IS PHYSICS CHARACTERIZED BY MATHEMATICS
ANIL MITRA © June 2016
These reflections began as a part of a response to a an answer to a question on a social network, Quora. The original question concerned how physics can be theoretical. My answer is here. Another answer suggested that “Physics is arguably that tiny subset of reality that is susceptible to mathematics.”
I wrote this piece as a response. I have placed it here because it is a little long for a comment on an answer on Quora.
The question was ‘How can physics be theoretical?’ and not ‘How can physics be mathematical?’ The latter is interesting and related to but is not the op’s question.
How is physics theoretical? Physics is not just theoretical but it is theoretical in that it refers to elements and patterns in the real and the theoretical or conceptual is about these elements and patterns.
Physics may be said to be about the elementary, original, and foundational forms of empirical reality. What do these terms mean? And where, how, and why does mathematics fit in?
Elementary means ‘simple’ and, though not atomic in the sense of indivisible, it includes the closest we have gotten to the indivisible so far. From the elementary character, conceptual representations would be among the simplest. It is therefore arguable to say that the concepts of the elementary forms generate mathematics just as much as that mathematics characterizes the elementary forms.
Original means ‘present at the beginning’ which is not true for all of conventional physics but is significantly true for the elementary part of it (in the big bang cosmology).
Foundational means that the ‘rest of reality’ is made up of the objects of physics – foundational means constitutional. We don’t know that this is true but regarding the objects of the sciences we have little or no reason to think that the constitution of the more complex objects of life, mind, and society is anything more than the physical. However it does not follow that physics is sufficient to describing those more complex objects. Still, even in the sciences of life, mind, and society there are, amid the complexities, elementary aspects that are significantly and usefully amenable to mathematics. And since the history of physics, mathematics, and the other sciences is not complete we do not yet know what the essence of the situation is.
Thus the ‘mathematics’ criterion does not essentially distinguish physics from the other sciences.
Does the ‘mathematics’ criterion fully characterize physics? It is certainly a partial characterization. Is the characterization complete and will it project into the future? There is a conceptual stage to the development of physics that must be at least superficially non mathematical – think of Faraday’s intuitions about the electromagnetic field and Einstein’s reflections in his fundamental Special Relativity paper on simultaneity. Will the important strong bond between mathematics and physics continue into the future? We might hope so because that is what gives physics much of its power. But it does not follow that the hope will be realized! And surely reality and history will be better judges. In any case, it might turn out that rather non mathematical computer simulation (and imagination) will be a powerful tool for the future. Of course, mathematics will probably be important in making such simulations efficient. And such developments would influence our notions of what mathematics and physics are.
That is, mathematics is important but not defining in the characterization of physics.