I must disagree with your assumption that we should not use anything not seen in nature in Mathematics. That may be a useful viewpoint in some sciences, but Mathematics often disconnects itself with reality only for the reality that belongs with it to be found later. In the 1500s, complex numbers were regarded as a ridiculous concept, yet now they are well accepted, their properties well documented, and natural cases have been found where they are used frequently. In electrical engineering, for example, heavy use is made of complex numbers. The concepts of infinitesimal and infinity are indeed quite necessary, as without it the idea of convergence and differentiation are less well-defined, perhaps impossible. So if your job, whether by its nature, its equipment, or otherwise relies on calculus, you do indeed need infinitesimals. Avogadro's number and the speed of light as values are solely decided by the measurements we use to find them, while π, e, γ, ρ, and many other Mathematical constant's don't change, regardless of measurement.
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