Soft logic: a new structure of dual numbers and Its topological interpretation

Artificial intelligence is bringing mathematicians back to questions of foundations. At the close of his 1900 Paris lecture, Hilbert stressed that the problems he had presented were only examples, behind which stood a vision of the organic unity of mathematics. Following this vision, we revisit the notion of the infinitesimal and show that the distinction between different multiples of zero carries meaning.

The soft numbers xO + y, with O^2 = 0, are algebraically Clifford's dual numbers; the novelty lies in their geometry. We present the soft coordinate system: a strip whose center is the zero axis and whose two edges carry the same real axis in opposite directions. Our main result is that the soft Möbius map (A, B) ↦ (1−|B|)A·O + BA, defined via a multivalued sign function, induces a homeomorphism between the open Möbius strip with its core circle collapsed to a point and the plane of soft numbers. Under this homeomorphism the absolute zero corresponds to the core circle, and the two non-commutative appearances of a soft number lie on a one-sided surface.

We then present applications. In probability, we prove that Kolmogorov's axioms together with the regularity principle (probability zero only for the impossible event) admit no real-valued model containing a continuous random variable. The soft probability Pₛ(X = x₀) = f(x₀)·O resolves this and yields a conservative extension of Kolmogorov's theory. In machine learning, soft multiplication underlies an optimizer that separates signal from noise during gradient-based training. We also discuss the dynamics of soft functions. Finally, we propose a direction toward a new model of computation: a soft extension of Turing's reaction–diffusion equations (1952), in which the real part describes the pattern and the soft-zero part describes the perturbation from which the pattern is born.