The Metrological Guillotine: The IS/AS Dichotomy as the Foundational Principle for Unified Physics

J. Rogers, SE Ohio

Abstract

We argue that the central crisis in foundational physics—the failure to unify General Relativity and Quantum Mechanics—stems from a foundational category error: the Metrological Naturalistic Fallacy. This fallacy involves the epistemological conflation of IS (Ontological Reality, defined by unitless ratios) with AS (Epistemological Perception, defined by conventional units and measurements). We introduce the IS/AS dichotomy as the necessary philosophical framework for scientific rigor, demonstrating that all dimensional "fundamental constants" (

$c,h,G,k_B$

) are not IS properties of the universe, but AS Jacobian coefficients required to translate unitless reality into arbitrary human coordinate systems. The restoration of foundational progress requires enforcing the Metrological Guillotine: the principle that AS cannot be derived from IS.

I. Introduction: The Crisis of Precisely Wrong Foundations

For over a century, theoretical physics has operated under a paradoxical condition: models provide extraordinary predictive precision while lacking conceptual coherence at the foundation. This state of being "precisely wrong" results from a breakdown in Natural Philosophy rooted in a failure to maintain the critical distinction between what is and how we describe it as.

We identify a structural parallel to Hume's famous distinction between descriptive statements (IS) and prescriptive statements (OUGHT). In physics, this manifests as the distinction between:

• IS (Ontology): The objective, invariant, unitless structure of physical reality.

• AS (Epistemology): The conventional, subjective, unit-laden system we use to measure and describe that reality.

The core argument is that modern physics has committed the Metrological Naturalistic Fallacy by reifying AS as IS—treating arbitrary human conventions (units, constants) as fundamental properties of objective reality. The observable consequence is a century of stagnation in unification attempts, as effort is spent trying to reconcile the arbitrary structure of our description (AS) rather than understanding the singular structure of reality (IS).

II. The Metrological Naturalistic Fallacy

II.1 Defining the Dichotomy

The distinction between IS and AS is absolute and non-negotiable:

IS (Reality / Unitless Substrate)	AS (Perception / Unit System)
Category: Ontology, Being, Fact	Category: Epistemology, Description, Convention
Content: Unitless Ratios (e.g.,  $\alpha \approx 1\mathrm{/}137$ ), Geometry	Content: Units (m, kg, s), Quantities, Coordinate Values
Status: Objective, Invariant, Exists	Status: Subjective, Coordinate-Dependent, Invented
Precedent: Newton's proportional laws ( $F\propto m_1{m}_2\mathrm{/}{r}^2$ )	Precedent: The work of the Royal Mint (defining the unit)

II.2 The Act of Reification

The fallacy occurs when a concept belonging to the AS category is treated as if it belongs to the IS category.

The Fallacy in Action (Reifying 

$c$

):

1. The IS (Fact): There exists a dimensionless maximum velocity ratio (

   $v_{max}\mathrm{/}{v}_{max}=1$

   ). This is a fundamental geometric constraint on the substrate.

2. The AS (Convention): We, the observers, choose to measure time in seconds and length in meters. This choice produces a numerical constant for this limit: 

   $c\approx 299,792,458m\mathrm{/}s$

   .

3. The Fallacy: Physics claims that 

   $c=299,792,458m\mathrm{/}s$

    IS a fundamental property of the universe.

This conflates the objective existence of the limit (IS) with the conventional numerical expression of that limit (AS). The number 

$299,792,458$

 is an artifact of the SI committee's choices, not a law of physics. The 2019 decision to fix 

$c$

 by definition confirms its status as AS (convention), yet the physics curriculum still treats it as IS (discovery).

II.3 The Metrological Guillotine

Just as Hume argued that no amount of factual statements (IS) can logically lead to a statement of value or duty (OUGHT), the Metrological Guillotine asserts:

$$\text{<span style="background-color: white;">Cannot derive AS from IS.</span>}$$

No amount of unit free ratios can logically determine the choice of human units. Therefore, the numerical values of the dimensional constants, which are products of unit choices, are themselves products of convention.   The constants are defined by the unit scaling relative to each other and the single physical scale of the universe. 

III. Epistemological Grounding: Constants as Jacobians

The IS/AS distinction formally proves that dimensional constants are not physical entities but mathematical operators.

