alt Hacker NewsEven just defining the hyperreals and showing why statements about them are also valid for the reals needs to go through either ultrafilters (which are some rather abstract objects) or model theory. Of course you can just handwave all of that away but then I guess you can also do that with standard analysis.
There 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”.