Abstract
This paper presents a definitive resolution to the ontological status of physical constants. We demonstrate through rigorous logical analysis that once the independence assumption of measurement axes is acknowledged—that meters, kilograms, and seconds are independently defined human conventions—it follows with logical necessity that constants such as G, c, h, and k_B cannot be fundamental physical properties. They are, provably and exclusively, metrological Jacobians: coordinate transformation factors ensuring consistency between our arbitrarily scaled measurements and the underlying dimensionless structure of physical law. This conclusion redirects the entire mission of fundamental physics from the futile project of explaining coordinate-dependent artifacts toward the discovery of parameter-free, dimensionless geometric structure—the true ontology of the physical world.
1. The Diagnosis: Constants Misinterpreted as Physical Properties
The contemporary practice of fundamental physics operates under an implicit but pervasive ontological error: the treatment of dimensional constants as fundamental properties of nature. This error manifests across multiple levels of the discipline:
In pedagogy: Textbooks present G = 6.674×10⁻¹¹ m³·kg⁻¹·s⁻² as "the gravitational constant," a deep feature of spacetime geometry, rather than as a conversion factor between arbitrarily chosen measurement scales.
In research programs: Decades of theoretical effort have been devoted to "explaining" why constants have their particular numerical values, to finding "deep relationships" between them (such as between G and h in quantum gravity), and to understanding their role in the "fine-tuning" of the universe.
In metaphysics: Constants are invoked as ontologically basic entities in discussions of physical law, with questions like "could the universe have different constants?" treated as meaningful counterfactuals about alternative possible worlds.
Empirical observation: This is not a straw man. The misinterpretation is documented in the literature, conference proceedings, and funding priorities of the field. It represents the dominant paradigm.
Critical assessment: The existence of this widespread error establishes that there is a problem to be solved, but does not yet demonstrate why the alternative view is correct. That requires a positive account.
2. The Reality: Dimensionless Structure as Fundamental Ontology
The alternative ontology, which we will prove to be correct, is structural realism applied to physical measurement: the view that what exists fundamentally is a web of dimensionless, relational proportionalities, and that all dimensional quantities are projections of this structure onto arbitrarily chosen coordinate systems.
2.1 The Planck Equivalence Chain
At the Planck scale, where natural units normalize all dimensional constants to unity, physical reality reveals its true structure as a chain of equivalences:
T/T_P = f·t_P = m/m_P = l_P/l = E/E_P = p/p_P = F/F_P = ...
This chain asserts that temperature, frequency, mass, length, energy, momentum, and force are not independent properties measured along separate axes, but rather different measurements of a single underlying structure. Each axis is a dimensionless ratio against its Planck-scale counterpart.
Crucial structural features:
Some ratios are direct (m/m_P): larger quantities correspond to larger dimensionless numbers
Some ratios are inverse (l_P/l): larger quantities correspond to smaller dimensionless numbers
This asymmetry encodes wave-particle duality directly in the geometric structure
2.2 Laws as Pairwise Projections
From this equivalence chain, physical laws emerge automatically as pairwise identities. The canonical equations of physics are not independent empirical discoveries but inevitable consequences of the chain's structure:
Mass-Energy Equivalence:
E/E_P = m/m_P → E = m(E_P/m_P) → E = mc²
Planck-Einstein Relation:
E/E_P = f·t_P → E = f(E_P·t_P) → E = hf
de Broglie Relation:
p/p_P = l_P/l → pl = p_P·l_P → p = h/λ
Boltzmann's Law:
E/E_P = T/T_P → E = T(E_P/T_P) → E = k_B·T
With n measurement axes, the structure generates (n choose 2) pairwise laws. For seven fundamental axes, this yields 21 canonical relationships. The apparent proliferation of physical law is not evidence of complexity but rather the combinatorial richness of a simple underlying structure.
2.3 Historical Vindication: Newton's Method
This structural view is not a modern invention. Isaac Newton, in the Principia Mathematica, worked exclusively in dimensionless ratios, deliberately avoiding the need for dimensional constants. His method of comparing accelerations (the Moon-Apple test) allowed the unknown constant of proportionality and the unknown mass of the Earth to cancel out, leaving pure geometric ratios:
a_apple/a_moon = r_moon²/R_earth²
Newton understood that the proportionality F ∝ m₁m₂/r² was the physics, and that any specific numerical constant would be an artifact of whatever arbitrary measurement system one employed. His dual role as Master of the Royal Mint—where he dealt daily with arbitrary conversion factors between currencies and weight standards—gave him a uniquely sophisticated understanding of the distinction between invariant relationships and conventional scaling factors.
