[username90]

The intuition that there is something in between isn't really wrong, it make sense and they work, otherwise physicists wouldn't be able to work with them. So that intuition is correct, it is mathematicians who just don't understand it fully yet. Maybe fully formalizing this is what unlocks the final piece keeping us from creating a unified theory in physics?

[steerablesafe]

Nonstandard analysis is a rigorous framework for working with infinitesimals (and infinitely large numbers).

https://en.wikipedia.org/wiki/Nonstandard_analysis