Object thinking is about naming things, naming meta things, and then using those name to describe those relationships. E.g. Fido is a dog, and a dog is an animal because I said so. Names are incredibly useful as abstractions: I know Fred, if I see him wearing different clothes and a haircut, I know he is still Fred; my other friends can tell me things about Fred, fred can have a criminal history. This is only possible because Fred has a name, otherwise, he would just be a stateless blob.

Note that nothing is really proven with names, just asserted. This is why theoreticians dislike them. Equational reasoning gets away from names by working purely with structure. Of course, we can apply labels to structure (since our brains are so reliant on names for reasoning), but they have no special meaning and we are careful not to let them bias the results.

Most programs involve heavy doses of object thinking, even if the language is not specifically OOP. It takes a real genius (or a Vulcan...joking) like the high end haskell crowd (SPJ, Conal elliot, etc) to leverage equational reasoning where most would otherwise use object thinking.