Quantity-Anchored Proof: Renormalization, Normalization, and the Limits of Unifying Physical and Mathematical Demonstration

Source abstract

The received distinction between physics and mathematics: that physics computes while mathematics proves, obscures the demonstrative character of physical computation. This paper develops a structural account of that character. I argue that both physical and mathematical demonstration presuppose a class of permissible transformations under which a claim must remain stable, but differ in what anchors those transformations to meaning: in physics, the anchor is the web of quantitatively interpreted observables (quantity-anchored proof); in mathematics, it is structural or purely formal relations. The theory is defended against a familiar reduction: the identification of physical scheme-independence with robustness reasoning in the sense of Levins, Wimsatt, or Woodward misses a constitutive distinction. Within perturbative quantum field theory (QFT), scheme-independence is not inductive evidence produced by running multiple calculations; it is a structural fact internal to the theory and conditional on its symmetry-preserving machinery, derivable within the theory. The centerpiece of the paper is a structural correspondence between renormalization and proof normalization: the Birkhoff decomposition of Connes-Kreimer on the Hopf algebra of Feynman graphs, which factors a raw amplitude into counterterm and finite components, is put alongside Gentzen-Prawitz cut elimination, which factors a derivation into redundant and normal components. The correspondence is real but partial: a family of structural kinships, not a formal isomorphism, and I use it to explain why renormalization carries the demonstrative feel physicists attribute to it, and why the unification of physical and mathematical demonstration faces a constitutive rather than merely practical barrier, illustrated by the standing tension between axiomatic and perturbative QFT.