Psychology feeds 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” was coined** to talk about the contributions of some authors.

We will see how two disciplines intertwine and what are the benefits that can be drawn from this relationship in order to develop different methodologies to carry out new research in the field of the study of the human mind.

Table of Contents

## 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 the processes of thought, perception or any other psychological process. The goal would be to be able to quantify the behavior and the stimuli that provoke it, to find the mathematical laws underlying this relationship.

Therefore, mathematical psychology is **a way to standardize psychological processes so that it is easier to measure them and to be able to work with the relationships between stimulus and response**, Thus obtaining much more precise and rigorous hypotheses and verifications. The way to be able to quantify the behavior of the individual goes through a procedure in which he must perform certain tasks.

The first connection between psychology and mathematics took place much earlier than it seems. They were extraordinary scientists like Galilego Galilei or Johannes Kepler, those who **in the 17th century, they tried to verify whether thought processes were governed by specific laws**, As was the case with physics. Logically, this approach was very diffuse, because psychology did not even exist as an independent science.

In the 18th century, some of the foundations on which mathematical psychology would later build. It is at this moment that Blaise Pascal develops the argument of Pascal’s bet, within the theories of probabilities. Shortly after, Nicolas Bernoulli, for his part, developed the Saint 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**, Proposing Bayes’ theorem, among many other contributions. Another author who went on to produce studies on which mathematical psychology would later be based is Robert Hooke. If this is the case, this English scientist is carrying out the first research on human memory, in search of predictive models.

## Contributions in the 19th century

The nineteenth century is when the great advances in psychology take place, taking its own identity as a scientific discipline, from the hand of the German Wilhelm Wundt, who founded **the first experimental psychology laboratory**. It was therefore when we began to try to explain human behavior in a scientific way and therefore that mathematics made their definitive appearance to form mathematical psychology.

**During these years, psychophysics also developed**, With authors like Ernst Weber or Gustav Fechner, who respectively develop Weber’s law and Fechner’s law. But even astrophysics has had some influence on mathematical psychology. How can this be? Due to studies in which the distance at which the stars were measured and for her was measured when they passed in front of the telescope.

The point is that it has been observed that the reaction time of different people responsible for taking the measurements is different. It was Friedrich Bessel the scientist who discovered these differences and developed personal equations for them to compensate for the characteristics of the observer who noted the recordings and obtained the most precise data on the distance of the stars. Another step towards mathematical psychology.

also, **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 perceive a specific part of the visible light spectrum, giving rise to the color vision that we have in humans.

Continuing his contributions to mathematical psychology, **Franciscus Cornelius Donders, Dutch author, conducted research to measure the time it takes for the brain to perform simple operations**. For his part, Johann Herbart also worked on mathematical models that could explain human consciousness, a truly ambitious job for his time.

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

## Mathematical psychology in the twentieth century

Another prominent author who encompasses the last decades of the 19th and early 20th centuries is Charles Spearman. He is neither more nor less than the creator of factor analysis, a statistical system that uses variance and covariance to fuel **study individual differences mathematically**. To this method are added two others, such as the modeling of structural equations on the one hand and the ANOVA, or the analysis of variance on the other hand.

The first is the result of researcher Sewall Wright and the second was developed by Ronald Fisher. Along with factor analysis, these methods represent a major breakthrough in the union between mathematics and psychology, crystallizing the branch of psychometry, which is related to mathematical psychology. Psychometrics was therefore officially developed in the mid-1930s.

With the advancement of the behaviorist stream, even greater emphasis is placed on variables such as reaction times. In those days, the outbreak of World War II, an event that **improves research related to mathematical sciences, logic or computer science**, Concepts that apply to other sciences, such as psychology. Of course, mathematical psychology emerges strengthened from 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 others. Some of them had been related in some way to psychology before, but the use of others meant 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 specializing in this branch began**, Which meant the consolidation of it and a new and fundamental role in psychology.

## Differences between mathematical psychology and psychometry

It is important not to confuse mathematical psychology with psychometry. **Psychometrics refers to statistical studies of quantitative measurements performed 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.

Moreover, psychometrics is particularly concerned with the explanation or classification of individual or population differences while mathematical psychology, in turn, seeks to generate models that can offer an explanation of the behavior of any average individual, that is. that is, it predicts psychological behavior under certain conditions.

Likewise, psychometrics attempts to discover the relationship between different population variables statistically analyzed. In contrast, mathematical psychology focuses on creating mathematical models into which all experimentally recorded psychological phenomena can fit.

Therefore, although mathematical psychology has some connection with psychometry in some respects, this connection 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 brains, evolutionarily, must be configured to deal with different problems in an optimized way that increases the chances of overcoming them satisfactorily and with minimal use of possible resources.

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

#### Bibliographical references:

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