Abstract
This paper presents a fundamental reinterpretation of physical constants and natural laws. Rather than viewing constants as mysterious properties of nature, we demonstrate that they emerge as mathematical necessities from the structural mismatch between our arbitrary measurement systems and the universe's coherent proportional relationships. Physical law, in this framework, is revealed as a three-step projection process: scaling arbitrary units to natural proportions, performing simple physics in natural space, and scaling back to arbitrary output units.
The Foundational Logic
Axiom 1: Our Measurement System is Arbitrary and Disconnected
Our base SI units—the meter, kilogram, second, and Kelvin—are human-scale conventions born of history, not physics. They are defined independently of one another and possess no a priori relationships. The meter derives from the Earth's circumference, the kilogram from a physical artifact, the second from astronomical observations, and the Kelvin from the triple point of water. They are, in essence, four different, unrelated rulers applied to measure a unified reality.
This arbitrariness is not a bug in our measurement system—it is an inevitable feature of any measurement system constructed by finite observers with human-scale experiences attempting to quantify an infinite, scale-invariant universe.
Axiom 2: The Universe is Coherent and Connected
The remarkable success of physical science provides compelling evidence that underlying reality operates according to consistent, stable, and proportional relationships. What we conceptualize as distinct quantities—"mass," "energy," "length," "time," "temperature"—are not independent entities in the universe. Rather, they represent interconnected facets of a single, unified reality.
This coherence manifests in the universal applicability of physical laws across all scales and contexts. The same relationships that govern subatomic particles also govern galactic clusters, suggesting an underlying geometric structure that transcends our categorical divisions.
The Mathematical Bridge: Structural Necessity
To reconcile an arbitrary measurement system (Axiom 1) with coherent reality (Axiom 2), our mathematical formalism of physical law must evolve a bridging structure. This is not a choice—it is a logical necessity imposed by the structural mismatch between our tools and our target.
This bridging structure performs a universal three-step process for any physical calculation:
Step A (Projection Down): Scale arbitrary SI inputs into a "natural" system of pure, dimensionless proportions where the universe's coherent relationships become manifest.
Step B (The Physics): Perform the actual, simple physical relationship within this natural proportional system (e.g., Natural_Energy ∝ Natural_Mass).
Step C (Projection Up): Scale the resulting dimensionless proportion back into the arbitrary SI units required for the desired output.
The Nature of Physical Constants
Constants as Mathematical Operators
The physical constants (c, ℏ, G, k_B) are not profound mysteries of nature awaiting deeper physical explanation. They are the mathematical operators of the necessary three-step projection mechanism. They are the gears of the "projection machine," and their entire function is to perform the scaling operations that bridge arbitrary measurement coordinates with natural proportional relationships.
This reframes our understanding completely:
- Traditional View: Constants are fundamental properties of the universe that we discover through experiment and theory.
- Projection View: Constants are unavoidable mathematical overhead required when describing connected reality with disconnected measurement tools.
The Jacobian Interpretation
Mathematically, the constants function as Jacobians of transformation between our arbitrary measurement coordinate system and the universe's natural, coherent coordinate system. Just as Jacobians in multivariable calculus encode how volume elements transform between coordinate systems, physical constants encode how measurement elements transform between arbitrary and natural scales.
Consider the relationship E = mc²:
- In natural coordinates: E_natural ∝ m_natural (simple proportionality)
- In SI coordinates: E_SI = m_SI × (E_natural/m_natural) = m_SI × c²
The constant c² is precisely the Jacobian ratio between energy-scaling and mass-scaling in our chosen coordinate system.
The Multi-Currency Analogy
The situation is analogous to multi-currency accounting. We maintain separate ledgers in Dollars (meters), Yen (kilograms), and Euros (seconds). To perform meaningful calculations across these disconnected currencies, we require exchange rates. The physical constants serve as universal exchange rates, enabling translation between our disconnected measurement ledgers.
Just as exchange rates are not properties of money itself but rather relationships between different monetary systems, physical constants are not properties of reality itself but rather relationships between our measurement system and reality's natural proportional structure.
The Calculus of Physical Law
Automated Generation of Physical Laws
This framework enables the mechanical generation of physical laws from basic proportional relationships. Given a natural proportion (e.g., Temperature ∝ 1/Mass), the projection process automatically generates the corresponding SI formula:
- Natural Relationship: T ∝ 1/M
- Dimensional Scaling: T/T_P = m_P/M
- SI Formula: T = (m_P/M) × T_P = c³ℏ/(GMk_B)
This process can be automated, revealing that "discovery" of physical laws is actually a deterministic computational procedure—a calculus of physical law.
