I'm generally a little skeptical about approaches that treat infinitesimals in a symbolic computation. The approach typically is to solve traditional analysis problems but throw in some infinitesimals and show that you can get the same answers. But if you approach infinitesimals from first principles I feel like you run into a lot of problems.
For example it is almost always the case that you can remove higher order infinitesimals, like (dx)^2, when computing things like derivatives. But this always necessitates a step where you translate from infinitesimals to "standard" reals, and then continue on your merry way. We happily round away the higher-order terms when we compute something like ((x + dx)^3 - x^3) / dx, but if we're continuing to do infinitesimal math they may become relevant again. Call that function, f_1(x) = 3x^2 + 3xdx + dx^2; we'll typically just call this f_2(x) = 3x^2, but these functions are not equivalent if we then proceed to compute (f(x) - 3x^2) / dx, which, presumably, we can just do because we've admitted this horror of a syntax into our formal language.
I'm very skeptical of this being useful outside of being able to reason about trivial limits for this reason.
I submit two claims basically, in response to what you are saying:
- The "proper" multiplicative calculus with limits is no more broken than its additive counterpart
- These multiplicative infinitesimals are no more broken than their additive counterparts
It seems like you were trying to hold these multiplicative infinitesimals to the standard of calculus with limits, and rejecting them on those grounds. To that rejection, I just say that these infinitesimals were never meant to meet that standard. :)
> you can remove higher order infinitesimals, like (dx)^2, when computing things like derivatives
I always found that iffy and a bit of a (completely legal) hack. It's a nice point that what enables this hack is promptly leaving the world of infinitesimals and retreating back to reals.
"we" who? You're projecting infinitesimals down onto reals, and then complaining that the infinitesimals are gone. That seems like a "you" problem. You can keep the ifinitesimals if you don't want to lose then.
For example it is almost always the case that you can remove higher order infinitesimals, like (dx)^2, when computing things like derivatives. But this always necessitates a step where you translate from infinitesimals to "standard" reals, and then continue on your merry way. We happily round away the higher-order terms when we compute something like ((x + dx)^3 - x^3) / dx, but if we're continuing to do infinitesimal math they may become relevant again. Call that function, f_1(x) = 3x^2 + 3xdx + dx^2; we'll typically just call this f_2(x) = 3x^2, but these functions are not equivalent if we then proceed to compute (f(x) - 3x^2) / dx, which, presumably, we can just do because we've admitted this horror of a syntax into our formal language.
I'm very skeptical of this being useful outside of being able to reason about trivial limits for this reason.