Psychology draws on many other sciences. In this case, mathematics offers us a new and interesting point of view, to the point that **the term “mathematical psychology” has been coined** to speak of the contributions of certain authors.

We are going to 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 innovative 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 the behavior and the stimuli that cause it, finding the mathematical laws that underlie this relationship.

Therefore, mathematical psychology is **a way of standardizing psychological processes so that it is easier to measure them and to be able to work with the relationships between stimulus and response** , thus achieving much more precise and rigorous hypotheses and verifications. The way to quantify the behaviors of the individual is through a procedure in which they have to perform certain tasks.

The first rapprochement between psychology and mathematics took place much earlier than it may seem. They were extraordinary scientists such as Galilego 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. Logically this approach was very diffuse, since psychology did not even exist as an independent science.

In the eighteenth century, some of the foundations were laid on which mathematical psychology would later be based. It is at this time when Blaise Pascal develops the argument of Pascal’s bet, within the theories of probability. Shortly after, Nicolas Bernoulli, for his part, develops the Saint Petersburg paradox, to try to explain decision-making from a mathematical point of view.

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

## Contributions during the 19th century

It was in the 19th century that the great advances in psychology took place, taking its own identity as a scientific discipline, at the hands of the German Wilhelm Wundt, who founded **the first experimental psychology laboratory** . It was therefore when it began to try to explain human behavior in a scientific way and therefore where mathematics made its final appearance to form mathematical psychology.

**During these years psychophysics also developed** , with authors such as Ernst Weber or Gustav Fechner , who developed Weber’s law and Fechner’s law, respectively. But even astrophysics had some influence on mathematical psychology. How can this be? Due to studies in which the distance to which the stars were measured was measured and for this 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 measures was different. It was Friedrich Bessel the scientist who discovered these differences and developed personal equations from them in order to compensate for the characteristics of the observer who recorded the records and obtain the most accurate data on the distance of the stars. Another step towards mathematical psychology.

Similarly, **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 eye cones perceived a specific part of the spectrum of visible light, giving rise to the color vision that we have. human beings.

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

As for the advances that came from England, the most noteworthy begin with Francis Galton, a benchmark in the study of individual differences. In fact, Galton is one of the fathers of psychometrics. 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 prominent author that encompasses the last decades of the 19th century and the first decades of the 20th century is Charles Spearman. He is nothing less than the creator of factor analysis, a statistical system that uses variance and covariance to be able to **study individual differences in a mathematical way** . Two others are added to this method, such as modeling of structural equations on the one hand and ANOVA, or analysis of variance on the other.

The first is the result of researcher Sewall Wright and the second was developed 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, is officially developed in the mid-30s of the last century.

With advances in the stream of behaviorism, variables such as reaction times are given even more importance. By then World War II also broke out, an event that **promoted research related to mathematical science, logic or computing** , concepts that are applied to other sciences, such as psychology. Of course, mathematical psychology comes out stronger from this interaction.

This can be observed 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 50s and 60s when **all the concepts of mathematical psychology were reflected in a series of volumes and the publication of a scientific journal specialized in this branch began** , which meant the consolidation of the same and a part new and fundamental in psychology.

## Differences between mathematical psychology and psychometry

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

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

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

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 aspects 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 that it encounters in an optimized way that increases the chances of successfully overcoming them and with the minimum use of resources possible.

Returning to cognitive psychology, some of its most important studies, such as those that have to do with the dichotomy between limited or unlimited processing capacity, or also the different types of processing (in parallel or in series, for example), are very present questions for the studies of mathematical psychology.