Politics, Power, and Science: The Concerning Blindness of Standard Dimensional Analysis: Why Units Carry Values, Not Just Types

Wednesday, January 7, 2026

The Concerning Blindness of Standard Dimensional Analysis: Why Units Carry Values, Not Just Types

J. Rogers, SE Ohio

Abstract

Standard dimensional analysis treats dimensions [L], [M], [T] as abstract algebraic symbols—mere typological placeholders ensuring dimensional homogeneity. This framework systematically obscures a fundamental truth: dimensions carry specific scaling values. These values are consistent for each unit type across all physics. We demonstrate that this blindness stems from confusing ratios with constants, and treating units as abstract categories rather than as coordinate scalings with definite Jacobian values. The correction of this error reveals that all "fundamental constants" are Jacobian coefficients encoding our arbitrary measurement choices from 1793 onward.

1. The Core Confusion: Ratios Are Not Constants

What Are Ratios?

Ratios are dimensionless relationships. They are the actual invariant physics. Examples:

F_nat ~ m_nat·M_nat/r_nat²

E_nat ~ m_nat

T_nat ~ 1/M_nat

These are pure proportionalities. No units. No constants. Just relationships between dimensionless quantities.

What Are Constants?

Constants have units. They are defined by the set of arbitrary scalings we chose for our unit definitions. Examples:

G = 6.674×10⁻¹¹ m³/(kg·s²)

c = 299,792,458 m/s

h = 6.62607015×10⁻³⁴ J·s

k_B = 1.380649×10⁻²³ J/K

Each of these numbers encodes how we chose to scale our measurement axes in 1793 and subsequent refinements.

The Fatal Confusion

Standard physics conflates these two completely different things:

Ratios (the actual physics, dimensionless, invariant)

Constants (coordinate artifacts, carrying units, arbitrary)

This confusion is why we think constants are "fundamental properties of nature" when they're actually just Jacobian transformation coefficients between our arbitrary coordinate system and natural dimensionless ratios.

2. What Standard Dimensional Analysis Gets Wrong

The Abstract Type Error

Standard dimensional analysis treats [L], [M], [T] as abstract algebraic symbols—mere type labels with no numerical content.

When we write [G] = [L³·M⁻¹·T⁻²], the framework says: "G has type length-cubed per mass per time-squared."

This treats dimensions like variable types in programming—just labels ensuring things match up correctly.

The Missing Truth

But [L], [M], [T] are not abstract. They carry specific scaling values:

L carries the value l_P ≈ 1.616×10⁻³⁵ m

M carries the value m_P ≈ 2.176×10⁻⁸ kg

T carries the value t_P ≈ 5.391×10⁻⁴⁴ s

When we say something has dimensions [L³·M⁻¹·T⁻²], we're actually saying:

It scales as l_P³/(m_P·t_P²) = 6.674×10⁻¹¹ m³/(kg·s²)

The dimensions carry specific numerical values. These values are the Jacobian scaling factors that relate our coordinate system to natural dimensionless ratios.

Why This Matters

Standard dimensional analysis says: "G has mysterious dimensions and mysterious value that we must measure."

The truth: G = l_P³/(m_P·t_P²)

The value of G is completely determined by our arbitrary 1793 choices of what we called a meter, kilogram, and second. There is no mystery.

3. The 1793 Bug: Arbitrary Unit Definitions

What Happened in 1793

The French Academy created the Metric System by defining:

Meter: 1/10,000,000 of the distance from equator to pole

Kilogram: Approximately the mass of 1 liter of water

Second: 1/86,400 of a mean solar day

These were completely arbitrary choices based on Earth-specific, human-scale phenomena.

The Consequence

By making these arbitrary choices, we fixed the Jacobian scaling factors:

l_P ≈ 1.616×10⁻³⁵ m (determined by our meter choice)

m_P ≈ 2.176×10⁻⁸ kg (determined by our kilogram choice)

t_P ≈ 5.391×10⁻⁴⁴ s (determined by our second choice)

These Planck scales are not fundamental properties of nature. They are the conversion factors between our arbitrary 1793 coordinate system and natural dimensionless ratios.

Every Constant Follows

Once we fixed meter, kilogram, and second arbitrarily in 1793, we fixed:

c = l_P/t_P = 2.998×10⁸ m/s

G = l_P³/(m_P·t_P²) = 6.674×10⁻¹¹ m³/(kg·s²)

h = m_P·l_P²/t_P = 6.626×10⁻³⁴ J·s

The numerical values of all constants are consequences of our 1793 arbitrary choices, not discoveries about nature. We embedded the Planck scales when we defined the arbitrary unit scales with an assumption that they were independent of each other.

