they correlate to physical events. To do this, psychologists employ introspection, self-reports of sensations, views, feelings, emotions, etc.

Psychology encompasses a vast domain, and includes many different approaches to the study of mental processes and behavior. Below are the major areas of inquiry that taken together constitute psychology. A comprehensive list of the sub-fields and areas within psychology can be found at the list of psychology topics and list of psychology disciplines.

Jean-Claude Falmagne Belgian mathematical psychologist

Jean-Claude Falmagne is a mathematical psychologist whose scientific contributions deal with problems in reaction time theory, psychophysics, philosophy of science, measurement theory, decision theory, and educational technology. Together with Jean-Paul Doignon, he developed knowledge space theory, which is the mathematical foundation for the ALEKS software for the assessment of knowledge in various academic subjects, including K-12 mathematics, chemistry, and accounting.


  1. 1 2 3 Leahey, T. H. (1987). A History of Psychology (Second ed.). Englewood Cliffs, NJ: Prentice Hall. ISBN   0-13-391764-9.
  2. 1 2 3 Batchelder, W. H. (2002). "Mathematical Psychology". In Kazdin, A. E. (ed.). Encyclopedia of Psychology. Washington/NY: APA/Oxford University Press. ISBN   1-55798-654-1.
  3. Bush, R. R.; Mosteller, F. (1951). "A mathematical model for simple learning". Psychological Review. 58 (5): 313–323. doi:10.1037/h0054388. PMID   14883244.
  4. Estes, W. K. (1950). "Toward a statistical theory of learning". Psychological Review. 57 (2): 94–107. doi:10.1037/h0058559.
  5. 1 2 Estes, W. K. (2002). History of the Society
  6. Luce, R. D., Bush, R. R. & Galanter, E. (Eds.) (1963). Readings in mathematical psychology. Volumes I & II. New York: Wiley.
  7. Luce, R. D., Bush, R. R. & Galanter, E. (Eds.) (1963). Handbook of mathematical psychology. Volumes I-III. New York: Wiley. Volume II from Internet Archive
  8. Luce, R. Duncan (1986). Response Times: Their Role in Inferring Elementary Mental Organization. Oxford Psychology Series. 8. New York: Oxford University Press. ISBN   0-19-503642-5.