Proof of identity A = A or 1 = 1

Question

Is 1 = 1 an assumption? I feel it's a very good assumption, but is there a proof for it? Imagine a world where people were contesting it, where equivalence wasn't a common sense concept. In reality no 2 things are exactly equivalent because even comparing an object to itself would least differ in temporal location, but in our minds we have the concept of perfect equivalence. Ask yourself, what if 1 ~ 1

Answer 1

In most ways to formalize logic with equality, "$t=t$" is an axiom for every expression (or "term", in the jargon of logic) $t$.

In other words, the validity of the claim $t=t$ is part of the conventions about what the symbol $=$ means. It's not really about any deep or philosophical truth -- simply an agreement among mathematicians that if you want to think about a relation that does not relate everything to itself, you're free to do so, but you must not use the $=$ symbol to write it (on pain of confusing everyone you speak to).

[IEEE-754 NaN comparisons nonwithstanding].