Infinitesimals without Logic

TL;DR

The paper introduces Fermat reals, a new extension of real numbers with nilpotent infinitesimals, enabling rigorous and classical logic-compatible infinitesimal calculus with applications in physics.

Contribution

It presents a constructive, classical-logic-compatible theory of Fermat reals with a geometric representation, expanding infinitesimal calculus without relying on logic-based frameworks.

Findings

Fermat reals include nilpotent infinitesimals with a clear order relation.

Every smooth function can be represented as a polynomial using Fermat reals.

Applications include rigorous derivation of the wave equation and formalization of physics calculations.

Abstract

We introduce the ring of Fermat reals, an extension of the real field containing nilpotent infinitesimals. The construction takes inspiration from Smooth Infinitesimal Analysis (SIA), but provides a powerful theory of actual infinitesimals without any need of a background in mathematical logic. In particular, on the contrary with respect to SIA, which admits models only in intuitionistic logic, the theory of Fermat reals is consistent with classical logic. We face the problem to decide if the product of powers of nilpotent infinitesimals is zero or not, the identity principle for polynomials, the definition and properties of the total order relation. The construction is highly constructive, and every Fermat real admits a clear and order preserving geometrical representation. Using nilpotent infinitesimals, every smooth functions becomes a polynomial because in Taylor's formulas the rest is now zero. Finally, we present several applications to informal classical calculations used in Physics: now all these calculations become rigorous and, at the same time, formally equal to the informal ones. In particular, an interesting rigorous deduction of the wave equation is given, that clarifies how to formalize the approximations tied with Hook's law using this language of nilpotent infinitesimals.