Are there absolutely no examples of cases where mathematicians disagreed about some theorem about some not-yet-axiomatized subject, and then it turns out the disagreement was because they were actually talking about different things?
There is such an example—rather more complicated than you’re describing, but the same sort of thing: Euler’s theorem about polyhedra, before geometry was formalised. This is the theorem that F-E+V = 2, where F, E, and V are the numbers of faces, edges, and vertices of a polyhedron. What is a polyhedron?
Lakatos’s book “Proofs and Refutations” consists of a history of this problem in which various “odd” polyhedra were invented and the definition of a polyhedron correspondingly refined, until reaching the present understanding of the theorem.
There is such an example—rather more complicated than you’re describing, but the same sort of thing: Euler’s theorem about polyhedra, before geometry was formalised. This is the theorem that F-E+V = 2, where F, E, and V are the numbers of faces, edges, and vertices of a polyhedron. What is a polyhedron?
Lakatos’s book “Proofs and Refutations” consists of a history of this problem in which various “odd” polyhedra were invented and the definition of a polyhedron correspondingly refined, until reaching the present understanding of the theorem.
Upvoted for “Proofs and Refutations” reference