Mathematics and the Dynamic Aspects of the Physical World
The physical world is throbbing with activity. Nothing ever remains the same. Change is the one pervasive feature in the universe. This inescapable fact has been recognized from the most ancient times. The Heraclitan phrase that all is flux and nothing is stationary has its echo in the thoughts and reflections of thinkers in other cultures also.
With the incessant passage of time, each view, every facet and part of the physical universe, from the microcosmic to the galactic, transforms into yet another. Science is interested in the dynamic aspects of the world.
Moreover, practically every change results from, and is the cause of, other changes. In other words, there is not only change, but also mutual and relative change. The length of a rod changes with change in temperature; the speed of a body is affected by the forces acting on it; an electromotive force is produced in a circuit when the magnetic flux through it changes; enormous gravitational pressures lead to extremely high temperatures, which in turn provoke nuclear fusion reactions, etc.
Since science explores the quantitative aspects of the world, one is also concerned with the mathematical aspects of changes. In fact, observable changes are expressed in terms of mathematical variables. Thus we may conceptually look upon the world as replete with invisible physical quantities that continuously keep changing with respect to one another. To understand the nature of physical phenomena we need to study the relative changes of these measurable quantities.
Hence it will be useful, indeed necessary, to express not only the quantities in question but also their mutual relationships in the symbolism of mathematics. The branch of mathematics that deals with changes is the differential calculus. Hence its importance in physics. If we lived in a static world where everything remained for ever the same, there would be little need for this kind of mathematics that is so pervasive in our physics.
The role of mathematics in science extends beyond the tagging of numbers to the measurable features of the world. Mathematics becomes indispensable in the analysis of the manner in which the various quantitative attributes of physically observable systems vary, and vary with respect to one another. Such analyses have two major consequences: First, they enable us to determine the precise functional dependence between or among the variables involved. Secondly, they also permit us to calculate and determine how these quantities will change from point to point and from instant to instant. In other words, the mathematical formulation of the precise manner in which mutually influencing changes occur with respect to both space and time in the context of a given phenomenon enables us to describe its evolution or development with a fair degree of precision. That is to say, the application of mathematics to the dynamic aspects of the world enables us to predict the evolution of systems.