The Principle of Mathematical Induction: Applications in Physical Optics

Abstract

The bare rudiments of the principle of mathematical induction as a method of proof date back to ancient times. In the contemporary university milieu, the demonstrative scheme is taught as part of a course in discrete mathematics, set theory, number theory, graph theory, group theory, game theory, linear algebra, logic, and combinatorics. In theoretical computer science, it bears the pivotal role of developing the appropriate cognitive skills necessary for the effective design and implementation of algorithms, assessing for both their correctness and complexity. Pure mathematics and computer science aside, the scope of its utility in the physical sciences remains limited. Following an outline of some elementary concepts from vector algebra and phasor analysis, the proofs by induction of a couple of salient results in multiple-slit interferometry are presented, viz., the fringe intensity distribution formula and the upper bound of the total fringe count. These specific optical instantiations serve to illustrate the versatility and power of the principle at tackling real-world problems. It thereby makes a welcome departure from the popular view of induction as a mere last resort for proving abstract mathematical statements.