Measurement as Abstraction: The Ontological Inversion in Physics

J. Rogers

Abstract

Modern physics suffers from a fundamental ontological inversion: we treat measurement as concrete reality and dimensionless physical relationships as mathematical abstractions. This paper argues for the opposite view—that dimensionless relationships constitute physical reality, while measurement is an abstract projection of that reality onto arbitrary coordinate systems. This inversion has profound implications for how we understand physical law, "fundamental constants," and the search for unification in physics.

1. The Standard View: Measurement as Reality

1.1 The Conventional Ontology

Physics education instills a particular ontological hierarchy:

Real (Concrete):

• Measured quantities: "365.25 days," "9.8 m/s²," "9.109×10⁻³¹ kg"
• Dimensional formulas: E = mc², F = GMm/r²
• Physical constants: h, c, G, k_B

Abstract (Mathematical):

• Dimensionless relationships: t ~ r^1.5 m^-0.5, E ~ m
• Natural units where ℏ=c=G=1
• Pure ratios: α ≈ 1/137, m_p/m_e ≈ 1836

1.2 The Operational Justification

This hierarchy appears justified operationally:

• Science proceeds through measurement
• Predictions must be testable
• Testing requires comparing predicted numbers to measured numbers
• Therefore, measured quantities are what physics is "about"

This reasoning seems unassailable. It is also completely backwards.

2. The Ontological Inversion

2.1 What Measurement Actually Is

Measurement is the process of:

1. Selecting an arbitrary reference scale (meter stick, clock, balance)
2. Comparing a physical quantity to that reference
3. Recording the dimensionless ratio (quantity/reference)
4. Labeling that ratio with a unit name

Example: "The table is 2 meters long"

• Reality: A spatial extent exists
• Abstraction: We compare it to an arbitrary stick and get ratio = 2
• Label: We call our stick a "meter" and write "2 m"

The number "2" and the label "meters" are artifacts of our measurement procedure. They are not properties of the table.

2.2 The Nature of Unit Systems

Unit systems (SI, CGS, Imperial, Natural) are coordinate charts - abstract reference frames we project reality onto.

When the SI system defines:

• The meter via the speed of light
• The kilogram via Planck's constant
• The second via cesium oscillations
• The Kelvin via Boltzmann's constant

This is not "discovering fundamental scales in nature." This is choosing coordinate axes by committee vote (literally, in the 2019 SI redefinition).

2.3 What Is Actually Real

What exists independent of our measurement choices?

Dimensionless relationships:

• Orbital period scales as t ~ r^1.5 m^-0.5
• Energy and mass are proportional: E ~ m
• Temperature and mass are inversely related: T ~ 1/M
• The fine structure constant: α ≈ 1/137

These relationships exist whether humans measure them or not, and they take the same form in every unit system. They are coordinate-independent.

The numbers "365.25 days" or "9.109×10⁻³¹ kg" are coordinate-dependent projections - they change when we change our reference scales.

3. Constants as Coordinate Artifacts

3.1 The Nature of Physical Constants

Consider the speed of light: c = 299,792,458 m/s

This is not telling us a property of light. It is telling us a property of our unit system - specifically, the relationship between our arbitrary meter scale and our arbitrary second scale.

The constant c is the Jacobian coefficient that converts between length and time coordinates in our measurement chart.

3.2 Dimensional Analysis as Unit Extraction

From the constants h, c, G, k_B we can "derive" the Planck units:

• m_P = √(ℏc/G)
• l_P = √(ℏG/c³)
• t_P = √(ℏG/c⁵)
• T_P = √(ℏc⁵/Gk_B²)

Physics textbooks present these as "nature's fundamental scales." But look at what we actually did:

We performed algebraic manipulation of our unit definitions to extract the scaling factors we embedded when we chose what a kilogram, meter, second, and Kelvin mean.

These are not properties of nature. They are properties of our coordinate chart.

3.3 Every Formula Already Operates at "Planck Scale"

Consider Einstein's E = mc².

