Bob K.,

Without abstract mathematics, there would be no quantum field theory, no theory of relativity and all of the other mathematical elaborations that make physics possible.

*the real Axiom of Principle of Identity is bound up with the underlying symmetries and invariants of physical reality. Rand's simplified approach is a mere cartoon this principle.

There probably is some valuable work being done in the field of mathematics, despite being rife with invalid concepts, flawed methodology and flimsy epistemological foundations. (This is equally true of the current state of physics.)  Much of it is barren gamesmanship, just as Robert M. said.  The context for his comment was the very prevalent nonsense referenced in prior posts.

Your critical comparison of the mathematical axiom of identity to Ayn Rand’s foundational principle confirms my own suspicion that you are the one who has “not taken the time and effort necessary to understand” Objectivism.  No one with an inkling of insight into Objectivist metaphysics would criticize a philosophic axiom for being “simplified” or lacking the complexity you describe (“bound up with the underlying symmetries and invariants of physical reality”). 

That’s fine.  No one is suggesting that you not continue posting here.  But the truth of the matter is that your absolute dearth of rudimentary knowledge makes one wonder what your purpose might be.  

These are good sources:

Non-Classical Logic

Graham Priest (Cambridge University Press 2001)

Relevant Logic

Edwin D. Mares (Cambridge University Press 2004)