4. What Planck Actually Found in 1899

Not "God's Rulers"

The standard narrative: Planck discovered fundamental scales built into spacetime—the smallest possible length, the smallest possible time, etc.

This is wrong.

The Jacobian Bridge

What Planck actually found was the bridge between SI unit measurements and natural dimensionless ratios. These are the identical natural ratios we see in Newton's Principia.

The Planck scales (l_P, m_P, t_P, etc.) are the missing values in dimensional analysis—the actual numerical content that standard framework treats as absent.

What Natural Units Really Mean

When physicists say "work in natural units where c = h = G = k_B = 1," they present it as a magic trick or computational convenience.

What's actually happening:

m_P = t_P = l_P = T_P = E_P = p_P = F_P =...= 1

You're harmonizing all unit scales to each other. This is an actual mathematical operation.

You Cannot "Just Set Constants to 1"

This is crucial: You cannot just arbitrarily set c = 1 or h = 1.

Constants are not singular things. They're made of various unit scales:

c = l_P/t_P

h = m_P·l_P²/t_P

G = l_P³/(m_P·t_P²)

k_B = m_P·l_P²/(t_P²·T_P)

When you harmonize m_P = t_P = l_P = 1, then constants go to 1 incidentally:

c = 1/1 = 1

h = 1·1²/1 = 1

G = 1³/(1·1²) = 1

The constants become 1 because the unit scales are harmonized, not the other way around.

Note: These are the non-reduced Planck units with h, not the reduced units with ℏ.

The Unity at Natural Ratios

When unit scales are harmonized, the real physics emerges:

T/T_P = f·t_P = m/m_P = l_P/l = E/E_P = p/p_P = F/F_P

All these ratios are the same dimensionless number.

This is unity. The fragmentation into "different types" (mass, time, length, temperature, energy) dissolves. There's just one dimensionless ratio that can be projected onto any measurement axis by multiplying by that axis's Jacobian.

5. Measurement as Tautology

The Actual Process

Measurement is multiplication by 1. Here's how it works:

Step 1 - Natural Ratio (the physics):

This is Newton's law in natural dimensionless form. No constants. Pure proportionality.

Step 2 - Multiply by 1 (in disguise):

We're multiplying by F_P/F_P and (m_P²·l_P²)/(m_P²·l_P²), which both equal 1.

Step 3 - Rearrange:

Step 4 - Isolate the SI formula:

Where we define: G ≡ F_P · l_P²/m_P²

What This Shows

G is not a fundamental property of gravity.

G is the coordinate transformation coefficient from natural dimensionless ratios to SI measurement coordinates.

The entire "complex" formula F = GmM/r² is just the simple ratio F ~ mM/r² decorated with Jacobian scaling factors to express it in SI coordinates.

Physical law is tautology decorated with coordinate artifacts.

6. Case Studies: Demystifying Constants

Case 1: Gravitational Constant (G)

Standard view: G = 6.674×10⁻¹¹ m³/(kg·s²) measures the strength of gravity—a fundamental mystery.

Actual structure:

The value encodes our 1793 arbitrary choices of meter, kilogram, and second.

Gravity's "strength" is invariant—the dimensionless ratio F_nat ~ m_nat·M_nat/r_nat². G merely converts between our coordinate system and that ratio.

Why does G have this exact value?

Because in 1793 we defined the meter as 1/10,000,000 of Earth's equator to pole, the kilogram as ~1 liter of water, and the second as 1/86,400 of a mean solar day. Those arbitrary choices fixed l_P, m_P, t_P, which fixed G.

No mystery.

Case 2: Speed of Light (c)

Standard view: c = 299,792,458 m/s is a fundamental property of spacetime geometry.

Actual structure:

The numerical value encodes the ratio of our arbitrary length scale to our arbitrary time scale.

After the 2019 SI redefinition, we fixed c exactly and then defined the meter in terms of the second using this fixed Jacobian.

The meter is now: "the distance light travels in 1/299,792,458 of a second."

We're not "measuring" c anymore. We stipulated it, then defined our length coordinate using that stipulation.

Case 3: Planck's Constant (h)

Standard view: h = 6.62607015×10⁻³⁴ J·s quantizes action—a fundamental discreteness of nature.

Actual structure: h defines the Jacobian:

This is the coordinate transformation coefficient between frequency measurements and mass measurements.

