Classically, $\exists$ and $\forall$ are dual. These don't seem to be the quantifiers Newton had quite in mind when differentiating. Suppose quantifiers attached separately to the *x*- and *h*-terms, "at least zero-many/much *x* or *h*." In this context, the *h*-terms are of a different sortal or typical aspect than the *x*-terms, and it's not that they themselves become 0 in the limit, but we consider the existential case where we toggle between "at least zero" and... what?
1. Are there many and varying such dualities? Like (at least zero, at most zero), (at least zero, exactly zero) (roughly equivalent to the preceding, at least in outcome), (almost zero-much, almost all) (where -much is stronger than -many, passing from discrete cardinal magnitude to continuous material), etc.
2. Is there a limit $(f: r \rightarrow 0) \circ (rh) = (0h)$? Is that one way to vanish the *h*-terms at the end of the derivative, by evaluating the model of the formula in which the *h*-subdomain is empty?
>- (For Newton, then: "kinematically," the *h*-variable could be any physical thing that admits of continuous determination in this sense. The *h*-terms, or Newton's counterparts, were not automatically overloaded with specific physical sense, like "*h*-much metal" or "*h*-many units of energy/force," but are simply from any sort "not merely numerical variables" in an at least two-sorted logic. So considering their empty subdomain is considering the case of the formula where its merely numerical factors "exist.")
Ripheus24
·
2026-05-21T23:02:19Z (4 months ago)