Post History
#2: Post edited
by
clemens
·
2026-05-08T03:54:26Z (4 months ago)
Added on the reason behind this distinction
- The reason is clear enough from what you stated. $-\epsilon < 0 < \epsilon$, so the ordering properties pick out which one is which. And so when one is defining left and right derivatives, one defines them with reference to the ordering properties of $\mathbb{R}$, and so one is able to distinguish between $\epsilon$ and $-\epsilon$.
Note that in contexts where left and right derivatives are not distinguished (e.g. in smooth infinitesimal analysis), one does not have this ability to "pick out" $\epsilon$ to uniquely distinguish it from $-\epsilon$.
- The reason is clear enough from what you stated. $-\epsilon < 0 < \epsilon$, so the ordering properties pick out which one is which. And so when one is defining left and right derivatives, one defines them with reference to the ordering properties of $\mathbb{R}$, and so one is able to distinguish between $\epsilon$ and $-\epsilon$.
- Note that in contexts where left and right derivatives are not distinguished (e.g. in smooth infinitesimal analysis), one does not have this ability to "pick out" $\epsilon$ to uniquely distinguish it from $-\epsilon$.
- Basically, there are symmetries in the underlying algebra, and to distinguish between a number (like $\epsilon$ or $i$) and its conjugates under these symmetries (like $-\epsilon$ or $-i$) one has to more or less explicitly "break" the symmetry. One can prove that this is the case by simple metalogical reasoning, systematically replacing all occurrences of a number by occurrences of its conjugates and verifying that the axioms remain invariant under this transformation.
#1: Initial revision
by
clemens
·
2026-05-08T03:27:01Z (4 months ago)
The reason is clear enough from what you stated. $-\epsilon < 0 < \epsilon$, so the ordering properties pick out which one is which. And so when one is defining left and right derivatives, one defines them with reference to the ordering properties of $\mathbb{R}$, and so one is able to distinguish between $\epsilon$ and $-\epsilon$.
Note that in contexts where left and right derivatives are not distinguished (e.g. in smooth infinitesimal analysis), one does not have this ability to "pick out" $\epsilon$ to uniquely distinguish it from $-\epsilon$.