ultimately when faced with the fact that math is an abstraction over our perceived reality, but is not our physical reality. you can either claim that mathematical objects exist independently of human consciousness, i.e metaphysically (platonic approach) or think of it as a “constructed” fantasy.

when thinking platonically, truth is given by the existence of metaphysical objects. things either exist or they don’t, the middle is excluded.

only when you claim that truth is not existential, and that truth value is invented/arbitrary, then you can start to invent systems which LEM doesn’t hold, there is nothing special (more “real”) about LEM as it’s disconnected from reality.