The Dimensional Analysis Confusion: How Physics Mistook Arithmetic for Abstraction

 J. Rogers, SE Ohio

Abstract

The standard treatment of dimensional analysis in physics presents dimensions [L], [M], [T] as "abstract categories" or "bookkeeping labels" without numerical values. We prove this framework is mathematically incoherent. By examining the actual equations used in dimensional analysis, we demonstrate that dimensions are not abstract—they are specific numerical values, dependent on the unit scaling of a unit chart, in SI it is equal to the Planck-scale quantities derived from h, c, and G in the SI unit chart. Specifically: [L] = 4.051×10⁻³⁵ m, [M] = 5.456×10⁻⁸ kg, [T] = 1.351×10⁻⁴³ s in SI units. This revelation exposes that dimensional analysis has always been substrate arithmetic in disguise, and the claimed "abstraction" is pedagogical obfuscation that obscures the relationship between measurement coordinates and physical reality.

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1. The Standard Dogma: Dimensions As Abstract Labels

1.1 What Students Are Taught

Physics education presents dimensional analysis as working with "abstract dimensions":

Standard Teaching:

• [L] represents "the dimension of length"—an abstract category
• [M] represents "the dimension of mass"—a type label
• [T] represents "the dimension of time"—a bookkeeping symbol
• These have no numerical values; they're merely labels that track what kind of quantity you're dealing with

Typical textbook language:

• "Dimensions are not the same as units. They're abstract."
• "We write [L] to mean 'length-like,' not any specific length."
• "Dimensional analysis is symbolic manipulation, not arithmetic."

1.2 How Dimensional Analysis Is Presented

Students learn procedures like:

Step 1: Write the dimensional formula

[Force] = [M L T⁻²]

Step 2: Check dimensional consistency

[M L T⁻²] = [M L T⁻²] ✓

Step 3: This proves the equation is dimensionally valid

The framework claims:

• This is purely symbolic checking
• No actual numbers are involved
• The dimensions are "abstract types" being matched

1.3 The Claimed Distinction

Dimension vs. Unit:

Standard pedagogy insists on a sharp distinction:

Concept	Nature	Example
Unit	Concrete, numerical, arbitrary	"5 meters"
Dimension	Abstract, symbolic, fundamental	"[L]"

Students are told:

• Units are specific scales we choose (meter, foot, etc.)
• Dimensions are abstract categories that transcend any specific choice
• Never confuse the two!

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2. The Planck Scale: What Dimensions Actually Are

2.1 Max Planck's 1899 Discovery

In 1899, Max Planck identified that three fundamental constants—h (Planck's constant), c (speed of light), and G (gravitational constant)—could be combined to form natural scales:

Planck length:

l_P = √(hG/c³)

Planck mass:

m_P = √(hc/G)

Planck time:

t_P = √(hG/c⁵)

Using the SI values:

• h = 6.62607015 × 10⁻³⁴ J·s
• c = 2.99792458 × 10⁸ m/s
• G = 6.67430 × 10⁻¹¹ m³/(kg·s²)

2.2 Computing The Planck Values

Planck length:

l_P = √(hG/c³)
    = √((6.626×10⁻³⁴ J·s)(6.674×10⁻¹¹ m³/(kg·s²)) / (2.998×10⁸ m/s)³)
    = √(4.423×10⁻⁴⁴ J·m³/(kg·s) / 2.694×10²⁵ m³/s³)

Converting J = kg·m²/s²:

    = √(4.423×10⁻⁴⁴ kg·m⁵/s³ / 2.694×10²⁵ m³/s³)
    = √(1.641×10⁻⁶⁹ m²)
    = 4.051×10⁻³⁵ m

Planck mass:

m_P = √(hc/G)
    = √((6.626×10⁻³⁴ J·s)(2.998×10⁸ m/s) / (6.674×10⁻¹¹ m³/(kg·s²)))
    = √(1.986×10⁻²⁵ J·m / 6.674×10⁻¹¹ m³/(kg·s²))

Converting J = kg·m²/s²:

    = √(1.986×10⁻²⁵ kg·m³/s / 6.674×10⁻¹¹ m³/(kg·s²))
    = √(2.976×10⁻¹⁵ kg²)
    = 5.456×10⁻⁸ kg

Planck time:

t_P = √(hG/c⁵)
    = √((6.626×10⁻³⁴)(6.674×10⁻¹¹) / (2.998×10⁸)⁵)
    = √(4.423×10⁻⁴⁴ / 2.408×10⁴²)
    = √(1.836×10⁻⁸⁶)
    = 1.351×10⁻⁴³ s

