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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.

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    $\begingroup$ Why shouldn't it work? The formula follows directly from the definition of dual numbers ($\varepsilon$ as a nonzero nilpotent element) and the assumption that $x$ (a real or complex-valued function) is infinitely differentiable at $t$... Where do you see a problem? $\endgroup$ Commented Oct 30, 2023 at 19:39
  • $\begingroup$ @KevinDietrich I didn't write that it doesn't work. I meant that time must have quantity but $\varepsilon$ has no quantity or value. $\endgroup$ Commented Oct 31, 2023 at 5:14

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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.

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