The Limits of Mathematical Freedom: Why the Independence of Mathematical Practice Does Not Establish the Existence of Infinite Objects

The independence of mathematics from physical application is one of the central achievements of modern mathematics. This paper accepts that independence fully — pure mathematics enjoys complete freedom from all external constraints including physics — and argues that this freedom does not extend across the border into physics. The paper's central contribution is the two-domain principle. The physical construction criterion applies only at the border: when a mathematical structure is imported into a physical theory as a physical axiom — as a claim about what the physical universe fundamentally contains — rather than as a formal rule generating useful computational consequences. On one side of the border, the continuum is a legitimate mathematical object. On the other, the continuum imported as physical ontology is a category error. The error is not in the mathematics. It is in the import. The argument establishes that the independence of mathematical practice is effective autonomy — genuine at the level of description, compatible with physical determination at the level of execution. Every proof is a finite causal sequence of physically executable operations. A practice physically bounded at the level of execution cannot bring into existence objects that exceed those bounds. The Axiom of Infinity posits a completed infinite set through a finite physical act. The positing is real. The object posited is not. The paper then identifies five pathologies of modern physics as structural consequences of this category error: ultraviolet divergences, the singularities of general relativity, the cosmological constant problem, the measurement problem in quantum mechanics, and the string theory landscape. In each case the pathology is absent by construction in QGD, which derives all of Newton, Maxwell, Einstein, and quantum mechanics from two axioms without importing completed infinite objects. This paper is a companion to Mathematics as a Subset of Physics [P12], the Physicality of Logic [P16], Physically Derivable Set Theory [P27], and Axiomatic Localism [P19], and is part of the Quantum-Geometry Dynamics (QGD) and Minimally Physically Derivable Theories (MPDT) programme (ORCID: 0000-0002-7966-4250, DOI: 10.5281/zenodo.20184952).