Please, list object-oriented systems with objects with identities and object-oriented systems without those. Count both. Discuss. ;)
Alan Kay, who coined term "object-oriented" described object-oriented system as an assembly of objects communicating by message passing. Message passing requires identities (and state, except in severely reduced form).
Message passing does require a sender, a receiver, and a message. You can have those in purely functional programming.
No identity needed. But if you really want one, you can have it: Two objects are identical, if they react to all messages in the same way. You can also explicitly define equality ==, and then say that a and b are identical if a == b.
My Ocaml code is not pure, but we can easy remove any occurrence of `ref`, and restricting oneself to pure methods, which produce a new fresh object instead of modifying `this` in place.
Sorry, I am not very fluent in OCaml. Also, I didn't find word "message" in the link above.
I am experiencing limitations of pure objects and message passing almost right now. In my spare time I am currently developing dynamic data flow CPU which is based on immutable "objects" and message sending. It is quite unusual style of programming, very unlike anything else, including Smalltalk, Erlang and everything.
Alan Kay, who coined term "object-oriented" described object-oriented system as an assembly of objects communicating by message passing. Message passing requires identities (and state, except in severely reduced form).