To What Extent Can Mathematical Modelling Capture the Complexity of Human Behaviour?

Imagine being able to model real life situations from falling off the stairs to lifting a dumbbell through a series of mathematical equations? Well, that's the essence of Mathematical Modelling and it extends beyond that as well. Essentially, Mathematical Modelling is the same as relating equations of motion to the real world, where we make assumptions, formulate a model of something like v = dS/dt in physics, solve the model and scrutinize it to understand its shortcomings. The concept of Mathematical Modelling came into existence more than 4000 years ago when the Babylonians (2000 BCE) used arithmetic models for astronomy. Over centuries, such modelling needs gave rise to what we now know as applied mathematics. Mathematical Modelling is seen almost in every industry nowadays, from STEM careers to social sciences, Mathematical Modelling is the base of almost everything. As Galileo stated, "Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and to read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it" (Galilei, 1623/1957, p. 183). If mathematics can explain the physical world, the question arises: can it do the same for human behaviour?

Human behaviour is perhaps one of the most complex domains of science to understand, which is influenced by both internal and external factors. Moreover, there are still some behaviours that cannot be explained and are beyond human understanding. An example of this includes the essence of human consciousness, it has been years since scientists have pondered upon this topic yet they haven't been able to come up with a definitive, scientific and concrete answer. Although scientists have been able to model human behavioural complexity to some extent they haven't been able to leverage these models to concretely make a breakthrough towards understanding the complexity of human behaviour.

Now the question that arises is: If scientists were able to model human behaviour to some extent, then to what extent is this, what models have they employed and what are the limitations that challenge them? Mathematical Modelling has been employed to test out many theories until now and to find subtle differences in between them, for example two theories that appear to make similar predictions when stated in words can be more readily compared and evaluated when they are put into mathematical form (Mazur, 2006). This essentially means that two theories which are not exactly the same but tend to give the same result can be differentiated from each other in a better way when represented and modelled mathematically. The extent of mathematical modelling is described metaphorically through this passage: "Mathematics can help illuminate complex systems, but it is important to evaluate critically why models work, recognize their limitations, and explore areas of uncertainty, especially when modeling human thought" (Rockmore, 2024). The extent of mathematical modelling can be thought of like this: as we try to include more and more complexities of the world into our model, the model itself becomes more complex, and the assumptions made are sometimes too far from reality, which makes the word 'modelling' lose its meaning in the process. To understand this better, we can take the example of the butterfly effect: when a butterfly flaps its wings in Tokyo, the forecast changes in New York. Before this effect, scientists thought they were able to model climatic conditions mathematically and were working on a new term called 'climatological warfare.' However, this proved the emerging concept of climatological warfare in the field of climatology to be unworkable. This illustrates a broader truth: while models can reveal fascinating patterns, they often break down when applied to systems as unpredictable as human behaviour and other complex phenomena such as climatic conditions.

When it comes to human complexity, going with an empirical approach is absolutely impossible as the amount of cases and experiments to be conducted would extend beyond millions, thus, going with an alternative approach saves and simplifies the process for us. This makes the use of mathematical modelling necessary in modelling human behaviour to some extent. The statement "I can calculate the motion of heavenly bodies but not the madness of people" by Sir Isaac Newton clearly implies that the actions a person does are completely independent of their previous actions in some cases. Unlike planetary motion, which follows stable physical laws, human behaviour is shaped by emotions, biases, and social contexts that resist reduction into neat equations. Moreover, the ways in which the human brain works is still not fully understood and is very complex. Mathematical models tend to work on the pretext of three main principles which are to make assumptions, borrow work and criticise your own work to understand the limitations of your model. However, in the case of human behavioural complexity, it is almost impossible to make models without a controlled environment and constraints, as this makes the model too difficult to solve and work upon.

The extent to which mathematical models can model are limited, for example, during the covid 19 pandemic the SIR based models were used to model and predict the amount of cases, however, their limitation of working only in equilibrium based settings wasn't tested which led to inaccurate predictions. Moreover, recent research has shown that a dramatic event like a lockdown can thwart the making of precise long-term predictions from S.I.R.-based models, even assuming perfect data collection (Rockmore, 2024).

In the world of sciences, there are many models that have been developed such as Drift Diffusion Model (DDM), Agent-Based Models (ABMs) and Behavioural game theory that model different domains of human decision making such as individual decision dynamics, group/emergent behaviour and strategic interactions. However, there is one limitation that is common to all of these models, which is that they all rely on simplifying assumptions about humans that strip away real-world complexity, essentially they ignore the other influential and concrete factors such as emotions, noise and other external factors. The limitation here forms the extent of mathematical modelling in our today's world, wherein we cannot accurately model the actual complexities and intricacies of human behaviour, even if we do they are based on assumptions that strip away the main aim of modelling human behaviour itself.

From these arguments we can establish a general answer that mathematical modelling works well when it is exposed to situations that aren't abrupt and complex like human behaviour which may or may not be influenced by external factors. These models can only model to the extent where the assumption made ignores the complexities such as emotions and external noise. However, once we fully understand the human brain, the essence of consciousness and emotions, modelling human behaviour would be comparatively easier.

Understanding human behaviour is complex and requires studying its basics empirically, theoretically, and mathematically, with realistic assumptions. In conclusion we need to understand that "Art is a lie that makes us realize truth" (Picasso), meaning that mathematical models are simplifications that help us uncover certain truths, but they never capture the full reality of human behaviour, because their assumptions remain too far removed from its essence.