After 2019, we fixed h exactly and then defined the kilogram in terms of frequency and length using this Jacobian:

A kilogram is now: "the mass corresponding to a specific frequency via the fixed m_P l_P^2 / t_P coefficient."

Case 4: Boltzmann's Constant (k_B) - The 100-Year Runtime Error

This is the most revealing case study.

The Setup (1793):

The French Academy defines Energy mechanically: Work = Force × Distance

They define Temperature phenomenologically: Water freezing (0°C) to boiling (100°C)

The Bug: They assume Heat and Energy are two different types

The Runtime (1793-1877):
For 84 years, scientists operate under the illusion that these are separate domains. Thermodynamics uses "Calories" or "Degrees" on one side and "Joules" on the other. The bug remains dormant because nobody looks at the microscopic code.

The Crash (1877):
Ludwig Boltzmann examines the microscopic structure. He realizes: Temperature is Energy per degree of freedom.

Nature's code: E ~ T (same type)

1793 code: E and T are incompatible types

The Patch (k_B):
To prevent the physics compiler from crashing, Boltzmann (and Planck) insert a "type-casting" coefficient to convert the arbitrary "Degree" into the arbitrary "Joule":

The Fatal Blindness:
Today, we teach students that k_B is a "fundamental constant of thermodynamics."

It is not.

It is the exchange rate between the mistake we made with the thermometer and the mistake we made with the kilogram.

If We Had Known in 1793:
If we had known that heat was motion, we could have measured Temperature in the same units we measure Energy. We would have defined:

Water freezes at: (some dimensionless natural ratio) × E_P

Water boils at: (some other dimensionless natural ratio) × E_P

In that system: k_B = 1 (by construction)

The Memorial Stone:
The existence of k_B is purely a memorial to the fact that for 84 years, we didn't know that "Hot" meant "Fast".

k_B = 1.380649×10⁻²³ J/K is carved into the SI system as a permanent reminder: "Here is where we made incompatible definitions for the same underlying quantity."

We could fix it. We won't. Because:

Institutional inertia

Every thermometer would need recalibration

Every textbook would need rewriting

We'd have to admit the mistake

So instead we keep the bug, call it "fundamental," and teach students to revere it.

7. The 2019 SI Redefinition: Accidentally Correct

What Happened

The International Bureau of Weights and Measures (BIPM) redefined SI by:

Fixing h, c, k_B, e exactly (no more uncertainty)

Defining base units (kg, K, A) in terms of these fixed constants

What They Thought They Were Doing

"Making the SI more fundamental by basing it on constants of nature."

What They Actually Did

They fixed the Jacobian coefficients between SI coordinates and natural dimensionless ratios, then defined the measurement axes using these fixed transformation coefficients.

This is the correct operational structure, even though the conceptual understanding remains confused.

The Modern Process (Since 2019)

To measure mass:

Measure frequency (we can do this very precisely with atomic clocks)

Apply the fixed Jacobian m_P t_P = h/c² = 7.372×10⁻⁵¹ kg/Hz

This defines what we mean by kilogram

The "fundamental constant" h is now the definition of the coordinate transformation from frequency to mass. It's not measured—it's stipulated.

Similarly for temperature:

Measure frequency

Apply the fixed Jacobian t_P/T_P = k_B/h = m_P = 2.084×10¹⁰ Hz/K

This defines what we mean by Kelvin

The constants are coordinate transformation definitions, not properties of nature.

8. Why Standard Dimensional Analysis Cannot See This

The Three-Layer Blindness

Layer 1: Type vs. Value Confusion

Standard framework: [L³·M⁻¹·T⁻²] is a type signature
Reality: L³·M⁻¹·T⁻² carries specific scaling value l_P³/(m_P·t_P²)

Layer 2: Constants as Discovered vs. Imposed

Standard framework: Constants are discovered properties of nature
Reality: Constants are Jacobian coefficients from our coordinate choices

Layer 3: Natural Units as Magic

Standard framework: "Set c=h=G=1 for convenience" This is a complete lack of rigor.

Reality: Harmonize m_P=t_P=l_P=...=1, constants become 1 incidentally. It is simply about aligning unit scales.

The Pedagogical Disaster

Students learn:

Dimensions are abstract types ❌

Constants balance dimensional equations ❌

Natural units are a computational trick ❌

Planck scales are fundamental ❌

The truth:

Dimensions carry specific scaling values ✓

Constants are coordinate transformation coefficients ✓

Natural units harmonize measurement axes ✓

Planck scales are Jacobian bridges ✓

Why the Blindness Persists

Institutional Inertia:

Every undergraduate textbook

Professional practice and notation

Decades of literature

Changing this requires admitting: "We've been systematically confused about dimensions for over a century."

