## Can Psychology Be a Mathematical Science? Can psychology be a mathematical science?

The simple answer is “Yes, and it has been from the very beginning.” Psychology became a recognized science in the mid 1800s. The key finding that shifted psychology from a philosophical pursuit to a scientific one was the discovery of a quantifiable mathematical relationship between an objectively measurable physical variable (the change in magnitude of a sensory stimulus such as the increase or decrease in the brightness of a light or the loudness of a sound)  and a subjective psychological  variable (how much the sensory stimulus is perceived to increase or decrease by the person experiencing it).

Not only was scientific psychology founded on a mathematical basis but there are many psychologists who engage in mathematical theory building and testing every day, and there are several sub areas within the general field of psychology, such as cognitive science for example, that are firmly grounded on mathematical modeling.

That being said, it is still the case that psychology has not been as thoroughly or securely placed on a mathematical basis as several other scientific disciplines. One of the reasons for this is that that the basic phenomena of interest in psychology – human mind and human behavior – are much more complex than the basic phenomena of these other sciences. Take Newtonian physics as an example. Suppose something is moving from point A to point B and you want to know the route it will take, how far it will travel and how long it will take to arrive.  If the something in question is a baseball or a cannonball, you can do a pretty good job of predicting these things based on the elevation above the ground from which the object was launched, the angle and the speed at which it was launched, and the gravitational constant. The situation is vastly more complex if the something in question is a person going home from work.  When predicting the route, distance and time taken by the cannonball you don’t have to take into account the conditions the cannonball thinks it might encounter along the route, how it thinks those conditions might affect its progress toward its destination, alternative paths it might consider and choose to take along the way, or a spur-of-the-moment decision to stop for a cup of coffee or pick up some ice cream for dessert. All of these things and more must be taken into account when we attempt to predict the route a person might take on her way home from work.

Putting a science on a mathematical foundation necessitates making the assumption that the phenomena that are the subject matter of that science have a structure that is mathematical at a fundamental level. Moreover, it isn’t enough to simply assume that the phenomena in question have a mathematical structure in some vague and general sense. The use of specific mathematical techniques requires that specific assumptions be met about the phenomena to which these techniques are applied. Determining whether these assumptions are met when the phenomena are as complex as they often are in psychology can be difficult. The mathematical techniques of regression and correlation are widely used to model psychological phenomena. Valid use of many of these techniques rests on the assumption that the variables that are used to measure psychological phenomena are linearly related. Variables are linearly related when a graph of the relationship between the two variables is a straight line. For example, human height and weight are linearly related as can be seen in the graph on the left. Many variables in psychology have a regular relationship but that relationship is nonlinear. For example, the relationship between the objectively measurable change in the strength of a stimulus and the perception of change in the strength of that same stimulus mentioned in the first paragraph is logarithmic as shown in the graph on the right.

There are many different types of nonlinear relationships. Determining whether complex psychological phenomena are linear or nonlinear, and if they are nonlinear, what kind of nonlinearity they exhibit can be a very daunting problem.

Drs. Michael Dougherty at the University of Maryland and Rick Thomas at Oklahoma University have proposed an interesting and important approach to this problem with the General Monotone Model or GeMM. GeMM allows a simplification of the assumptions that are needed to model either linear or nonlinear relationships in psychological theory. Because the model relies on less stringent assumptions, it can be applied validly in a wider variety of cases.

All linear and many nonlinear relationships share the property of being monotonic or monotone. A relationship between two variables is monotonic when as one variable changes in a single direction (it either increases or decreases) the other variable also changes in only one direction (again, it either increases or decreases). Monotonic relationships can be either positive or negative. Both the linear relationship between weight and height and the nonlinear relationship between the perceived and actual increase in the strength of a stimulus shown above are positive monotonic relationships. The relationship shown in the graph on the left between the amount of information remembered and the time that has elapsed since learning the information is an example of a negative monotonic relationship that is also nonlinear. The dose-response curve shown on the right is an example of a nonmonotonic and nonlinear relationship.

GeMM rests on the assumption that the variables that are used to measure psychologically meaningful phenomena are monotonically related. Under many circumstances it is much easier to provide evidence that supports this assumption than it is to provide evidence that variables are linearly related. In fact, one of the strengths of GeMM is that it can be successfully used as a basis for theory in psychology when you can’t tell whether the variables of interest have a linear or nonlinear relationship but you can provide evidence that the relationship is monotonic.

Having a model like GeMM that is easier to validate and can be applied in a wider variety of circumstances is only of value if the results that GeMM produces are as accurate as the models it replaces. When tested on linear data, GeMM performed as well or better than all of the strictly linear models that were tested on the same data. This means that when the variables of interest are in fact linearly related you don’t lose anything, and in some cases you actually do better, using GeMM than using models that are specifically designed for use with linear data.

When tested on nonlinear data GeMM performed better than all of the linear models that were tested on the same data. This is a major benefit because linear models, such as correlation and regression, are often used in psychology in cases where the relationship between the variables of interest may not be linear.

GeMM has several advantages over many of the linear models that are commonly used to capture the relationships between variables that measure psychological phenomena. First, GeMM rests on the assumption that variables are monotonically related and this assumption is often easier to verify than the assumption that variables are linearly related. Second, GeMM has a wider range of application because it can be used to model data that are either linear or nonlinear. Finally, GeMM performs as well or better than the linear models when the data are linear, and performs better than the linear models when the data are nonlinear.

Psychology has always been a mathematical science but applying mathematics properly in psychology is often more difficult than it is in some of the other mathematical sciences.  Theories like GeMM show promise of establishing psychology more firmly on a mathematical foundation by expanding the range of phenomena to which math models can be validly applied. 