As for the 1 does not equal 1 disproof, here is a very simple equation:
z=x+y, thus x=z-y, substitute z=z-y+y, z=z. However this is true under the condition that z=z, x=x, and y=y. If that condition is not true, thus the axiom of x=x not true, then z may or may not equal z.
The reason there is no way to prove or disprove this is that this is the fundamental axiom of mathematics, and the only way to disprove that z=z as our friend Daymare here is proposing is to assume that z=z thus creating a contradiction as to his proof.
There is no way to prove the axiom either, and that's why pure mathematics is USEFUL if it is applied. That's the proof, physical proof, but it's not mathematical proof, and thus it is an axiom. I think there are systems where x does not equal x, and such axioms are unprovable too. Such as 10/10=1/2, this statement cannot be proven true or untrue.
Whether a nature can be based on such a rule is under the extent of your imagination, since it could have some arbitrary value, say "v" meaning that all of it exists if and only if only half of it exists as the statement 10/10=1/2 suggests
In pure mathematics these two systems are both valid under their axioms, and there's no need to ridicule a person who's seeing the fact that axioms are unproven mathematically. It is only in nature that they apply.