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What does it mean in mathematics for two things to be equal, or for two things to have the same "identity"?
For example, because anything divided by zero is "undefined", can we say that 1/0 = 2/0?
What about the relational database concept of "null" which is supposed to stand for "unknown"? In relational algebra, they say NULL is not equal to NULL, but doesn't that violate the law of identity that everything is equal to itself?

I think it is important to distinguish here between the meanings of expressions and the things that those expressions denote. Peter is right that the expressions "2+2" and "4" are different expressions, and they are not synonymous. But they both denote the same thing, namely the number 4. Now, in the equation "2+2=4", is the equal sign being used to express a relationship between meanings of expressions, or does it express a relationship between what is denoted by those expressions? (In other words, is this an intensional context or an extensional context?) I would say it expresses a relationship between what is denoted by the expressions, and the relationship is identity: what is denoted by the two sides of the equation is one and the same thing, namely the number 4. So I disagree with Peter's conclusion about what "=" means in mathematics. I would say "=" in mathematics means "is identical with". I would say that the situation here is very much like the situation in the sentence "The morning...

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