https://aeon.co/videos/western-logic-has-held-contradictions-as-false-for-centuries-is-that-wrongThe CPU bus logic of my computer and yours, is three-state, not two-state ... so the law of excluded middle doesn't apply. The notion that 2+2=4 is true in some number systems, but not in others ... 2+2=11 in base 3. One can claim that is just a notational technicality, but in number theory, that isn't so ... in arithmetic there are some things that are true in all number systems, some which are true in some number systems and false in others, and there are things that are false in all number systems. Some abstractions are so difficult, that there is no proof at present, that such and such is true in all number systems ... this may be because some proofs are undecidable (aka no Turing machine can simulate a proof, because the simulation doesn't necessarily terminate). A valid proof is one that can be shown to terminate after no more than X steps ... with confirmation or denial.