I have always been amused by the similarities between the different systems around us. It is a true joy to examine the remarkable resemblance between models of electrical, mechanical, thermal, hydraulic, and even biological systems. Throughout the ages, scientists and mathematicians have been developing tools to deal with these systems in a unified manner; tools which fill the gaps between various disciplines and which provide us with a set of equations and models in which one can replace the corresponding elements and variables of a certain system by those of another, and use the same model to analyze and predict the performance of the new system.
Similarly, on a more elementary and physical level, physicists have been striving for centuries now in an attempt to reach an understanding of the universe we live in. Step by step, the different forces of physics are being grouped and joined together in a beautiful manner (General Relativity, Quantum Mechanics,…). But each theorem functions only within a limited scope and on a certain scale. The ultimate goal would be reaching a grand theory of everything, which governs the behavior of everything in the universe, and of which different, simple systems are nothing but special cases.
In the engineering sense, mathematics and Control Theory have played crucial roles in unifying various systems and disciplines in terms of analysis and design methodologies. Control Theory has been tailored to deal with most systems, whether linear or non-linear, and it utilizes almost every field in mathematics to do so. Highly complex and non-linear systems have even been compared to existing complex biological models in an attempt to find analogies when classical techniques fail. An example of this would be Neural Networks or Fuzzy Logic, which use the structure of the brain and human reasoning techniques, respectively, in order to model and analyze, and hence control, complex systems. But all those techniques are still developing, and the journey continues for control theorists to reach a valid set of equations and analysis techniques that unify such systems in terms of stability analysis and design, and hence develop a solid science.
It is marvelous how it seems to be embedded within us. Man has always tried to unify things around him and group them in a grand category. It simply makes sense. And as a result of this quest of making analogies, theories have been validated, while others falsified, and new sciences were born. In that sense, the achievements of a certain field will nourish the other fields and so on. Thus is the beauty of mathematics; a universal language of all disciplines.
Filed under: Mathematics, Philosophy, Science and Technology
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