History of the abstract method in mathematics Recently I have "finished" a 13-year on and off research on the history of the mathematical notion of equivalence. At the end of which, I learned that we owe the nowadays rather elementary process of "Definition by abstraction" (taking equivalence classes as new objects")  not to a person, a mathematical theory, a domain 
of study, but to the acceptance of what Timothy Gowers (2002, p, 18) calls “the 
abstract method in mathematics”, that " a mathematical object is what it does.” 
The following comment on the definition of cardinal numbers, written by Hausdorff about a century ago, shows the influence of this attitude on our understanding of equivalence.   

This formal explanation says what the cardinal numbers are supposed
  to do, not what they are. More precise definitions have been
  attempted but they are unsatisfactory and unnecessary. Relations
  between cardinal number are merely a more convenient way of expressing
  relations between sets; we must leave the determination of the 
  “essence” of the cardinal number to philosophy. (Hausdorff 1914, pp.
  28-29)

I am not sure "the abstract method" to be a common name for this attitude/philosophy. To be honest, I've believed in it since my undergraduate days without having a name for it. Has it ever had any name? What is its story? Why and how did it start and spread? Who were the main advocates? The main antagonists? Asking a single MO-style question: what is the history or a history of the abstract method in mathematics? I would be more than happy to have some references for study.   
 A: The question is interesting, but presupposes that we understand and accept Amir's conclusion, namely that "we owe Definition by abstraction (taking equivalence classes as new objects) to the the abstract method in mathematics, that a mathematical object is what it does.” I do not fully understand it, and as I understand it now, I am not convinced. 
"Taking equivalence classes as new objects" seems essentially as old as human thought or at least human use of language. A word never denotes a "primitive" object, always an equivalence class, and a baby between one and two spends a lot of his time actively learning how to manipulate equivalence classes. By the way, Grothendieck says something similar somewhere in Recoltes et Semailles (this is not intended as an argument by authority, however it may seem :-) about the word "maman" (French version of the cross-language word "ma" or "mama"), in general one of the first word that a baby utters, at first used for what it does (calling Maman for cuddling, fooding, etc.) but which soon becomes a name for the class of all persons that are in the same relation to another baby (then another person or even animal) as the baby's own mother is to itself. (Grothendieck's point by the way is that the "method of abstraction" is universal and that he just pushed it maybe a little further than others) 
That seems different, and in a complicated relationship, with the idea that "a mathematical object is what it does", for which I think the proper name is functionalism. 
A: It sounds like you were talking about formalism as a philosophy of mathematics https://en.wikipedia.org/wiki/Formalism_%28philosophy_of_mathematics%29  To me, abstraction would be a different matter, and is as old as mathematics itself -- the very concept of (natural) numbers is a form of abstraction.
Formalism may be most identified with Hilbert, though Dedekind's construction of real numbers may be a prominent precursor.