Summary - The Planck Scale in SI:

l_P = 4.051 × 10⁻³⁵ m
m_P = 5.456 × 10⁻⁸ kg
t_P = 1.351 × 10⁻⁴³ s

2.3 The Gravitational Constant Equation

Now consider the fundamental constant G expressed dimensionally and numerically:

Dimensional formula:

[G] = [L³ M⁻¹ T⁻²]

Planck-scale expression:

G = l_P³/(m_P·t_P²)

Numerical verification:

l_P³/(m_P·t_P²) = (4.051×10⁻³⁵)³ / ((5.456×10⁻⁸)(1.351×10⁻⁴³)²)
                 = 6.647×10⁻¹⁰⁴ / (5.456×10⁻⁸ × 1.825×10⁻⁸⁶)
                 = 6.647×10⁻¹⁰⁴ / 9.957×10⁻⁹⁴
                 = 6.674×10⁻¹¹ m³/(kg·s²)
                 = G_SI ✓

2.4 The Equation That Destroys The Dogma

Equating these expressions:

[L³ M⁻¹ T⁻²] = l_P³/(m_P·t_P²)

This equation is not symbolic. It's arithmetic.

For this equation to be true:

[L] must equal l_P [M] must equal m_P [T] must equal t_P

Otherwise:

• If [L] ≠ l_P, then [L³] ≠ l_P³, and the equation is false
• If [M] ≠ m_P, then [M⁻¹] ≠ m_P⁻¹, and the equation is false
• If [T] ≠ t_P, then [T⁻²] ≠ t_P⁻², and the equation is false

The equation is true.

Therefore, in SI units:

[L] = l_P = 4.051 × 10⁻³⁵ m
[M] = m_P = 5.456 × 10⁻⁸ kg
[T] = t_P = 1.351 × 10⁻⁴³ s

The dimensions are not abstract. They are specific numerical values defined by h, c, and G in the chosen unit chart.

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3. Every Dimensional Formula Is Substrate Arithmetic

3.1 The Speed of Light

Standard presentation:

[c] = [L T⁻¹]

Claimed meaning: "c has the abstract dimension of length divided by time."

Actual meaning:

[c] = l_P / t_P
    = (4.051×10⁻³⁵ m) / (1.351×10⁻⁴³ s)
    = 2.998×10⁸ m/s
    = c ✓

The speed of light.

When you write [c] = [L T⁻¹], you're not doing abstract symbol manipulation.

You're calculating c using Planck-scale values derived from h and G.

3.2 Planck's Constant

Standard presentation:

[h] = [M L² T⁻¹]

Actual meaning:

[h] = m_P × l_P² × t_P⁻¹
    = (5.456×10⁻⁸ kg) × (4.051×10⁻³⁵ m)² × (1.351×10⁻⁴³ s)⁻¹
    = (5.456×10⁻⁸) × (1.641×10⁻⁶⁹) × (7.401×10⁴²)
    = 6.626×10⁻³⁴ J·s
    = h ✓

Planck's constant itself.

Every time you write h's dimensional formula, you're referencing these specific Planck values.

3.3 Force Dimension

Standard presentation:

[F] = [M L T⁻²]

Claimed meaning: "Force has the abstract dimension of mass times length divided by time squared."

Actual meaning:

[F] = m_P × l_P × t_P⁻²
    = (5.456×10⁻⁸ kg) × (4.051×10⁻³⁵ m) × (1.351×10⁻⁴³ s)⁻²
    = (5.456×10⁻⁸) × (4.051×10⁻³⁵) × (5.477×10⁸⁵)
    = 1.210×10⁴⁴ N

This is the Planck force—a specific numerical value.

When you write [F] = [M L T⁻²], you're not doing abstract symbol manipulation.

You're calculating the Planck force using h, c, and G.

3.4 Energy Dimension

Standard presentation:

[E] = [M L² T⁻²]

Actual meaning:

[E] = m_P × l_P² × t_P⁻²
    = (5.456×10⁻⁸) × (4.051×10⁻³⁵)² × (1.351×10⁻⁴³)⁻²
    = (5.456×10⁻⁸) × (1.641×10⁻⁶⁹) × (5.477×10⁸⁵)
    = 4.904×10⁸ J

This is the Planck energy (from non-reduced h, c, G).

Every time you write an energy dimension, you're referencing this specific number.

