A New Way to Ask If Math Is Invented or Found

In the paper's words

Whether mathematics is invented or discovered remains a live dispute with practical consequences for how research, education, and patent law treat mathematical objects. We assume the 2020 PhilPapers survey reports the field split, with nominalism at 41.85 percent and Platonism at 38.38 percent of the distribution and a fifth of respondents rejecting both labels (Section 3.3), yet both camps argue from the same celebrated cases while no principled criterion separates what practitioners invent from what they discover. Every anchor fits the split: representations are culturally variable and partly invented, while the constraint structure they navigate behaves as discovered.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. We emphasize what this number is and is not: it is a documented count from the cited assessment, not a measured rate over all formalisms; with $n = 3$ the one-sided 95 percent Clopper–Pearson upper bound on the underlying adoption probability $p$ solves $(1-p)^3 = 0.05$, giving $p \leq 1 - 0.05^{1/3}$.

   from 4.3 Survivorship control on Wigner's puzzle

2. Even this weak bound is informative: a formalism built today has, on this class, at most a ~63 percent chance of decade-scale adoption even under the most pessimistic reading of three failures—and the point estimate is far lower.

   from 4.3 Survivorship control on Wigner's puzzle

3. The documented historical reference class for formalism adoption gives 0 of 3 closest analogues achieving mainstream adoption within a decade [10]; the one-sided 95 percent Clopper–Pearson upper bound on the per-formalism decade adoption probability from three failures is $p \leq 1 - 0.05^{1/3} \approx 0.632$, with the point estimate 0.

   from 5. Results

4. And the philosophy profession itself is split: we assume the 2020 PhilPapers survey reports 41.85 percent of respondents endorsing or leaning toward nominalism about abstract objects, 38.38 percent toward Platonism, and 20.81 percent choosing neither (Section 3.3), so no position commands even a plurality majority among experts.

   from 1. Introduction

5. A quantified extension result: the generating scheme behind the tablet's fifteen rows predicts five additional rows [6], so the recorded corpus covers 75 percent of a twenty-row extension, computed in Section 4.1.

   from 1. Introduction

How it is done

1. This paper's contributions are: A principled representation/constraint criterion: axioms, notation, and definitions are invented representations; theorem-space consequences and structural constraints are discovered, in the precise sense that they are invariant under the choice of representation.

   from 1. Introduction

2. The earlier version of the same argument documents a concrete case: an algorithm in computational group theory that the author came into possession of independently of its publication—by John Bray's discovery, with the author disclaiming priority—built on Brauer's observation that two involutions generate a dihedral subgroup and are conjugate by an element of $\langle uv\rangle$ when $uv$ has odd order, with the principal ideas traceable to Burnside's 1900 paper and remaining forgotten for the good part of the twentieth century [2].

   from 2. Background and Related Work

3. We define a piece of mathematics as a pair $(R, C)$: $R$ (representation layer): the choice of axioms, definitions, notation, primitives, and descriptive language.

   from 3.1 The representation/constraint criterion

4. The paper's method is documentary analysis plus explicit arithmetic.

   from 3.2 Analytical method

What would falsify it

1. We conclude that weak structural realism states the applicability puzzle precisely rather than dissolving it, and we outline the archival search that would falsify the Plimpton anchor.

   from Abstract

2. If the dichotomy is false—if the correct split cuts across it—then each of these decisions is currently being made against a badly drawn map.

   from 1. Introduction

3. The real-world problem, stated plainly: Wigner's puzzle—that mathematics developed for internal reasons keeps turning out to describe physics, with the standard dating taken as given—general relativity's light-bending prediction of 1915 confirmed in 1919 and the Dirac equation's surplus residue becoming the positron in 1932 (we assume these uncontested historical dates; no conclusion rests on them beyond their qualitative role as celebrated hits)—is standardly cited as evidence that mathematicians discover a mind-independent structure.

   from 1. Introduction

Where the claim stops

1. The observation that would falsify the main result is a counterexample to the invariance criterion: a mathematical element that is demonstrably constraint-like (it drives proofs and applications in one community) yet fails to survive every admissible translation to an independent representational scheme, or conversely a representational artifact that is invariant under all admissible translations and therefore indistinguishable from a constraint.

   from 6. Discussion

2. The result rests on the following assumptions, with sensitivity notes: The invariance criterion itself.

   from 6. Discussion

3. The Plimpton 322 triple assumption.

   from 6. Discussion

Where to read

1. Abstract240 words · 1 min
2. 1. Introduction663 words · 3 min
3. 2. Background and Related Work1,211 words · 5 min
4. 3. Methods530 words · 2 min
5. 4. Analysis1,345 words · 6 min
6. 5. Results357 words · 2 min
7. 6. Discussion635 words · 3 min
8. 7. Practical Implications and Engineering Pathways761 words · 3 min
9. 8. Conclusion332 words · 1 min
10. Provenance and AI Involvement240 words · 1 min