Rigor Without Rigidity V2 with Table1.pdf

Measuring rigorousness.

1. Rigor Without Rigidity: Quantifying Creative Coherence in
Mathematical Frameworks
Version 2 (revised)
Adrian Cox
Independent Researcher, Lincoln, United Kingdom
[email protected]
2025-10-20
Public Domain Dedication (CC0 1.0). You may copy, modify, distribute, and perform this work,
even for commercial purposes, without asking permission. See
https://creativecommons.org/publicdomain/zero/1.0/.
Keywords
rigor, porosity, permeability, rigidity, coherence, creativity, philosophy of mathematics,
methodology
Introduction
Mathematical “rigor” is universally prized yet operationally under-specified. This paper
offers a quantitative lens for assessing the creative health of mathematical frameworks via
four quantities: porosity (φ), permeability (κ), rigidity (ρ), and rigor (R). The key move is to
decouple rigor from rigidity and reinterpret rigor as coherent flow relative to openness. In
response to earlier feedback, this revised version adds: (i) a structured methodology for
estimating φ and κ using observable indicators and an inter-rater protocol; (ii) narrative
justifications for the comparative examples; and (iii) an expanded literature context
connecting the framework to work in the philosophy and sociology of mathematics. Our
aim is not to replace proof, but to supply a transparent vocabulary and procedure for
evaluating inclusivity, coherence, and developmental maturity in both classical and
emerging mathematical programs.
Core Definitions
Definition 1 (Porosity and Permeability).
Let (φ,κ)∈¿ with 0≤κ≤φ≤1. Porosity φ measures openness (tolerance for variation,
generalization, alternative structures). Permeability κ measures realized, coherent flow
through the available openness (empirical structure, theorem-density, integrability of new
ideas).

2. Definition 2 (Rigidity and Rigor).
Define the rigidity ρ:=1−φ. The rigor is
R:={
κ/φ,φ>0,
0,φ=0,)
so R∈¿ quantifies coherence-of-flow relative to openness and is logically independent of ρ
.
Remark 1 (Residual Porosity and Transmissive Capacity).
Define the residual porosity ´R:=φ−κ≥0 (open capacity not yet coherently integrated) and
the transmissive capacity T:=φκ∈¿ (combined openness and realized flow).
Elementary Properties
Lemma 1 (Bounds).
For all admissible (φ,κ), we have 0≤ρ,R,´R,T≤1.
Proposition 1 (Monotonicity).
At fixed φ>0, R increases monotonically with κ. At fixed κ, increasing φ decreases both ρ
and R.
Related Work
This framework intersects several traditions. Lakatos’ Proofs and Refutations emphasizes
the evolution of mathematical concepts under criticism and repair; our R captures the
degree to which such evolution remains coherent within available openness. Davis and
Hersh discuss the lived practice of mathematics, where judgments of “rigor” are social as
well as formal. Ernest and the social-constructivist strand similarly stress negotiated
standards. Our contribution is a compact quantitative vocabulary and procedure that can
be used alongside—not instead of—proof to reduce opaque, taste-based objections.
Representative sources:
•I. Lakatos, Proofs and Refutations. Cambridge Univ. Press, 1976.
•P. J. Davis and R. Hersh, The Mathematical Experience. Birkhäuser, 1981.
•P. Ernest, Social Constructivism as a Philosophy of Mathematics. SUNY Press, 1998.

3. Methodology: Operationalizing φ and κ
To guard against circularity and subjectivity, we propose observable indicators, a scoring
rubric, and an inter-rater protocol.
Indicators for Porosity φ (Openness)
1.Generalizability.
2.Plurality of Models.
3.Interface Width.
4.Tolerance for Variation.
Indicators for Permeability κ (Coherent Flow)
1.Theorem Density.
2.Integrative Links.
3.Problem-Solving Efficacy.
4.Counterexample Discipline.
Scoring Procedure and Reliability
Each indicator is rated in {0,1,…,5} with published rationale. Normalize to ¿ to obtain ^φ
and ^κ:
^φ=
1
4
∑
i=1
4
P
i
5
,^κ=
1
4
∑
i=1
4
K
i
5
,^κ≤^φ.
Compute ρ=1−^φ and R=^κ/^φ if ^φ>0 (else R=0).
Comparative Examples with Justifications
Illustrative comparative scores (normalized).
Framework φ κ ρ=1−φ R=κ/φ
Euclidean
Geometry
0.20 0.20 0.80 1.00
Real Analysis
(ε–δ)
0.25 0.25 0.75 1.00
Abstract
Algebra
0.55 0.51 0.45 0.93
Topology 0.85 0.67 0.15 0.79
Category
Theory
0.90 0.72 0.10 0.80

4. Framework φ κ ρ=1−φ R=κ/φ
Riemann
Hypothesis
(program)
0.90 0.58 0.10 0.64
Exsolvent
Numbers
(program)
0.90 0.62 0.10 0.69