>mathematics is a system of assumptions and generative rules, and more generally a discipline around creating and operating such systems. This is essentially t

>mathematics is a system of assumptions and generative rules, and more generally a discipline around creating and operating such systems. This is essentially the standard mathematical approach developed during the early last century. From a few basic axioms (which are not really justifiable) new statements are proven and structures are built up. All notions of truth and provability are relative to that system. In standard ZFC (those are the standard axioms) mathematics "1+1=2" is, like all other statements, a statement about sets. The statement is true by the definitions of 1,2, "=" and "+". In an alternative system with different axioms or definitions the statement is false. This is not the view of intuitionists though. For them the symbols 1,2 or + aren't formalized objects (e.g. sets in this case). They are just symbols, which transfer some (hopefully) shared meaning to another person, who then (potentially with some addition arguments) might also accept the truth of that statement. In the former the question of "truth" is fully formal there can be no "interpretation" in any meaningful sense. Intuitionism places mathematics fully inside the mind of the mathematician and "truth" can only be found there. >Can something relating to an imaginary system can be true, or is it false because truth can only apply to objective reality? In formalized mathematics "truth" is fully formal. There can not be any "external" truth derived from it, as any such statement is non-sensical.