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RichardKennaway comments on [LINK] Steven Landsburg "Accounting for Numbers" - response to EY's "Logical Pinpointing" - Less Wrong
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So, they were lucky. It could have been that that-which-Pythagoras-calls-number was not that-which-Fibonacci-calls-numbers.
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?
(I know of no such examples, but I would be surprised it none exist)
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