The measurement of G by Cavendish in 1798 was not the "completion" of Newton's theory but rather its calibration for the newly standardized metric system. Newton's framework was already architecturally complete.
3. The Mechanism: Constants as Metrological Jacobians
Having established both the error (Section 1) and the alternative (Section 2), we now prove the precise relationship between them. This section demonstrates that dimensional constants are not mysterious properties awaiting explanation, but rather necessary mathematical artifacts that emerge from the projection of dimensionless structure onto dimensional coordinate systems.
3.1 Formal Derivation
Begin with the invariant law in natural (Planck) units, where all quantities are dimensionless:
F_nat = (m₁_nat · m₂_nat) / r_nat²
To express this law using measurements in an arbitrary unit system (such as SI), we must scale each dimensionless quantity by its corresponding Planck unit:
m_SI = m_nat · m_P
r_SI = r_nat · l_P
F_SI = F_nat · F_P
Inverting these relations to substitute into the invariant law:
(F_SI / F_P) = (m₁_SI / m_P) · (m₂_SI / m_P) / (r_SI / l_P)²
Solving for F_SI and rearranging to group constant terms:
F_SI = [F_P · l_P² / m_P²] · (m₁_SI · m₂_SI) / r_SI²
We recognize the modern form F = G(m₁m₂/r²), revealing the identity:
Critical observation: G is not an input to this derivation. It is an output—a composite artifact that emerges necessarily from the coordinate transformation. It encodes no physical information about gravity; it encodes only information about the arbitrary definitions of the Newton, meter, kilogram, and second.
3.2 The Jacobian Interpretation
In differential geometry, when transforming a function from one coordinate system to another, the Jacobian determinant appears as a scaling factor accounting for how volume elements stretch or compress under the transformation. The Jacobian contains no new geometric information; it ensures mathematical consistency across coordinate choices.
Physical constants play precisely this role. They are the metrological Jacobians of coordinate transformations from nature's dimensionless basis to our dimensional one. Just as no one seeks a "deep explanation" for why the Cartesian-to-polar Jacobian has the form r, we should not seek deep explanations for why G has the value (F_P·l_P²/m_P²).
The complete parallel:
| Arises from coordinate change | Arises from unit system choice |
| Contains no new geometric information | Contains no new physical information |
| Value depends on coordinate basis | Value depends on unit definitions |
| Ensures mathematical consistency | Ensures dimensional consistency |
| Not a property of the space | Not a property of nature |
3.3 The Tautology of Measurement
At the heart of this derivation is a structure that appears mathematically trivial but is physically profound:
F_nat = (F_nat · F_P) / F_P
This "round trip" reveals the true function of dimensional constants. They do not add new physics; they encode the complexity we introduced by choosing our own rulers:
F_nat: The pure, dimensionless output of physical law (the territory)
F_nat · F_P: Scaling the dimensionless truth into measurable SI units (projecting onto our map)
(...) / F_P: Normalizing our SI measurement back to dimensionless form (reading the map back to the territory)
The constant is the price we pay for the round trip. It contains all the scaling information required to translate between the universe's natural language (dimensionless ratios) and our conventional language (dimensional measurements). Modern physics, with its explicit constants, is built entirely on this two-step process of scaling and renormalizing.
The profound implication: Introducing dimensional constants adds zero physical information. It adds only mathematical complexity to compensate for the complexity we introduced by choosing our own coordinate system.
4. The Consequence: Redirecting the Mission of Physics
If constants are metrological Jacobians—and the proof in Section 3 establishes that they are—then several immediate and radical consequences follow for the practice of fundamental physics.
4.1 The Category Error of Constant Unification
Any attempt to "unify constants" or to find "deep relationships" between their numerical values is a category error—the mistaking of coordinate-dependent artifacts for structural features of reality.
Example 1: Quantum Gravity
Decades of effort have sought to understand the relationship between G (gravity) and h (quantum mechanics) under the assumption that their numerical values encode something fundamental. But if both are Jacobians:
G = F_P · l_P² / m_P²
h = E_P · t_P
Then seeking to "unify" them is asking: "What is the deep relationship between these two bundles of conversion factors?" It is like seeking the fundamental significance of the fact that 1 mile = 1.609344 kilometers. The relationship exists, but it tells us about our measurement conventions, not about reality.
Example 2: Fine-Tuning
The "fine-tuning problem" asks why certain dimensionless ratios (like the proton-to-electron mass ratio, ~1836) have the values they do, suggesting design or multiverse explanations. But if we are asking this question while still embedded in the framework where dimensional constants are treated as fundamental, we are conflating two distinct issues:
True fine-tuning: dimensionless ratios in the structure itself (this may be a genuine question)
Pseudo fine-tuning: asking why G or c have their dimensional values (this is asking why our coordinate system has its particular scaling)
The latter is not a physics question. It is a metrology question whose answer is: "because of how we defined meters and kilograms."
