# Mathematical psychology: what it is, and its main representatives

**Let's see what is and what has been mathematical psychology and how research area in sciences.**

Psychology is nourished by many other sciences. In this case, mathematics offers us a new and interesting point of view, so much so that the term "mathematical psychology" has been coined. **to the point that the term "mathematical psychology" has been coined** has been coined to speak of the contributions of certain authors.

Let's see how both disciplines are intertwined and what are the benefits that can be obtained from this relationship in order to develop different methodologies to achieve novel research in the field of the study of the human mind.

## What is mathematical psychology?

Mathematical psychology is **a way of conducting research in psychology based on the use of mathematical models.** in order to explain and predict thought processes, perception or any other psychological process. The objective would be to quantify behavior and the stimuli that provoke it, finding the mathematical laws that underlie this relationship.

Therefore, mathematical psychology is **a way of standardizing psychological processes to make them easier to measure and to be able to work with the relationships between stimulus and response, thus obtaining hypotheses and verifications of the relationship between stimulus and response.**Thus, we can obtain much more precise and rigorous hypotheses and verifications. The way to be able to quantify the behaviors of the individual is through a procedure in which he has to execute certain tasks.

The first rapprochement between psychology and mathematics took place much earlier than it may seem. It was extraordinary scientists such as Galileo Galilei or Johannes Kepler, who **in the seventeenth century tried to verify whether thought processes were governed by specific laws, as was the case with physics.**as was the case with physics. Logically, this approach was very diffuse, since psychology did not even exist as an independent science.

In the 18th century, some of the foundations on which mathematical psychology would later be based were laid. It was at this time that Blaise Pascal developed the argument of Pascal's wager, within the theories of probability. Shortly afterwards, Nicolas Bernoulli, for his part, developed the St. Petersburg paradox, in an attempt to explain decision making from a mathematical point of view.

**Thomas Bayes also made important advances in the statistical studies of the time by**Bayes' theorem, among many other contributions. Another author who continued to generate studies on which mathematical psychology would later be based is Robert Hooke. In his case, this English scientist conducted the first research on human memory, in search of predictive models.

## Contributions during the 19th century

It was in the 19th century that psychology made its greatest advances, taking on its own identity as a scientific discipline, thanks to the German Wilhelm Wundt, who founded the first laboratory of experimental psychology. **the first laboratory of experimental psychology**. It was therefore when they began to try to explain human behavior in a scientific way and therefore where mathematics made its definitive appearance to form mathematical psychology.

**During these years, psychophysics was also developed.**with authors such as Ernst Weber and Gustav Fechner, who developed Weber's law and Fechner's law, respectively. But even astrophysics somehow influenced mathematical psychology. How can this be? Because of studies in which the distance to the stars was measured and for this purpose it was measured when they passed in front of the telescope.

The point is that it was observed that the reaction time in the different people in charge of taking the measurements was different. It was Friedrich Bessel the scientist who discovered these differences and developed from them the personal equations to compensate for the characteristics of the observer who noted the records and get the most accurate data on the distance of the stars. Another step towards mathematical psychology.

Likewise, **Hermann von Helmholtz was a prolific author who studied the speed of nerve impulses.**. Together with Thomas Young, he developed the Young-Helmholtz theory or trichromatic theory, in which they explained how the three types of cones in the eyes perceived a specific part of the visible light spectrum, giving rise to the color vision that we humans have.

Continuing with the contributions to mathematical psychology, **Franciscus Cornelius Donders, a Dutch author, led an investigation to measure the time it took for the brain to perform some simple operations.**. For his part, Johann Herbart also worked on mathematical models that could explain human consciousness, a truly ambitious work for his time.

As for the advances that came from England, the most notable ones began with Francis Galton, a reference in the study of individual differences. In fact, Galton is one of the fathers of psychometry. Likewise, many of the studies on the psychology of intelligence in England are based on the pioneering studies of Francis Galton.

