When we say $a=a$ means that are absolutely the same right? What does equal mean in $3+2=7-2$? Does it mean that they are absolutely the same or that the value is just the same?

I've been tasked with explaining to a group of people what the notion of equality means in mathematics, I've come up with a working explanation, but would appreciate some input, suggestions etc. Mathematical equality $X=Y$ is a binary relation between two mathematical expressions $X$ and $Y$ whi...

An identity is a relation that means that whatever the number or value may be, the answer stays the same. But is it possible to have equality without identity?

I recently pulled out my old Real Analysis textbook and noticed something that didn't stand out when I was taking the class all those years ago. When the book is listing out the axioms it seems to assume we understand what equality is. Consider the first axiom $(A)$, which defines commutivity o...

This is my first post here so I'm not sure about the proper etiquette, but I have quite a few questions all pertaining to one underlying concept, and I hope it's okay that I've grouped them all into one post. I'm completely new to logic, or at the least the academic study of logic (we all make u...

Sometimes in real analysis, we write an equality and specify that is true in the sense of the $L^2$ norm. My question is, when in mathematics we write '=' and don't specify anything, what we really mean? Do we mean uniform convergence, more than that, or less than that?

what is the definition of $=$? Above is the question that I would like to be answered, below are some of my thoughts. I've been thinking about what it means to say $A = B$ I came to this from thinking about why it makes no sense to say something like $5 = \begin{pmatrix}1, 2, 3\end{pmatri...