HISTORIAS                    Manuel Espejo

Catastrophe Theory

 

The term "catastrophe theory" may initially appear misleading, as it might suggest a connection to events like earthquakes, floods, or hurricanes. However, it does not carry a negative connotation, as we'll explore further in the following discussion.

From a mathematical perspective, a catastrophe represents a significant change in a system's state—a discontinuous shift where reality undergoes a radical, non-linear transformation, subject to new rules. It is distinct from the gradual, continuous variations that occur in any system, such as the aging process in individuals. When a catastrophe occurs, the continuous changes in a system's parameters eventually lead to a rupture, propelling it into a qualitatively different state.

For example, consider the shift from liquid water to ice at around 0 degrees Celsius or from liquid water to a gaseous state at 100 degrees Celsius. In both cases, the resulting state is entirely new, governed by different rules than the previous one. Other examples of catastrophes include a balloon bursting due to excessive pressure, a stock market crash, a beam suddenly failing to support its load, a manic-depressive episode in a mental patient, or a prison riot. Countless such examples exist, all representing changes of state.

Small, continuous alterations—such as changes in temperature, weight, age, or pressure—accumulate within a system until they culminate in a discontinuous change of state. At this point, various bifurcations can occur, signifying that, after a catastrophe, reality can embark on different paths.

In the context of catastrophe theory, change is not regarded as an anomaly; instead, it constitutes the very essence of the mathematics involved. This mathematics stands in contrast to deterministic approaches, which explain phenomena using mathematical functions that consistently yield the same results. Traditional mathematics, taught at all levels of education up to the university level, portrays a world where change is absent, as functions unfold uniformly and predictably. It represents a static world.

Conversely, catastrophe theory introduces a dynamic world where everything is comprised of possibilities and bifurcations that lead to uncertain new scenarios. It distinguishes between state variables, which define a specific situation between one catastrophe and another, and control variables, which designate points where changes of state become possible.

The reality we know is a composite of various states, each eventually transitioning when it reaches a control variable, thereby transforming into something entirely different. This new reality adheres to laws and regulations distinct from those governing the previous state.

Catastrophe theory furnishes valuable mathematical models applicable to both the natural and human sciences. What makes this mathematical theory intriguing for non-mathematicians is the following:

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• Reality is in a constant state of flux, characterized by perpetual motion. Change is the sole constant.

• Every reality undergoes continuous transformations, and at some juncture, it may experience a discontinuous shift—a rupture leading to a new reality.

• ​This new reality operates under laws and principles distinct from those of the preceding state.