In natural units (where c=1), this becomes: E = m

But we can also write this as: E/E_P = m/m_P

The "Planck scale" is just the normalization point of our coordinate system. Every equation in physics already operates at this scale - we just multiply by conversion factors (the "constants") to project into SI coordinates.

The search for "Planck scale physics" is looking for physics at the coordinate origin - we're already there, we just can't see it through our coordinate projections.

4. The Case of Orbital Mechanics

4.1 The Complete Physics

Kepler's third law in dimensionless form:

t ~ r^1.5 m^-0.5

This is complete. This tells you everything about how orbital period relates to orbital radius and central mass. The exponents encode all the geometric and dynamic content.

4.2 The Coordinate Projection

To express this in SI units, we write:

T² = (4π²/GM) r³

This adds:

• The factor 4π² (geometric)
• The constant G (coordinate transformation coefficient)

Neither adds physical content. They are required only because we chose to measure distance in meters, time in seconds, and mass in kilograms - three arbitrarily scaled axes.

4.3 The Illusion

Students are taught that the second formula is "the real equation" while the first is "just a proportionality."

But the truth is inverted:

• t ~ r^1.5 m^-0.5 is the physics
• G is the telescope calibration

The constant G doesn't describe planetary motion. It describes the relationship between our kilogram scale, meter scale, and second scale.

5. The Two-Tier System

5.1 The Public Narrative

For undergraduates, the public, and funding agencies:

• "Physical constants are fundamental properties of nature"
• "The Planck scale is where quantum gravity becomes important"
• "We're searching for unification at high energies"
• "Understanding why constants have their values is a deep mystery"

5.2 The Working Reality

Graduate students are told on day 2:

• "Set ℏ=c=G=1 for convenience"
• "Work in natural units from now on"
• All actual calculations are performed in dimensionless form

Theoretical physicists already work in the dimensionless substrate. They use natural units constantly. They know, operationally, that the constants are "just units."

5.3 The Gap

But this knowledge is never formalized or made explicit:

• Students learn to use natural units, not what this reveals about physical law
• The constants are still called "fundamental"
• No rigorous treatment of the coordinate transformation between unit systems
• No acknowledgment that "Planck scale" is just a normalization choice

The gap between what physicists do (work in dimensionless form) and what they teach (constants are fundamental) is not accidental.

6. Why the Inversion Persists

6.1 Institutional Incentives

If physics is just dimensionless relationships plus coordinate projections, then:

• What justifies the complexity?
• What requires specialized expertise?
• What needs billion-dollar facilities?
• What generates papers and grants?

Problems that evaporate under proper ontology:

• "Unifying the constants" - they're just coordinate coefficients
• "Physics at the Planck scale" - we're already there
• "Why constants have their values" - we voted on them in 2019
• "Quantum gravity at high energies" - there is no special energy scale

6.2 The Profit Model

Unsolved problems generate funding. The current framework maintains the appearance of deep mysteries requiring continued research.

Making explicit that constants are coordinate artifacts and physical law is already unified would eliminate entire research programs.

6.3 Guild Protection

A graduate student who insists on seeing the rigorous mathematical formulation - the Jacobian transformations, the fibration structure, the coordinate-independence - would be dismissed:

"We already know this. It's just units. Stop doing philosophy and calculate."

But if they already know it, why isn't it taught explicitly? Why is the rigor forbidden?

Because explicit rigor would make clear what the implicit practice obscures: we're studying our measurement apparatus, not nature.

7. Implications for Physical Theory

7.1 What Physics Actually Studies

If we accept measurement as abstraction, then:

Physics studies: Dimensionless relationships in the substrate of reality Theoretical physics: Discovering coordinate-independent proportionalities
Experimental physics: Projecting those relationships onto measurement apparatus Constants: Coordinate transformation coefficients, not properties of nature

7.2 The Unity We've Been Seeking

Physics searches for:

• Unification of forces
• A theory of everything
• The fundamental scale

But physics is already unified in dimensionless form:

• All relationships are proportionalities in the substrate
• The fragmentation only appears in coordinate-dependent projections
• There is no "fundamental scale" - there is only dimensionless unity

We broke the unity by projecting onto separate measurement axes, then spent a century trying to reunify what we artificially separated.

