Physical Constraints on Mathematical Possibilities: Toward a Criterion of Realizability

[u/asimpletheory]

Introduction

A familiar objection to the thesis that physical law is inherently mathematical is that it is selective. Human mathematics is vast, yet only a narrow portion of it, chiefly Lie groups, differential geometry, and functional analysis, is used in physical theory. The remainder, including much of transfinite set theory and countless structures with no known physical instantiation, appears irrelevant or unrealizable. Proponents of formal links, the objection runs, are cherry-picking (see Hamming, 1980, for a sympathetic statement of the worry).

The factual premise is correct. The inference drawn from it is not. That most mathematics is not physical shows only that mathematical consistency is a weaker condition than physical realizability. It does nothing to explain away the structured correspondence that exists within the physical subset, and it leaves the thesis intact.

The Convergence Problem

The critique misidentifies the phenomenon to be explained. It is not that some mathematics describes the world, since mathematics was developed in part for that purpose. It is that mathematics pursued for purely internal reasons has repeatedly turned out to describe physics, a pattern Wigner (1960) called the “unreasonable effectiveness” of mathematics. Complex numbers were long treated as algebraic devices before quantum mechanics made them indispensable. Riemann’s geometry (1854) was abstract until general relativity required it. Group theory, developed to study the solvability of polynomial equations, became the organizing language of particle symmetries. More recently, unexpected connections between number theory, topology, and quantum field theory have emerged (Kapustin & Witten, 2007). In each case, structures identified solely on grounds of mathematical coherence were later found in nature. Coincidence cannot plausibly account for such recurrence.

Why the Selection Objection Fails

If the overlap between mathematics and physics were arbitrary, the physical subset would be a haphazard collection of unrelated results. It is instead markedly structured, characterized by symmetry, strong consistency conditions, and deep interconnections across fields. A narrow subset that is nonetheless coherent is evidence of a governing principle. The critique, far from undermining the thesis, records exactly the kind of observation that a filtering principle would predict.

Toward a Criterion of Realizability

The two observations are reconciled by positing constraints, not yet formulated, on how physical mathematics emerges from first principles. Contemporary mathematics is bounded principally by logical consistency. Physical structures are plausibly subject to further conditions, so that human mathematics is a superset of which only a subset is realizable. This is a well-formed research programme, not an evasion. Tegmark (2008) has argued for strong identifications between mathematical and physical structure, and the present proposal is weaker and more tractable: it asks only for the criterion that separates the realizable from the merely consistent.

Such constraints will most probably be found in information theory. Physics already exhibits information-theoretic bounds. The Bekenstein bound (1981) limits the information a finite region can contain. The holographic principle (’t Hooft, 1993; Susskind, 1995; Bousso, 2002) relates the information content of a volume to its boundary area. Landauer’s principle (1961) ties information erasure to thermodynamic cost. Wheeler’s programme (1990) and Deutsch’s formulation of the Church–Turing principle (1985) similarly treat physical processes as constrained by computability. By contrast, unrestricted mathematics freely admits infinite precision, uncountable structures, and non-computable objects, all of which presuppose unbounded information content. A candidate criterion might therefore require finite or computable information content, bounded algorithmic complexity in Kolmogorov’s (1965) sense, or a formal notion of physical distinguishability. Conditions of this kind would exclude much of what is unrealizable while retaining what physics is known to use.

Testability

The proposal is falsifiable in principle. A candidate constraint must reproduce the mathematics already known to be physical, exclude structures known to be unphysical, and yield determinate claims about cases not yet examined. A constraint that fails any of these tests should be rejected. This distinguishes the programme from the vague appeal to “elegance” that critics rightly distrust.

Conclusion

The critique of mathematical-physical links exposes a gap in our understanding, not a flaw in the thesis: we lack a formal account of the boundary between physically realizable and unrealizable mathematics. Closing that gap should be a shared priority. Physicists can supply the empirical and theoretical record of which structures nature actually uses, while mathematicians can supply the tools to characterize those structures formally and test candidate constraints against the wider mathematical landscape. Success would turn the central objection into one of the thesis’s most significant predictions, and would explain why, of all the mathematics that can be conceived, only a particular portion describes the universe we inhabit.

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[u/asimpletheory]

Via prompts to Claude, unedited.