Is it possible to say that $dx$ indistinguishable from zero in classical analysis as $\varepsilon$ in Smooth Infinitesimal Analysis?

Question

$dx$ isn't nonzero infinitesimal in classical analysis.

Is it possible to say that $dx$ indistinguishable from zero in classical analysis as $\varepsilon$ in Smooth Infinitesimal Analysis?

Thanks.

Answer 1

Well in all types of infinitesimal analysis, the notation $dx$ is just a notation for another infinitesimal. So there is no difference between $\epsilon$ and $dx$ on this account. Usually one uses $dx$ when one gets ready to define derivatives (or integrals). One doesn't typically use $dx$ when defining the continuity of a function via infinitesimals ("every infinitesimal increment leads to an infinitesimal change in the function", as Cauchy defined it).

"Indistinguishable" in Bell's sense means "not provably zero and not provably nonzero". Classical analysis does not enable such entities since the classical logic incorporates the law of excluded middle: $\epsilon$ is either zero or provably nonzero.