Comments on "General Fourier solution and causality in attenuating media" by R.E. Duren

Abstract

For original paper, see IEEE Trans. Electromagn. Compat., vol.35, no.1, p.43-8 (1994). When applying mathematics to physical problems one must satisfy the mathematical axioms and in addition the physical laws. Neither the mathematical axioms nor the physical laws can be listed comprehensively, but one does not need to concern oneself too much with mathematical axioms as long as one does not make a mathematical mistake. The physical laws must be introduced into a mathematical model of a physical process, which implies that one must know which physical laws are important in any particular case and one must be sure to actually introduce them. Confusion is caused by the practice of using physical sounding terms in applied mathematics. For instance, R.E. Duren calls a function of time "causal" if it is zero for values of the time variable below a certain threshold. No law prevents one from doing so, but such "causal" functions have no evident connection with the causality law of physics and no results about physical causality should be derived from them. How does one know that the causality law is a physical law that has to be introduced and not a mathematical axiom that is satisfied as long as one calculates correctly? To answer this question the present author considers the causality law in the following form: every effect requires a sufficient cause that occurred a finite time earlier (Harmuth, 1993). The use of the term "time" shows that one is dealing with a physical concept. Pure mathematics has no time variable or spatial variable but it has complex variables.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>