Clarke-Doane, Justin (2025) Logical Dependence of Physical Determinism on Set-theoretic Metatheory. [Preprint]
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My Revisions of Logic and Determinism -- PhilSci Archive - 8.15.25 -- UNBLINDED.pdf Download (295kB) |
Abstract
Baroque questions of set-theoretic foundations are widely assumed to be irrelevant to physics. In this article, I challenge this assumption. I argue that even the fundamental physical question of whether a theory is deterministic—whether it fixes a unique future given the present—can depend on one’s choice of set-theoretic axiom candidates over which there is intractable disagreement. Suppose, as is customary (Earman 1986), that a deterministic theory is one whose mathematical formulation yields a unique solution to its governing equations. Then the question of whether a physical theory is deterministic becomes the question of whether there exists a unique solution to its mathematical model—typically a system of differential equations. I argue that competing axiom candidates extending standard mathematics—in particular, the Axiom of Constructibility (V = L) and large cardinal axioms strong enough to prove Projective Determinacy—can diverge on all the core dimensions of physical determinism. First, they may disagree about whether a given physical system is well-posed, and so whether a solution exists. Second, even when they agree that a solution exists, they can differ on whether that solution is unique. Finally, even when they agree that a system has a solution, and agree that this solution is unique, they may still dispute what that solution is. Whether a theory is deterministic—and even which outcome it deterministically predicts—can depend on one’s choice of set-theoretic metatheory. I indicate how the conclusions extend to discrete systems and suggest directions for future research. One upshot of the discussion is that either physical theories must be relativized to set-theoretic metatheories (in which case physics itself becomes relative), or, as Quine (1951) controversially argued, the search for new axioms to settle undecidables may admit of empirical input.
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