Multiplicative Infinitesimals

I find infinitesimals more intuitive than the formal 'limits-based' approach. I'm currently studying my old degree material but using a fairly interesting book:

"Full Frontal Calculus: An Infinitesimal Approach" by Seth Braver.

I like his readable style. His poetic intro finally gave me an intuition why infinitesimals might be useful, compared to the good old reals:

"Yet, by developing a "calculus of infinitesimals" (as it was known for two centuries), mathematicians got great insight into `real` functions, breaking through the static algebraic ice shelf to reach a flowing world of motion below, changing and evolving in time."

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.

> Infinitesimals are liked, despite their formal rigor (in most settings), are liked in some settings, like informally solving differential equations, and other applied tasks.

There seems to be one too many "are liked" in that sentence. Deleting either one of them makes the sentence read a lot better. I think deleting the second one reads better than deleting the first one.

> Despite their formal rigor, Infinitesimals are liked in some settings, ...

Even better without a split clause. "In some settings" was also repeated.

You might be more interested in the linked page on proper multiplicative calculus then, https://github.com/Ericson2314/baccumulation/blob/main/math/... . That, in turn, is mostly just a retelling of doi:10.1016/j.jmaa.2007.03.081

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. :)