Physics Is Ontology, and So Is Mathematics: On the Physicality of Mathematical Practice

Tim Maudlin has argued that physics constitutes ontology while mathematics does not. His argument rests on four related considerations: that mathematical objects lack causal powers; that they lack spatiotemporal location; that mathematical truths are independent of physical law and therefore require non-physical truthmakers; and that the perfect objects of mathematics — the perfectly straight line, the exact geometric square — belong to a Platonic intelligible world accessible only through thought. This paper argues that all four considerations, while capturing something genuine, misidentify what mathematical objects actually are. The key analytical move is the distinction between a prescription — a physical notation encoding an unexecuted construction process — and a result — the completed physical output of that process. All four of Maudlin's considerations correctly describe prescriptions pointing toward not-yet-produced outputs. None correctly describes a completed physical construction. Under Physical Constructivism within the framework of Quantum-Geometry Dynamics (QGD) and the Minimal Physically Derivable Theories (MPDT) programme, completed physical constructions are the only mathematical objects that exist. The causal inertness of mathematical objects dissolves: completed constructions are physical events embedded in the causal order of the universe, interacting gravitationally with all other matter at every subsequent moment of causal succession. The spatiotemporal absence dissolves: every completed construction occupies a definite position in the global causal succession. The invariance of mathematical truth under changes of physical law reflects not the non-physicality of its truthmakers but the depth of their physical grounding — they are grounded in structural constraints of any physically constructible universe, more fundamental than any contingent law. The perfect square is not a non-physical original: it is an unrealizable prescription, a construction instruction that exceeds the finite resolution of the discrete physical substrate. QGD's Theorem of Emergent Euclidean Space establishes that Euclidean geometry is the large-scale limit of discrete preonic structure — physical squares at large scale are the structural reality, not approximations to a Platonic original. The paper further argues that Maudlin's conclusion is in internal tension with his own physical realism. Every mathematical act — every counting operation, every proof step — is a physical process performed by a physical system using physical resources, producing physical outputs. A programme that grounds all ontological claims in physics cannot consistently exempt mathematical practice from the physical substrate in which it is entirely embedded. Landauer's principle (that information processing is physical) provides independent empirical support. Physical Constructivism applies Maudlin's own evidential standard consistently: the result is not the elimination of mathematical truth but its deeper grounding. The positive account replaces the binary physical/mathematical distinction with a three-tier ontology: prescriptions (physically real notations encoding construction instructions), executions (physical processes running those instructions), and results (completed constructions, which are the mathematical objects that exist). Not all prescriptions produce results: realizable prescriptions produce objects; resource-exhausted prescriptions encode constructions that exceed the finite information capacity of the universe; axiomatically forbidden prescriptions encode constructions impossible in any physically constructible universe. The Uniqueness Theorem establishes that the structural conditions on any physically constructible universe are unique and minimal, grounding mathematical necessity in physics rather than in an independent Platonic realm. Wigner's puzzle about the unreasonable effectiveness of mathematics in physics evaporates: the mathematics that is effective in physics is precisely the mathematics abstracted from physical structure. The mathematics that generates persistent foundational problems — the real number continuum, infinite-dimensional Hilbert spaces, completed infinite sets — does so because non-constructivist mathematics is being applied beyond its domain of physical validity. Mathematics is not a separate ontological domain. It is the subset of physics concerned with the abstract structural properties of physically constructible processes. Physics is ontology. So is mathematics. Part of the QGD / MPDT publication programme. Related papers: Mathematics as a Subset of Physics P12, DOI: 10.5281/zenodo.19610265; The Physicality of Logic P16, DOI: 10.5281/zenodo.19491912; The Limits of Mathematical Freedom P28, DOI: 10.5281/zenodo.20184952; The Unrealizable Prescription: PDST P27, DOI: 10.5281/zenodo.20088508; On the Uniqueness of Minimal Physically Derivable Theories P1, DOI: 10.5281/zenodo.19380764.