I thought he was asking if it had ever happened in any not-yet-axiomatised subject, presumably looking for examples other than arithmetic.

Yes. I think the mathematicians were lucky that it didn't happen on the sort of integers they were discussing (there was, after all, great discussion about irrational numbers, zero , later imaginary numbers, and even Archimedes' attempt to describe big integers was probably not without controversy ).

1JoshuaZ7yHmm, in that case, it might be relevant to point out examples that don't quite fit Plasmon's situation but are almost the same: There are a variety of examples where due to a lack of rigorous axiomatization, statements were believed to be true that just turned out to be false. One classical example is the idea that of a function continuous on an interval and nowhere differentiable. Everyone took for granted that for granted that a continuous function could only fail differentiability at isolated points until Weierstrass showed otherwise [http://en.wikipedia.org/wiki/Weierstrass_function].

[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.