Mathematics as a Subset of Physics: Physical Constructivism and the Reasonable Effectiveness of Mathematics in Quantum-Geometry Dynamics

Mathematics as a Subset of Physics: Physical Constructivism and the Reasonable Effectiveness of Mathematics in Quantum-Geometry Dynamics In 1960, Eugene Wigner described the applicability of mathematics to the natural sciences as "unreasonable effectiveness" — a puzzle that has remained unresolved for over six decades. This paper argues that the puzzle dissolves once its generating premise is rejected. Mathematics is not independent of physical reality. It is a subset of physics: a body of abstract structures abstracted from physical reality, constrained by the laws of physics in its very practice, and meaningful only to the extent that its objects are physically constructible. On this view — which I call physical constructivism — the effectiveness of mathematics in physics is not unreasonable but necessary: the mathematics that works is precisely that which was built from physical structure in the first place. I develop this thesis through the framework of Quantum-Geometry Dynamics (QGD), an axiomatic theory of physics derived from the discreteness of space and the kinetic nature of matter. QGD provides both the clearest positive instantiation of physically grounded mathematics and a diagnostic tool for identifying where non-constructivist mathematics has generated artificial problems in physics. The paper proceeds on three fronts. First, I reject the two positions that generate Wigner's puzzle: mathematical Platonism, which locates mathematical objects in a non-physical realm and generates the access problem; and formalism, which treats mathematics as a free-standing formal game and makes the applicability of mathematics to physics inexplicable rather than expected. Second, I develop the positive thesis of physical constructivism through three claims: that the practice of mathematics is a physical process consuming matter, energy, and causal time; that mathematical objects exist only to the extent that they are physically constructible, entailing that infinite sets and the mathematical continuum have no physical referent; and that propositions are meaningful in physics only when the mathematical objects they invoke are physically constructible. QGD's discrete structure operationalises these claims by placing hard limits on what mathematical operations are physically meaningful: singularities, ultraviolet divergences, and the mathematical continuum are not physical phenomena but artefacts of applying continuous mathematics beyond its domain of validity. Third, I deliver a unified diagnosis of three major unresolved problems in fundamental physics — renormalisation, the cosmological constant problem, and the quantum measurement problem — as consequences of importing non-constructivist mathematical structures into physics. All three dissolve when the non-constructivist assumptions generating them are replaced with physically grounded discrete structure. Gödel's incompleteness theorems are shown to support rather than threaten this picture: formal mathematics cannot ground itself, confirming that the ground must be physical. The prime number distribution is developed as a case study in physical constructivist reframing, and is connected to companion papers delivering a full physical constructivist account of prime distribution and a physical constructivist proof of Fermat's Last Theorem. This paper is part of the Minimal Physically Derivable Theory (MPDT) programme.