Brian Bi

What is the difference between mathematics and physics?

Many fields of study have overlap. In the case of physics and mathematics, the overlap is particularly large. So if you are going to point to a specific topic, like "representation theory of SU(2)", and ask whether that topic belongs to mathematics or physics, and how you can tell the difference in general, you might not get a good answer.

The crucial difference is that the two fields differ in objectives and, because of this, they have different ways of evaluating ideas.
  • Mathematics is, in some sense, the study of all concepts that can be described precisely. Mathematicians focus their attention on specific classes of problems, structures, relationships between structures, etc. that are interesting for their own sake or likely to reveal insight into other concepts. However, the only requirement for something to be good mathematics is that it is logically correct.
  • Physics is the study of the physical universe. Physicists aim to understand the properties of physical entities and how they interact with each other. Physicists seek and employ mathematical tools in order to describe the physical universe precisely, so they focus their attention on mathematical concepts that are useful for describing objects or phenomena that might actually exist or occur physically. Experiment is the final arbiter of truth and no matter how interesting or insightful a theory is, it must be discarded when it is contradicted by physical evidence.
It follows that there are many purely mathematical topics that are nevertheless studied by physicists. If you believe that physics is what physicists do, then those topics are part of both mathematics and physics. Those topics, however, belong to mathematics and physics for different reasons, and this is the essential difference between mathematics and physics.