3.5 The Pattern

Every dimensional formula [M^α L^β T^γ] literally equals:

m_P^α × l_P^β × t_P^γ

Where m_P, l_P, t_P are solved as a system of equations from h, c, and G.

This is not abstraction. This is exponentiation and multiplication of specific numbers.

Dimensional analysis is Planck-scale arithmetic using h, c, and G.

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4. The Dimensional Matrix: Numbers, Not Symbols

4.1 The Standard Presentation

Dimensional analysis textbooks present the dimensional matrix:

       h   c   G
    ⎡  1   0  -1 ⎤  M
D = ⎢  2   1   3 ⎥  L
    ⎣ -1  -1  -2 ⎦  T

Claimed interpretation: "This shows how the abstract dimensions combine to form the constants."

4.2 What The Matrix Actually Contains

The matrix entries are exponents of specific numerical values:

                h                   c                   G
    ⎡  m_P^1                 m_P^0               m_P^(-1)         ⎤  
D = ⎢  l_P^2                 l_P^1               l_P^3            ⎥  
    ⎣  t_P^(-1)              t_P^(-1)            t_P^(-2)         ⎦  

Where m_P, l_P, t_P are derived from h, c, and G via Planck's 1899 formulas.

Substituting the actual values:

For h:

[h] = m_P^1 × l_P^2 × t_P^(-1)
    = (5.456×10⁻⁸) × (4.051×10⁻³⁵)² × (1.351×10⁻⁴³)⁻¹
    = (5.456×10⁻⁸) × (1.641×10⁻⁶⁹) × (7.401×10⁴²)
    = 6.626×10⁻³⁴ J·s ✓

For c:

[c] = m_P^0 × l_P^1 × t_P^(-1)
    = 1 × (4.051×10⁻³⁵) × (1.351×10⁻⁴³)⁻¹
    = (4.051×10⁻³⁵) × (7.401×10⁴²)
    = 2.998×10⁸ m/s ✓

For G:

[G] = m_P^(-1) × l_P^3 × t_P^(-2)
    = (5.456×10⁻⁸)⁻¹ × (4.051×10⁻³⁵)³ × (1.351×10⁻⁴³)⁻²
    = (1.833×10⁷) × (6.647×10⁻¹⁰⁴) × (5.477×10⁸⁵)
    = 6.674×10⁻¹¹ m³/(kg·s²) ✓

The dimensional matrix is a numerical computation table.

Not abstract. Arithmetic with Planck values from h, c, and G.

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5. Why The Confusion Persists

5.1 The Pedagogical Convenience

Teaching dimensions as abstract serves several purposes:

1. Avoids Natural Units Early: Students aren't ready to understand that all physics reduces to dimensionless substrate ratios.
   ( We disagree strongly with this, it is easy for kids to understand unity. )

2. Postpones Deep Questions: "What is length really?" is deflected by "It's just an abstract category."

3. Makes Checking Easier: Symbol matching [M L T⁻²] = [M L T⁻²] is simpler than computing Planck values.

4. Hides The Jacobian: If students knew dimensions equal Planck values, they'd ask why—leading to uncomfortable questions about constants being coordinate artifacts that the standard frame tries super hard to ignore.

5.2 The Textbook Language

Careful examination of textbook claims reveals hedging:

They say:

• "Dimensions are abstract categories" (vague)
• "Think of [L] as representing 'length-ness'" (metaphorical)
• "Dimensions track the type of quantity" (avoiding numerical commitment)

They never say:

• "Dimensions have no numerical values" (too explicit—would be provably false)
• "[L] ≠ any specific length" (false, it equals l_P in the si unit chart they are doing the formula in. )

The language is carefully chosen to suggest abstraction while never explicitly claiming it.

5.3 The Cognitive Dissonance

Students who question this are told:

Student: "If dimensions are abstract, how do we compute with them?"

Professor: "We're doing symbolic manipulation, not arithmetic."

Student: "But G = l_P³/(m_P·t_P²) is arithmetic..."

Professor: "That's just one representation. The dimensions themselves are abstract."

Student: "Then how does [L³ M⁻¹ T⁻²] equal a specific number?"

Professor: "You're conflating dimensions with units. Let's move on..."

The questions are deflected, not answered.

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6. The Proof: Dimensional Analysis IS Planck Arithmetic

6.1 Test Case: Checking Force Units

Standard procedure:

Given F = ma, verify dimensions:

[F] = [M] × [L T⁻²]
    = [M L T⁻²] ✓

Claimed process: "We matched abstract symbols."

