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 &quo... (read more)

Upvoted for "Proofs and Refutations" reference

[LINK] Steven Landsburg "Accounting for Numbers" - response to EY's "Logical Pinpointing"

by David_Gerard 1 min read14th Nov 201247 comments


"I started to post a comment, but it got long enough that I’ve turned my comment into a blog post."

So the study of second-order consequences is not logic at all; to tease out all the second-order consequences of your second-order axioms, you need to confront not just the forms of sentences but their meanings. In other words, you have to understand meanings before you can carry out the operation of inference. But Yudkowsky is trying to derive meaning from the operation of inference, which won’t work because in second-order logic, meaning comes first.

... it’s important to recognize that Yudkowsky has “solved” the problem of accounting for numbers only by reducing it to the problem of accounting for sets — except that he hasn’t even done that, because his reduction relies on pretending that second order logic is logic.