How is it possible that time can be represented as abstract element?

Question

I read about differentiation using dual numbers (and about NSA/SDG approaches to differentiation) and I have question.

Let we have function that represents position $x$ at time $t$: $x(t)$

If we use dual numbers or SDG approach to differentiation then we have: $x(t+\varepsilon)=x(t)+\varepsilon x'(t)$

For time derivative we get: $$x'(t)=\frac{x(t+\varepsilon)-x(t)}\varepsilon$$ Now we have time $\varepsilon$ in denominator but $\varepsilon$ is abstract nilsquare element.

How is it possible that time can be represented as abstract element? How to understand it?

Thanks.

Answer 1

I will comment in the context of Synthetic Differential Geometry (SDG) because it is not clear what $x(t+\varepsilon)$ would mean for the dual numbers. Your formula $$x'(t)=\frac{x(t+\varepsilon)-x(t)}\varepsilon$$ is meaningless because you cannot divide by $\varepsilon$, which is a nilsquare infinitesimal in SDG. Rather, the derivative must be defined as the unique number, denoted $x'$, such that your other equation $x(t+\varepsilon)=x(t)+\varepsilon x'(t)$ is satisfied for all such $\varepsilon$. Acting together, the nilsquare infinitesimals enable the existence of a unique value for the derivative in this approach.

All mathematical theories are "abstract", whether classical analysis or SDG. Representing an element of time by a nilsquare $\varepsilon$ is therefore no different from any other application of an abstract mathematical concept to physical entities.