What is the formal problem caused by interpreting dx as an infinitesimal?

Question

My mother is a physicist. One evening, I told her $dx$ is a linear mapp $\mathbb{R}^3\to\mathbb{R}$ taking $\hat u_x=e_1$ to 1 and the other canonical vectors to 0 (if considered on $\mathbb{R}^3$). She answered, «but why do you have to make it so complicated, come on, $dx$ is the infinitesimal step in the $x$ direction!». I'm sure such an interpretation formally makes no sense, but could you explain why? I mean, what problem comes about with that interpretation of $dx$?

PS I have seen a post about this but I have no time to read all of that, so could you please give me a simple argument to answer my mum with :)?

Answer 1

The Archimedean Property forbids the existence of infinitesimal values. It states: $$ (\forall x > 0)(\exists N \in \mathbb{N})\left( \frac{1}{N} < x \right). $$

If $ x $ is an infinitesimal, then the above statement isn’t true. Hence, if you want the Archimedean Property to hold, then you can’t have infinitesimals. Note that the Archimedean Property is a characteristic property of $ \mathbb{R} $, being one of the axioms of the real-number system.

Note: There are real-number systems with infinitesimals. Have a look at non-standard analysis with the set of hyperreal numbers. It’s thus possible to have infinitesimals, and there are no formal problem with them. However, you’ll lose the Archimedean Property once you impose the existence of infinitesimals...