I don’t know if this is true, but I once had a lecturer tell me that there used to be considerable debate over the question of whether a derivative of a differentiable function was necessarily continuous, which ultimately boiled down to the two sides having different definitions of continuity, but not realising it since neither had ever fully set down their axioms and definitions.
I’ve heard that before also but haven’t seen a source for it. But keep in mind that that’s a question about functions on the real numbers, not about what is being called “numbers” here which seems to be a substitute for the integers or the natural numbers.
Hmm, 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.
For another one, Russell’s paradox seems like it was a consequence naively assuming our intuitions about what counts as a ‘set’ would necessarily be correct, or even internally consistent.
I don’t know if this is true, but I once had a lecturer tell me that there used to be considerable debate over the question of whether a derivative of a differentiable function was necessarily continuous, which ultimately boiled down to the two sides having different definitions of continuity, but not realising it since neither had ever fully set down their axioms and definitions.
I’ve heard that before also but haven’t seen a source for it. But keep in mind that that’s a question about functions on the real numbers, not about what is being called “numbers” here which seems to be a substitute for the integers or the natural numbers.
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 ).
Hmm, 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.
For another one, Russell’s paradox seems like it was a consequence naively assuming our intuitions about what counts as a ‘set’ would necessarily be correct, or even internally consistent.