Definition of definition

I was wondering if there is a good way to "define" what definition means exactly in mathematics. Since the answers may be subjective or philosophical, I want to ask only for references on this topic. So I am looking for references which answer the question

"What does definition mean in mathematics" in a concise and commonly accepted way.

and also for references which discuss philosophical problems connected with this question (if there are any).

Solution 1:

I would summarize my personal views about what "definition" means in mathematics as follows:

"[M]eaning is use — words are not defined by reference to the objects they designate, nor by the mental representations one might associate with them, but by how they are used. (source)

and

"Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only." - Henri Poincaré

For example consider the symbol "$\mathbb{Z}$". If I wanted to tell someone what I meant by it, I might write $$\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$$ or if were being highbrow, perhaps $$\mathbb{Z}\text{ is the infinite cyclic group.}$$ or $$\mathbb{Z}\text{ is the initial object in }\mathsf{Ring}.$$ If I spoke another language, or used a different method of writing integers, or used a different notation for sets, these would appear quite different. But such differences are irrelevant; it doesn't even matter if someone else's mental conception of the integers is radically different from my own. What matters is that our usages agree - if it is the case that, any time I write a statement about $\mathbb{Z}$ that I consider true, anyone else agrees that (modulo differences of language / notation) that is a true statement about whatever it is they think of when they see "$\mathbb{Z}$", then functionally, our "definitions" agree. So, I don't think of "definition" as a formal concept in math (I know almost nothing about logic / set theory / metamathematics - I am just expressing my opinions). Even in formal logic, how can we hope to define parentheses? Or "$\in$"? We just start using them, and if people agree what we've written makes sense to them, that's the best we can hope for - we can try to use natural language to convey our mental conceptions to other people, but we can't dive into their heads and check that their mental conception is actually the same. (Obviously, intuitions / mental conceptions are of the utmost importance in doing mathematics - we won't get anywhere with blind manipulation of symbols. I'm just saying that all we can check our agreement on are external expressions such as equations or sentences.)

Finally, I'd just like to add this comic from SMBC:

Solution 2:

How can there be 6 other answers, yet no one has so far mentioned conservative extensions and/or extensions by definition? This is the correct framework in which to view definitions.