III.1 The Role of the Jacobian

In the language of coordinate geometry, a Jacobian matrix is the transformation required to map coordinates between two different systems. In the context of physics:

$$\text{<span style="background-color: white;">Dimensional Constant </span>}\mathrm{<span\; style="background-color:\; white;">J</span>}<span\; style="background-color:\; white;">=</span>\frac{{\text{<span style="background-color: white;">Quantity</span>}}_{\mathrm{<span\; style="background-color:\; white;">A</span>}\mathrm{<span\; style="background-color:\; white;">S</span>}}}{{\text{<span style="background-color: white;">Quantity</span>}}_{\mathrm{<span\; style="background-color:\; white;">I</span>}\mathrm{<span\; style="background-color:\; white;">S</span>}}}$$

The "fundamental constants" (

$G,c,h,k_B$

) are the numerical components of this coordinate transformation matrix.

$$\mathrm{<span\; style="background-color:\; white;">c</span>}<span\; style="background-color:\; white;">=</span>\frac{{\text{<span style="background-color: white;">Length</span>}}_{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">I</span>}}}{{\text{<span style="background-color: white;">Time</span>}}_{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">I</span>}}}$$

$c$

 is not a speed; it is the numerical coefficient that converts the arbitrary unit of Time into the arbitrary unit of Length, making them coherent within the SI system.

III.2 The Failure of Dimensionality

Dimensional analysis (using dimensions like 

$[M],[L],[T]$

) belongs entirely to the AS category. Dimensions are part of the conventional framework we invent to organize our measurements.

The pursuit of unifying the constants (

$G,c,h$

) is the pursuit of mathematically unifying the three arbitrary Jacobians required to translate the unitless reality into the three dimensions of human measurement. This effort is fundamentally misguided because it is trying to solve an AS (metrological) problem using IS (ontological) tools.

IV. Formal Structure: The Grothendieck Fibration Model

The mathematical proof for the IS/AS dichotomy is found in Category Theory, specifically via a Grothendieck Fibration 

$\pi :\mathcal{E}\to \mathcal{B}$

.

1. The Base Category 

   $\mathcal{B}$

    (IS): The category of unitless conceptual axes and ratios. Objects are Mass, Time, etc.; morphisms are pure proportionalities (e.g., 

   $T\propto 1\mathrm{/}M$

   ). This is the realm of IS.

2. The Total Category 

   $\mathcal{E}$

    (AS): The category of measured quantities, where objects are pairs (value, unit). This is the realm of AS.

3. The Fibration 

   $\pi$

   : The projection functor 

   $\pi :\mathcal{E}\to \mathcal{B}$

   , mapping the measured quantity back to its unitless conceptual origin.

In this model, a physical law is a Cartesian lifting of a simple proportional morphism in 

$\mathcal{B}$

 up into the measurement space 

${\mathcal{E}}_{SI}$

. The "fundamental constants" are the cocycle data—the transition coefficients required to ensure that this lifting is coherent with the arbitrary choices of the SI unit basis.

The 

$\mathcal{B}$

 (IS) layer is the fixed point of reality. The complexity and numerical values of the constants are generated entirely by the choice of projection into the 

${\mathcal{E}}_{SI}$

 (AS) layer.

V. Conclusion: Restoration of Natural Philosophy

The philosophical rigor of Natural Philosophy demands that theory must both work (match empirical reality) and explain why (provide axiomatic clarity). Physics' success with the former while abandoning the latter led to the conflation of IS and AS.

The prolonged failure to unify fundamental theories is not evidence that nature is hard, but evidence that the AS framework is flawed. Progress can only resume by:

1. Acknowledging the IS/AS Dichotomy: Recognizing that unitless ratios are IS and all dimensional constants are AS.

2. Enforcing the Metrological Guillotine: Ceasing all attempts to derive the value or meaning of Jacobians (constants) from ontological principles.

3. Refactoring the Axioms: Defining foundational theories (Layer 1) entirely in terms of unitless ratios, treating the SI system (Layer 2) purely as an arbitrary coordinate transformer.

By respecting the IS/AS dichotomy, physics can shed the metrological baggage that has constrained it for a century, finally allowing the true simplicity of unitless reality to be revealed. The search for a Theory of Everything must pivot from explaining the arbitrary values of the Jacobians to discovering the full, coherent, unitless geometry of the IS substrate.