alt Hacker NewsThere are theories like SPOT and Internal Set Theory that don’t require filters.
Plus the ancient mathematicians did very well with just their intuition. And more to the point, I cared much more about building (hyper)number sense than some New Math “let’s learn ultrafilters before we’ve even done arithmetic”.
> Plus the ancient mathematicians did very well with just their intuition.
They did. But they also got things wrong, such as thinking that pointwise limits are enough to carry over continuity (see here for this and other examples: https://mathoverflow.net/a/35558). Anyway, mathematics has changed as a discipline, we now have strong axiomatic foundations and they mean that we can, in principle, always verify whether a proof is correct.