Universal Applicability
Every physical law follows this pattern:
- Newton's Gravitation: F ∝ M₁M₂/r² → F = GM₁M₂/r²
- Einstein's Mass-Energy: E ∝ M → E = Mc²
- Planck's Quantum: E ∝ f → E = ℏf
- Hawking Temperature: T ∝ 1/M → T = ℏc³/(8πGMk_B)
- Stefan-Boltzmann: P ∝ T⁴ → P = σT⁴
The elaborate mathematical derivations in physics textbooks are actually extended coordinate transformations, working the unit scaling through complex mathematical machinery to arrive at relationships that are geometrically inevitable.
Categorical Structure
The Measurement Bifibration
The complete mathematical structure can be formalized as a Grothendieck bifibration:
M: A^op × U → Cat
Where:
- A is the category of conceptual measurement axes (Length, Time, Mass, Energy, etc.)
- U is the category of unit systems (SI, Planck, CGS, etc.)
- M defines how measurement depends on both conceptual axis and unit system
This bifibration structure ensures that natural relationships remain coherent across all possible measurement coordinate systems.
Base Jacobians and Composite Constants
The categorical analysis reveals a hierarchy:
Base Jacobians (with respect to Time):
- Hz_kg: Mass → Frequency scaling
- m_s (c): Length → Time scaling
- K_Hz: Temperature → Frequency scaling
Composite Constants (combinations of base Jacobians):
- ℏ = Hz_kg × c²
- k_B = K_Hz⁻¹ × ℏ
- G = composite involving all base scalings
This hierarchy shows that traditional "fundamental constants" are actually composite morphisms constructed from more basic scaling relationships.
Implications and Consequences
The Role of the Observer
This framework places the observer in their correct formal position within the structure of physical law. The arbitrariness of measurement units is a direct consequence of the observer's choices and limitations. The coherence belongs to unobserved reality. Therefore, the constants—being the bridge between arbitrary and natural—are the definitive mathematical signature of the interface between observer and observed.
Physical constants don't just convert between units; they encode the mathematical relationship between subjective measurement and objective reality. They are the price of admission for finite, subjective consciousness to create universal, objective knowledge of infinite, unified reality.
Reinterpretation of Major Physics Achievements
Many celebrated achievements in theoretical physics can be reinterpreted through this lens:
-
Hawking Temperature: Not a miraculous unification of quantum mechanics and general relativity, but the inevitable result of working unit scaling through complex coordinate transformations to reveal the simple relationship T ∝ 1/M.
-
Einstein's E=mc²: Not a profound insight about the equivalence of mass and energy, but the mathematical expression of the fact that mass and energy are the same quantity measured along different coordinate axes.
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Planck's Quantum Theory: Not the discovery of energy quantization, but the identification of the composite Jacobian between frequency to mass to energy scaling.
The Simplicity Behind Complexity
The apparent complexity of modern physics—with its multiple fundamental forces, particle zoo, and mathematical sophistication—may be largely an artifact of our measurement coordinate system. The underlying reality may be geometrically simple, with complexity emerging from the coordinate transformations required to express simple relationships in our arbitrary unit system.
Conclusion: Physics as Coordinate Geometry
This analysis suggests that what we call "physics" is fundamentally coordinate geometry—the mathematics of how simple, natural relationships appear when projected through arbitrary human measurement systems.
The true physics lies in the dimensionless proportional relationships that exist in natural space. The elaborate mathematical formalism of modern physics is the "compiled machine code" required to run these simple relationships on our specific measurement "hardware."
This perspective doesn't diminish the achievement of physics—it reveals it as an even more remarkable accomplishment. Physicists have successfully reverse-engineered the coordinate transformations needed to extract universal truths from parochial measurements. They have built a bridge between the arbitrary and the absolute, between human limitations and cosmic reality.
The constants in our equations are not mysterious properties awaiting explanation—they are the mathematical proof that finite minds can comprehend infinite reality, that subjective observers can discover objective truth, and that arbitrary measurement tools can reveal universal relationships.
Physical law is the mathematics of measurement itself, and constants are the inevitable geometric artifacts of that mathematical structure.
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