## Mathematical psychology during the 20th century

Another outstanding author that encompasses the last decades of the 19th century and the first decades of the 20th century is Charles Spearman. He is none other than the creator of factor analysis, a statistical system that uses variance and covariance to study individual differences in a mathematical way. **to study individual differences in a mathematical way.**. In addition to this method, there are two others: structural equation modeling on the one hand, and ANOVA, or analysis of variance, on the other.

The former was developed by the researcher Sewall Wright and the latter by Ronald Fisher. Together with factor analysis, these methods represent an important advance in the union between mathematics and psychology, crystallizing the branch of psychometry, which is related to mathematical psychology. Psychometry, therefore, was officially developed in the mid-1930s.

With the advances in the current of behaviorism, even more importance is given to variables such as reaction times. By that time, World War II broke out, an event that boosted research related to mathematical science. **The Second World War, an event that boosted research related to mathematical science, logic or computation, concepts that were applied to the rest of the world.**These concepts are applied to the rest of the sciences, such as psychology. Of course, mathematical psychology was strengthened by this interaction.

This can be seen in the increasingly frequent use in psychology of mathematical concepts such as game theory, signal processing, filter theory, information theory or stochastic processes, among many others. Some of them had already been related in some way to psychology before, but the use of others represented a revolution in the field and a new way of doing science in the study of the human mind.

It was between the 1950s and the 1960s when all the concepts of psychology were **all the concepts of mathematical psychology were embodied in a series of volumes and the publication of a scientific journal specialized in this branch was begun.**This meant the consolidation of mathematical psychology and a new and fundamental part of psychology.

## Differences between mathematical psychology and psychometry

It is important not to confuse mathematical psychology with psychometry. **Psychometrics refers to the statistical studies of quantitative measurements in psychology studies.**. On the other hand, mathematical psychology, as we have already seen, refers to the use of mathematical models that attempt to predict psychological phenomena such as cognitive processes.

In addition, psychometry is especially concerned with explaining or classifying individual or population differences, while mathematical psychology, for its part, tries to generate models that can offer an explanation for the behavior of any average individual, i.e., that predict psychological behavior under given conditions.

Similarly, psychometry tries to find out the relationship between different variables of the population analyzed statistically. In contrast, mathematical psychology focuses on the creation of mathematical models into which all experimentally recorded psychological phenomena can be fitted.

This is why, although mathematical psychology has a certain relationship with psychometry in some aspects, this link is more powerful with other branches of this science such as cognitive psychology and experimental psychology. **It is also related to other branches such as econometrics or computational neuroscience, as it has in common with them.**It is also related to other fields such as econometrics or computational neuroscience, since it has in common with them the use of statistical optimization.

This question is generated by the premise that our brain, evolutionarily, must be configured to be able to face the different problems it encounters in an optimized way that increases the probabilities of overcoming them satisfactorily and with the minimum possible use of resources.

Returning to cognitive psychology, some of its most important studies such as those related to the dichotomy between limited or unlimited processing capacity, or also the different types of processing (parallel or serial, for example), are very present issues for the studies of mathematical psychology.

Bibliographical references:

- Busemeyer, J.R., Wang, Z., Townsend, J.T., Eidels, A. (2015). The Oxford handbook of computational and mathematical psychology. Oxford University Press.
- Gras, J.A. (1977). Use of mathematical models in psychology. Anuario de psicología/The UB Journal of psychology.
- Luce, R.D. (1997). Several unresolved conceptual problems of mathematical psychology. Journal of mathematical psychology. Elsevier.
- Rasch, G. (1960). Studies in mathematical psychology: I. Probabilistic models for some intelligence and attainment tests.
- Townsend, J.T. (2008). Mathematical psychology: Prospects for the 21st century: A guest editorial. Journal of mathematical psychology. Elsevier.

(Updated at Apr 12 / 2024)