Knowledge Question: If we treat mathematics as a kind of game with it’s own set of rules. Then to what extent can the rules of mathematics be changed, but can still be treated as the same kind of “game”?

Scope/Applications:

The scope of this knowledge question (KQ) is mathematics and is the study of quantity, space, shape and change. By applying the important knowledge in mathematics regarding quantities, spaces, shapes and changes, we can learn more definite truths about the world around us. Some practical problems that maths helps to solve can be as simple as how much change you should receive after paying for something, to something as difficult as black holes. Currently some of the main questions that mathematicians are trying to find answers to are the remaining unanswered millennial prize problems.

Concepts/Language:

The foundation of mathematics, mathematical axioms are the initially concepts that mathematicians use for their mathematical proofs. However in order for mathematicians to be able to solve new and harder problems, they have been forced to bend some of the fundamental axioms in order to solve these problems. However in doing so, have mathematicians stepped into a new “game” of mathematics; or are they simply playing the same “game” with revised rules?

Methodology:

Answering this question would require one to fully understand all the different mathematical axioms there are, and then look at all the different “games” of mathematics that are a result of changing the basic Euclidian axioms. Then one should see whether the kind of mathematics done in these new “games” are fundamentally the same as regular, traditional mathematics; or are something entirely different.

Historical Development:

Axioms in Euclidian geometry can be considered to be the fundamental “rules” for the game of mathematics. Much later on in time, Hyperbolic and Elliptic geometry modified these “rules” so that they can work in other situations. One can look into these relatively newer forms of geometry and compare the mathematics done in Hyperbolic, Elliptic and Euclidian geometry to see whether they are fundamentally the same; or completely different.

Links to Personal Knowledge:

As of right now, I am only versed in the axioms of Euclidian axioms (ie Things that coincide with one another equal one another) and the mathematics behind Euclidian geometry (ie the sum of the interior angles of a triangle is 180º).