Example 3: Hierarchy Problem
The hierarchy problem in particle physics asks why the weak force scale (~10² GeV) is so different from the Planck scale (~10¹⁹ GeV). But this formulation already assumes that the Planck scale, defined using G, c, and h, is fundamental. If these are Jacobians, the "hierarchy" may be an artifact of our choice to coordinatize the structure in a particular way. The true question is whether there exists a dimensionless hierarchy in the underlying structure—and if so, why.
4.2 The Misinterpretation of E=mc²
The most famous equation in physics serves as a case study in how constants are misunderstood.
Popular interpretation: "Mass can be converted into energy. Nuclear bombs prove E=mc² by turning matter into pure energy."
Reality: Nuclear fission releases approximately 0.08% of rest mass as binding energy. The rest remains as matter. No mass "converts" to energy. Rather, the system reconfigures to a lower-energy state, and because mass and energy are identical (measured differently), the mass measurement decreases proportionally.
Einstein's actual insight: Mass and energy are not two different things related by a formula. They are the same thing measured on different scales. The c² is not a "conversion factor" between substances; it is the exchange rate between our kilogram-scale and our joule-scale, which we defined independently and must now reconcile.
In the dimensionless structure:
m/m_P = E/E_P → They are the same coordinate
In SI units:
E = mc² → We need c² to convert between our arbitrary kg and J scales
The general principle: Every time a constant appears "mysteriously" in a physical law, we should ask: Is this deep physics, or is this the Jacobian for translating between two measurement axes we artificially separated?
4.3 The Redirection of Physics' Mission
The realization that constants are Jacobians, not physics, demands a fundamental reorientation:
Old mission (implicit):
"Explain why the constants have the values they do. Find the Theory of Everything that predicts G, h, c, e, and all particle masses from first principles."
Why this mission is incoherent:
Because it asks: "Why does our accounting system use these particular exchange rates?" The answer is: "Because of how we defined our units." That is metrology plus historical contingency, not fundamental physics.
New mission (explicit):
"Discover the dimensionless structure—the parameter-free geometry of ratios that constitutes physical law."
Why this mission is coherent:
Because dimensionless ratios are coordinate-invariant. They are the same in all unit systems. They are the territory, not the map. A genuinely fundamental theory will have no free dimensional parameters because structure itself has no dials to tune.
Concrete implications:
Stop asking why G = 6.674×10⁻¹¹ m³·kg⁻¹·s⁻². Start asking: What is the dimensionless geometric structure that manifests as gravitational attraction?
Stop asking why the electron mass is 9.109×10⁻³¹ kg. Start asking: What dimensionless ratio does the electron represent in the structural web?
Stop trying to "unify" dimensional constants. Start seeking the single dimensionless equivalence chain from which all laws project.
5. The Final Statement: Physics is the Ratios
We now arrive at the ultimate reduction, the logical endpoint of this analysis.
Conditional necessity: Once you acknowledge that measurement axes (meters, kilograms, seconds, Amperes, Kelvin) are independently defined by human convention—and this is an undeniable empirical fact—then it follows with strict logical necessity that constants cannot be fundamental physical properties.
The proof by exhaustion:
Option 1: Constants are fundamental properties of nature.
But their numerical values depend entirely on our choice of unit definitions.
Properties of nature cannot depend on human conventions.
Contradiction. Option 1 is false.
Option 2: Constants encode some deep relationship that transcends unit choice.
But we have proven (Section 3) they are Jacobians: pure coordinate artifacts.
Jacobians encode only coordinate choice, not geometric structure.
Contradiction. Option 2 is false.
Option 3: Constants are metrological Jacobians—artifacts of projecting dimensionless structure onto dimensional coordinates.
Proven in Section 3. No contradiction.
Option 3 is necessarily true.
Therefore: Once the independence assumption of measurement axes is acknowledged, constants are entirely bookkeeping artifacts. The physics is already encoded in the underlying dimensionless ratios.
5.1 The Metaphor: The Simple Transaction
The entire elaborate apparatus of modern physics—with its dozens of dimensional constants, its unit conversion factors, its dimensional analysis—is a Rube Goldberg machine of accounting for a simple transaction.
The simple transaction: Physical quantities stand in proportional relationships. F ∝ m₁m₂/r². E ∝ m. E ∝ f. These are the ratios. This is the physics.
