"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.

So, they were lucky. It could have been that that-which-Pythagoras-calls-number was not that-which-Fibonacci-calls-numbers.

Are there

absolutely noexamples 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 aboutdifferentthings?(I know of no such examples, but I would be surprised it none exist)

Consider also the parallel 'postulate', reluctantly introduced as an axiom in Euclid. People tried to prove it as a theorem for two thousand years, until it was realized that its negation defined entirely different kinds of geometry.