7.3 Real vs. Fake Questions

Real physics questions:

• Why is α ≈ 1/137? (dimensionless ratio)
• Why is m_p/m_e ≈ 1836? (dimensionless ratio)
• What determines coupling strengths? (dimensionless parameters)

Fake questions (coordinate confusion):

• Why does the electron have mass 9.109×10⁻³¹ kg? (arbitrary reference comparison)
• What physics exists at Planck energy? (coordinate normalization point)
• Why do constants have their values? (asking about our 2019 vote)

We've spent enormous resources on the second category while neglecting to ask: what are the dimensionless relationships that actually structure reality?

8. The Path Forward

8.1 Pedagogical Reform

Physics education should:

• Teach dimensionless relationships as primary
• Introduce unit systems as coordinate charts
• Make explicit the Jacobian transformations between systems
• Frame constants as coordinate artifacts from the start
• Present natural units not as "convenience" but as revealing the substrate

8.2 Theoretical Clarity

Research should:

• Focus on dimensionless parameters and ratios
• Stop reifying coordinate choices as physical scales
• Acknowledge that working in natural units reveals actual structure
• Formalize the fibration structure of physical law
• Make explicit what is currently implicit in practice

8.3 Epistemological Honesty

The physics community should:

• Stop maintaining separate narratives for different audiences
• Acknowledge that constants are metrological, not ontological
• Admit that "Planck scale physics" is coordinate confusion
• Recognize that unification already exists in dimensionless form
• Allow rigorous questioning of foundations without dismissal

9. Conclusion: Reality vs. Representation

We have inverted reality and representation:

• Reality: Dimensionless relationships in physical substrate
• Representation: Coordinate-dependent projections onto measurement scales

We treat the representation (measured quantities with units) as real and the reality (dimensionless relationships) as abstract mathematics.

This inversion is not a harmless pedagogical choice. It:

• Obscures the actual structure of physical law
• Generates fake research problems
• Maintains institutional complexity
• Prevents clear thinking about fundamentals
• Confuses coordinate artifacts for natural properties

The correction is simple but radical: measurement is the abstraction, not the reality.

Physical relationships exist independent of our measurement choices. The numbers and units we write down are artifacts of projecting those relationships onto arbitrary reference scales we selected by convention and committee vote.

Understanding this doesn't make physics less rigorous - it makes the rigor visible. It doesn't eliminate prediction - it clarifies what we're predicting. It doesn't reject measurement - it properly situates measurement as a practical tool for accessing coordinate-independent truth.

The substrate is real. The coordinates are abstract. Physics lives in the substrate. Measurement projects onto coordinates.

Once we see this clearly, the mystifications dissolve. The unity appears. And we can finally study reality directly, rather than through the distorting lens of our arbitrary measurement apparatus.

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Appendix: The Friend's Question

A colleague, upon seeing the dimensionless form t ~ r^1.5 m^-0.5, asked:

"Don't we have to convert at some point into a measurable unit system?"

This question perfectly encapsulates the ontological inversion. He believes:

• The dimensionless relationship is "incomplete"
• We must "convert" to make it "real"
• Only measured quantities are actually physical

But the truth is inverted:

• The dimensionless relationship is the complete physics
• "Converting" means projecting onto our arbitrary coordinate chart
• Measured quantities are abstractions derived from the real relationship

His question reveals he's been trained to believe measurement defines reality, when measurement is merely one possible defined representation of reality.

The fact that this seems paradoxical to him - that reality could exist independent of measurement - shows how deeply the inversion has been internalized.

Physics has successfully convinced people that reality is what appears on their meter sticks and stopwatches, rather than recognizing that meter sticks and stopwatches are arbitrary tools for projecting reality onto human-convenient scales.

The orbit doesn't become "real" when we measure it in days. The orbit is t ~ r^1.5 m^-0.5, whether we measure it or not. The "365.25 days" is just how our particular arbitrary calendar projects that reality into coordinates we find convenient.

Mistaking the projection for the reality is the fundamental error from which all other confusions flow.