# 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.

## Contents

The application of math in psychology could be traced back to at least seventeenth century when scientists like Kepler and Galileo were investigating the laws of mental process. [1] At that time, psychology had not even been recognized as an independent subject of science. The applications of mathematics in psychology can be roughly classified into 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. [2] [3]

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. [3] [4]

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

### 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. [5] In the 18th century, Nicolas Bernoulli proposed the St. Petersburg Paradox 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", 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 (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 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, 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 to scientifically infer the elements of complex cognitive activity by measurement of simple reaction time [6]

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 psychometrics.

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 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 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 invented factor analysis 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 (ANOVA). Since factor analysis unable to make causal inferences, the method of structural equation modeling was developed by Sewall Wright 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. [6] 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. [6] [7]

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, [8] and a paper by Estes that started the stimulus sampling tradition in psychological theorizing. [9] 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.

Signal processing and detection theory are broadly used in perception, psychophysics and nonsensory area of cognition. Von Neumann's book The Theory of Games and Economic Behavior establish the importance of game theory and decision making. R. Duncan Luce and Howard Raiffa contributed to the choice and decision making area.

The area of language and thinking comes into the spot light 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. [10] Coombs constructed formal cognitive models of the respondent in a measurement situation rather than statistical data processing algorithms, for example the unfolding model. [11] [12] 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. [13]

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. [7] 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. [7] [14] 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 [15] and three handbooks. [16] 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. [14]

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.

## Important theories and models [17]

### Stimulus identification

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

### Simple decision

• 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.

## Related Research Articles

Psychology is the science of mind and behavior. Psychology includes the study of conscious and unconscious phenomena, as well as feelings and thought. It is an academic discipline of immense scope. Psychologists also seek an understanding of the emergent properties of brains, linking the discipline to neuroscience. As a social science, psychologists aim to understand the behavior of individuals and groups.

Psychological statistics is application of formulas, theorems, numbers and laws to psychology. Statistical Methods for psychology include development and application statistical theory and methods for modeling psychological data. These methods include psychometrics, Factor analysis, Experimental Designs, and Multivariate Behavioral Research. The article also discusses journals in the same field.

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 specialist fields within psychology and education devoted to testing, measurement, assessment, and related activities.

Wilhelm Maximilian Wundt was a German physiologist, philosopher, and professor, known today as one of the founders of modern psychology. Wundt, who distinguished psychology as a science from philosophy and biology, was the first person ever to call himself a psychologist. He is widely regarded as the "father of experimental psychology". In 1879, at University of Leipzig, Wundt founded the first formal laboratory for psychological research. This marked psychology as an independent field of study. By creating this laboratory he was able to establish psychology as a separate science from other disciplines. He also formed the first academic journal for psychological research, Philosophische Studien, set up to publish the Institute's research.

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.

Louis Leon Thurstone was a U.S. pioneer in the fields of psychometrics and psychophysics. He conceived the approach to measurement known as the law of comparative judgment, and is well known for his contributions to factor analysis. A Review of General Psychology survey, published in 2002, ranked Thurstone as the 88th most cited psychologist of the 20th century, tied with John Garcia, James J. Gibson, David Rumelhart, Margaret Floy Washburn, and Robert S. Woodworth.

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".

Level of measurement or scale of measure is a classification that describes the nature of information within the values assigned to variables. Psychologist Stanley Smith Stevens developed the best-known classification with four levels, or scales, of measurement: nominal, ordinal, interval, and ratio. This framework of distinguishing levels of measurement originated in psychology and is widely criticized by scholars in other disciplines. Other classifications include those by Mosteller and Tukey, and by Chrisman.

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, which is based on a logarithmic relationship between stimulus and sensation, because the power law describes a wider range of sensory comparisons, down to zero intensity.

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 things, 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.

Pairwise comparison generally is any process of comparing entities in pairs to judge which of each entity is preferred, or has a greater amount of some quantitative property, or whether or not the two entities are identical. The method of pairwise comparison is used in the scientific study of preferences, attitudes, voting systems, social choice, public choice, requirements engineering and multiagent AI systems. In psychology literature, it is often referred to as paired comparison.

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 is measured by the elapsed time between stimulus onset and an individual’s response on elementary cognitive tasks (ETCs), which are relatively simple perceptual-motor tasks typically administered in a laboratory setting. Mental chronometry is one of the core methodological paradigms of human experimental, cognitive, and differential 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 in humans and other species.

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 and 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.

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 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.

Experimental aesthetics is a field of psychology founded by Gustav Theodor Fechner in the 19th century. According to Fechner, aesthetics is an experiential perception which is empirically comprehensible in light of the characteristics of the subject undergoing the experience and those of the object. Experimental aesthetics is the second oldest research area in psychology, psychophysics being the only field which is older. In his central work Vorschule der Ästhetik Fechner describes his empirical approach extensively and in detail. Experimental aesthetics is characterized by a subject-based, inductive approach.

Computational Psychometrics is an interdisciplinary field fusing theory-based psychometrics, learning and cognitive sciences, and data-driven AI-based computational models as applied to large-scale/high-dimensional learning, assessment, biometric, or psychological data. Computational psychometrics is frequently concerned with providing actionable and meaningful feedback to individuals based on measurement and analysis of individual differences as they pertain to specific areas of enquiry.

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