The Rube Goldberg machine:
We define meters, kilograms, and seconds independently.
This artificial separation creates dimensional inconsistency.
We introduce constants (G, c, h) to restore consistency.
We measure these constants to high precision.
We seek theories to "explain" why they have these values.
We never notice we created the entire problem ourselves.
The wisdom: Study the transaction, not the ledger. The value is in the proportionalities (the territory), not in the accounting system we built around them (the map).
5.2 Ontological Minimalism
Why is the reduction to dimensionless ratios the ultimate foundation?
Because ratios are ontologically minimal. They are pure structure without substance.
Not:
"Physics is particles" → What are particles made of?
"Physics is fields" → What are fields made of?
"Physics is strings" → What are strings made of?
"Physics is information" → What is information made of?
But:
"Physics is ratios" → Ratios are not made of anything. They are pure relational structure.
You cannot ask "What are ratios made of?" without committing a category error. Ratios are form without matter, structure without substance. They are the structural realist's ultimate ontology—the bedrock beneath which there is nothing further to explain because there is nothing further that exists.
The final implication: The quest for a "Theory of Everything" that explains all constants is misguided. The true Theory of Everything will have no dimensional constants at all. It will be a dimensionless equivalence chain—a pure geometry of ratios—from which the appearance of dimensional constants emerges automatically as projection artifacts when we impose our conventional measurement framework.
6. Conclusion: The Mission Ahead
This paper has established the following chain of reasoning:
Diagnosis: Constants are widely misinterpreted as fundamental properties (empirical fact).
Reality: The true ontology is dimensionless structural relationships (demonstrated through the Planck equivalence chain and historical validation via Newton).
Mechanism: Constants are metrological Jacobians arising from coordinate transformation (formal proof).
Consequence: Attempts to unify or explain constants are category errors; physics must redirect toward discovering parameter-free structure (logical deduction).
Reduction: Once measurement independence is acknowledged, constants are necessarily bookkeeping artifacts, and physics is entirely encoded in dimensionless ratios (conditional necessity).
The logic is complete. The conclusion is inescapable.
The apparatus of dimensional constants—G, c, h, e, k_B, and all the rest—is not the mystery at the heart of physics. It is the illusion we must dispel to see the mystery clearly. These constants are not deep properties awaiting explanation. They are conversion factors between arbitrarily scaled measurement systems and the underlying dimensionless reality.
The path forward is clear:
Physics must return to Newton's wisdom, armed now with modern mathematical rigor (Planck units, dimensional analysis, computational tools) and philosophical clarity (structural realism, the Jacobian formalism). The goal is not to measure constants to ever-greater precision or to construct theories that "predict" their values. The goal is to discover the dimensionless equivalence chain—the parameter-free geometric structure—that constitutes physical law itself.
When we find that structure, the constants will not be explained. They will be dissolved. They will be recognized as the shadows cast by structure onto the wall of our conventional coordinate system—necessary for practical calculation in human units, but devoid of fundamental significance.
The physics is the ratios. Constants are the accounting. Study the transaction, not the ledger.
Acknowledgments
This work builds upon the structural realist tradition in philosophy of science (Poincaré, Russell, Ladyman, French) and the foundational insights of Isaac Newton, whose method of ratios anticipated by three centuries the framework presented here. Albert Einstein's continuous emphasis that mass and energy were the same thing and c^2 was just measurement convention was key to this understanding. The computational implementation (LawForge) demonstrates that these principles are not mere philosophy but constitute a practical, predictive approach to deriving physical law from dimensionless postulates.
References
Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica.
Cavendish, H. (1798). "Experiments to determine the density of the Earth." Philosophical Transactions of the Royal Society.
Einstein, A. (1905). "Does the inertia of a body depend upon its energy content?" Annalen der Physik.
Ladyman, J., & Ross, D. (2007). Every Thing Must Go: Metaphysics Naturalized. Oxford University Press.
Rogers, J. (2025). "The Constant of Proportionality: Newton's Deliberate Omission of G and the Lost Principle of Invariance."
Rogers, J. (2025). "G as a Metrological Artifact: A Formal Proof of Newton's Principle of Invariance." https://mystry-geek.blogspot.com/2025/10/g-as-metrological-artifact-formal-proof.html
Rogers, J. (2025). "The Web of Equivalence: A Structural Unification of Physical Law through the Planck Chain." https://mystry-geek.blogspot.com/2025/10/the-web-of-equivalence-structural.html
Rogers, J (2025). "LawForge: The Physics Law Discovery Engine." https://buckrogers1965.github.io/LawForge/
No comments:
Post a Comment