State and Change: On the Common Root of Mathematics and Physics

This paper proposes a new answer to Wigner's classic puzzle of "the unreasonable effectiveness of mathematics in the natural sciences." Unlike Platonism, Kantianism, and empiricism, it argues that mathematics and physics share a single transcendental root. The method is transcendental reduction: through a performative experiment of "trying to affirm everything," we recognize two irreducible categories—state and change. State answers "what something is"; change answers "what something becomes." State unfolds into the process of classification (P1), which exhaustively articulates all possibilities in state-space. Change unfolds into the process of determination (P2), which actualizes one possibility among many, generating order, irreversibility, and the transcendental root of temporal experience. P1 and P2 are not two independent processes but two mutually constitutive dimensions of a single process of self-determination—related by definitional entanglement rather than mere necessary co-occurrence. Discrete and continuous structures are both possibilities within state-space. Continuous structure's capacity to be determined as actual rests on its geometrically intrinsic direction, which terminates the infinite regress of rule-execution separation that afflicts merely definitionally intrinsic discrete structures. Whether other structures could equally carry determination is an open question beyond the scope of transcendental argument. Thermodynamic irreversibility and quantum measurement irreversibility share a single transcendental root: the irreversibility of P2 determination, operating respectively in classical phase space and Hilbert space. Physical mechanisms explain how irreversibility occurs; transcendental logic explains why. The common root of mathematics and physics is thus explained: mathematics studies the formal structure of state-space; physics studies the regularities of determination. The physical world cannot violate mathematics because no actual state can lie outside state-space. Mathematics' predictive power is rooted in the prior marking-out, in the dimension of state, of logically prior structures—ontologically, structure and determination are always synchronous; mathematical "precognition" is epistemological. This paper establishes common-rootedness through transcendental argument alone; precise structural correspondence is delegated to the Emergence-Convergence Framework (ECF) for formal verification. At the limit, the always-already-existing identity of classification and determination becomes fully manifest—but the deepest root of Wigner's problem lies in common-rootedness itself: mathematics and physics are, at every stage, two sides of one and the same unfolding.