Why the Domains and Ranges of Functions in Physics Are Often Ambiguous

Unlike mathematics, where humans can decide the domain and range, the definitions of various concepts in physics are often difficult to understand because, in a sense, that is the very essence of physics as a discipline. The fact that physics is the study of the principles of things means that we are exploring precisely because the definitions and essence of those 'things' are not yet fully understood. Unlike mathematics, where one can construct logic by strictly defining the subject in advance, physics belongs to the natural sciences, where one must face the singular, complex, and mysterious reality of nature. While the objects of mathematics do not need to be related to the real world, physics cannot afford such luxury.

Concepts in physics are defined through provisional, operational definitions based on a solid understanding of the results obtained from experiments and observations up to that point. Furthermore, as technological progress expands the range of what can be experimented upon, the perpetual endeavor of continuously improving existing definitions and theories based on new results is what constitutes physics and the natural sciences.

The reason why the domain of the variable x and the range of the function f(x) appearing in physics textbooks are often not explicitly stated is that both the domain and the range are the 'natural world.' Physicists wish to discuss the natural world, about which there is still much that remains unknown.

Therefore, unless otherwise specified, functions f considered in physics are always discussed based on the common understanding that 'f: natural world → natural world.' For the domain and range, we provisionally set values that are valid within the regions known from experiments and observations to date, and we will investigate the details from here on. This, too, illustrates the essence of what kind of discipline physics is.

In mathematics, as long as there is no logical self-contradiction, humans can freely set domains and ranges to consider various functions, but in physics, which deals with a single, unique nature, one cannot do so arbitrarily. If, at the end of physics research, the 'domain' and 'range' in the mathematical sense were to become clear, it would mean that the physical object has been completely understood, that the physics of that theme has fully matured, and that it has reached a state of senile decay with nothing left to do.

It is the physicist's job to track down and narrow down the mysterious domains and ranges currently hidden in the natural world through experiments and observations. In this sense, I believe physics students should recognize that their work is significantly different from that of mathematicians.

Physicists are very similar to the detective Sherlock Holmes, who says, 'Watson, explanations later. Come, let's go!' It is important to catch the culprit (in physics, the mechanism by which a physical phenomenon occurs) quickly through reasoning based only on the evidence available so far. I hope that physics students will grow into such great detectives. Formalization using rigorous mathematics can be left to the experts later. I believe that the most important thing is to catch the culprit for certain first.