A fundamental theorem of mathematics would be a powerful thing to have in your toolkit. This course makes a good case that symmetry, which plays important roles in both math and physics, may be a key concept that gets us closer to such a theorem.
A basic but powerful truth that unites much of mathematics is a seductive idea. Certainly worth setting as a goal for 21st century mathematics. Never know what you’ll discover.

Math is both inherent in nature and a human construct. In it’s simplest form it is born out of a desire of the human mind to quantify elements of the physical world we live in. One sheep, two shoes, three blind mice etc. It is a tool with which humans can quantify and measure the world and this, in turn allows us to find useful relationships between these quantities. From these basics we can derive further relationships by applying the rules of logic to ensure that from these basic truths (relationships) any newly derived relationships would also be true. And so we build a system of demonstrative mathematics – a deductive system where truth is demonstrated by proof. Euclid’s “Elements” is the template (or more accurately the Bible) of such mathematical systems.

From this we can see why math is so effective in describing the world around us. It builds rigorous systems of truth through proof (via logic). Whether or not these systems ever find expression in the real world, they none the less generate systems which are internally consistent. If the basic axioms are true then the rest of the system that follows from these is also true.

Does math exist by itself apart from physics. Yes. It is grounded in the real world but it has matured to the point where we can generate systems which may or may not exist in the real world. Some may turn out to exist in the real world, as with the Riemann-Einstein example that we have been given, but others may not. But that does not mean that we will not find or invent a way to make use of such abstract knowledge in the future. Part of the excitement is that we can not predict what will suddenly become useful.