(I hope a real mathematician like Noether will correct me on this one.)

In the course of the 19th century, mathematicians studying Euclidean geometry became aware that you can entirely separate formal deductive reasoning (mathematical logic) from the subject matter it is about.

The case in point: as long as all reasoning in Euclidean geometry is logical argument, concerned with proving logical statements from the basic postulates of Euclidean geometry, it isn't really specific to the subject of geometry at all! Rather, it is applicable to any subject that satisfies these postulates. The reasoning itself can be thought of as a set of rules by which postulates can be transformed or combined into more postulates.

Mathematicians tried to omit some of Euclid's axioms and replace them with others, and they found many an instance of non-Euclidean geometry that turned out to be interpretable in the real world - for example, geometry on a sphere can be described in this way.

So the next question was: what is Euclidean geometry fundamentally, is it the subject matter that we attempt to capture with Euclid's axioms, or is it the formal mathematics created by these axioms? And, since the same approach can be extended to all other areas of mathematics, what is the subject of mathematics in general? What is mathematics fundamentally, is it the formal game of playing with axioms and deduction, or is it fundamentally a science, that expresses truths about the (Platonic) world in terms of mathematics? Is a triangle something real, or merely a convenient vehicle of thought for what is really a game with deductive rules applied to logical statements?

The world's foremost mathematician, David Hilbert, defended the view that mathematics is fundamentally the game of mathematical logic, the game of formulating and proving formal propositions. The founder of topology, L.E.J. Brouwer, defended the view that mathematical logic is merely a tool to capture the mathematician's discoveries about the mathematical world.

To Hilbert, you can pose any set of axioms you want; as long as they are consistent, the results are equally meaningful to mathematics. Euclidean geometry is the set of rules that Euclid happened to use; if Euclidean geometry is applicable in the real world, that's great, but it is of no concern to mathematicians. To Brouwer, all axiom systems are no more than tools to capture the real laws of real subjects; in the case of Euclidean geometry, the axioms capture Euclid's observations about the laws that apply in land measurement.

What makes this mathematically relevant is Brouwer's insistence that certain reasoning rules don't make mathematical sense; he insists that they must be constructive, they must define a method to construct the mathematical objects they define. The resulting mathematics is known as intuitionism. Other constructive mathematicians impose even stricter rules of meaningfulness than intuitionism does.