The Mystique Protection:
Treating constants as "fundamental mysteries" serves purposes:

Maintains professional mystique

Justifies ongoing measurement programs

Protects narrative of "discovering nature's secrets"

Avoids admitting arbitrary basis of coordinate choices

The Hand-Wave:
When pressed, physicists say: "It's something to do with units" and move on. This dismissal protects against examining what "setting constants to 1" actually means.

9. Example: Hawking Temperature Demystified

The Standard Presentation

Hawking temperature of a black hole (ignoring geometric factors which are the actual physics):

T = (hc³)/(GMk_B)

This looks profound and complex. Multiple "fundamental constants" governing the temperature-mass relationship.

The Actual Structure

Step 1 - The dimensionless physics: Multiply by 1

    T_nat (1) = (1)/M_nat

This is the actual relationship: temperature scales inversely with mass. Simple proportionality. This is the real physics.

Step 2 - Express in SI coordinates:

Step 3 - Substitute Planck scales in SI with their "definitions":

What This Reveals

The "complex" formula is just the simple ratio T ~ 1/M decorated with Jacobian scaling factors. SI units are just decoration in our unit chart. In a different unit chart the scaling would be different.

The constants (hc³)/(GMk_B) are pure coordinate artifact. They encode our arbitrary 1793 measurement choices, nothing more.

The physics is: T ~ 1/M
The rest is coordinate bookkeeping.

10. Implications

For Physics Education

Dimensional analysis must be taught as:

Coordinate transformation theory

With explicit Jacobian structure

Where Planck scales are bridge values

And constants are transformation coefficients

Not as:

Abstract type-checking

Where constants mysteriously appear

And natural units are magic tricks

For Fundamental Physics

Any "Theory of Everything" must explain:

How unified reality fragments into measurement axes

How arbitrary scales get imposed on those axes

How this fragmentation generates "physical laws" and "constants"

The observer is not peripheral—the observer's coordinate choices generate the visible form of physical law.

For Space Exploration

When humanity expands beyond Earth:

"1/10,000,000 of Earth's equator to pole" becomes absurd

"Mean solar day" makes no sense on Mars

Earth-centric scales are parochial

We need unit systems that are:

Universal (not tied to one planet)

Practical (human-scale convenience)

Mathematically clean (decade scaling, clean Jacobians)

11. Conclusion: We've Been Chasing Our Tail Since 1793

Physical law reduces to:

Natural dimensionless ratio × (Jacobian decoration) = Formula in coordinates

Or more explicitly:

Physics × (Arbitrary 1793 choices) = What we call physical laws

Standard dimensional analysis treats the left side as abstract and the middle term as fundamental. Both are backwards.

The actual structure:

Natural ratios are the physics (dimensionless, coordinate-free)

Jacobians are our arbitrary measurement choices (Planck scales in SI)

Constants are those Jacobians (transformation coefficients)

Formulas are decorated ratios (physics expressed in coordinates)

Measurement is multiplication by 1.

Constants are that "1" expressed usefully.

Physical law is tautology decorated with coordinate artifacts.

The mystery isn't why G = 6.674×10⁻¹¹. The mystery is why we ever thought this was mysterious. We assumed Mass, Length, and Time were independent variables. That was the Original Sin of 1793. The constants are simply the error codes the universe returns when we try to compile that false assumption.

Appendix: The Rogers Rational Unit Chart

As a practical demonstration that constants are coordinate choices, consider defining:

c = 1 × 10¹⁰ m_r/s_r

h = 1 × 10⁻³⁰ J_r·s_r

k_B = 1 × 10⁻²⁰ J_r/K_r

G = 1 × 10⁻⁶ m_r³/(kg_r·s_r²)

This "Rogers rational chart" combines:

Mathematical clarity (mantissa = 1 for all constants)

Human convenience (powers of 10, practical scales)

Honest structure (the "ugly" irrational factors are in the unit definitions where they belong)

In this system, 4×8 foot construction sheets become ~30.1×60.2 m_r² sheets, and 16 inches on center becomes about 10 m_r. The divisibility of 30 and 60 provides practical advantages while maintaining decade-based scientific scaling.

This isn't proposed as a replacement for SI, but as a demonstration: constants are coordinate choices. We could choose cleaner Jacobians any time we wanted. We don't because we're chasing the tail we stuck on ourselves in 1793.