Actual process:

[F] = m_P × (l_P × t_P⁻²)
    = m_P × l_P × t_P⁻²
    = (5.456×10⁻⁸) × (4.051×10⁻³⁵) × (1.351×10⁻⁴³)⁻²
    = 1.210×10⁴⁴ N

[M L T⁻²] = m_P × l_P × t_P⁻²
          = 1.210×10⁴⁴ N

[F] = [M L T⁻²] ✓

We verified that both sides equal the same numerical value: 1.210×10⁴⁴ N.

This is arithmetic with Planck values from h, c, G—not symbolic matching.

6.2 Test Case: Energy-Mass Equivalence

Given E = mc²:

Standard dimensional check:

[E] = [M] × [L T⁻¹]²
    = [M L² T⁻²] ✓

Actual calculation:

[E] = m_P × (l_P/t_P)²
    = m_P × l_P² × t_P⁻²
    = (5.456×10⁻⁸) × (4.051×10⁻³⁵)² × (1.351×10⁻⁴³)⁻²
    = 4.904×10⁸ J

[M L² T⁻²] = m_P × l_P² × t_P⁻²
           = 4.904×10⁸ J

[E] = [M L² T⁻²] ✓

Both sides equal the Planck energy (from h, c, G): 4.904×10⁸ J.

We're computing with specific numbers derived from h, c, and G—not manipulating abstract symbols.

6.3 The General Proof

For any dimensional formula [M^α L^β T^γ]:

Step 1: Substitute the Planck values (from h, c, G)

[M^α L^β T^γ] = m_P^α × l_P^β × t_P^γ

Step 2: Compute the numerical result

= (5.456×10⁻⁸)^α × (4.051×10⁻³⁵)^β × (1.351×10⁻⁴³)^γ

Step 3: This yields a specific number with SI units

Dimensional analysis is exponentiation and multiplication of Planck-scale values derived from h, c, and G.

There is no "abstraction." It's all arithmetic.

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7. The Unit Chart Dependency

7.1 Planck Values Are Chart-Specific

The Planck values change with unit chart choice:

In SI:

• l_P(SI) = √(hG/c³) = 4.051×10⁻³⁵ m
• m_P(SI) = √(hc/G) = 5.456×10⁻⁸ kg
• t_P(SI) = √(hG/c⁵) = 1.351×10⁻⁴³ s

In RUC (Rogers Rational Unit Chart with c=1×10¹⁰, h=1×10⁻³⁰, G=1×10⁻⁶):

• l_P(RUC) = √(hG/c³) in m_r units
• m_P(RUC) = √(hc/G) in kg_r units
• t_P(RUC) = √(hG/c⁵) in s_r units

In any unit chart U:

• l_P(U) = √(hG/c³) in U's length units
• m_P(U) = √(hc/G) in U's mass units
• t_P(U) = √(hG/c⁵) in U's time units

The formulas use h (Planck's constant from 1899), not the reduced constant.

7.2 Dimensions Are Chart-Relative

The dimensions [L], [M], [T] in a given chart equal that chart's unit "Planck"  values (really unit jacobians):

In SI:

[L]_SI = l_P(SI) = 4.051×10⁻³⁵ m
[M]_SI = m_P(SI) = 5.456×10⁻⁸ kg
[T]_SI = t_P(SI) = 1.351×10⁻⁴³ s

In RUC:

[L]_RUC = l_P(RUC) in m_r
[M]_RUC = m_P(RUC) in kg_r
[T]_RUC = t_P(RUC) in s_r

Different charts, different dimensional values.

But both represent the same substrate ratio when units cancel.

7.3 The Invariant Substrate

What remains constant across all charts:

The dimensionless ratio when you divide by the chart's Planck scale:

Electron mass in any chart:

m_e / m_P(chart) = dimensionless invariant ratio

In SI:

m_e = 9.109×10⁻³¹ kg
m_P(SI) = 5.456×10⁻⁸ kg
m_e/m_P(SI) = 1.670×10⁻²³

In RUC:

m_e in kg_r / m_P(RUC) in kg_r = 1.670×10⁻²³

The substrate value is the invariant truth.

The dimensions [M], [L], [T] are the chart-specific Planck values derived from h, c, and G in that chart.

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8. Conclusion: The Deception Exposed

8.1 The Central Proof

We have proven rigorously that the "abstract dimensions" taught in physics education are not abstract:

[L] = l_P = √(hG/c³) = 4.051×10⁻³⁵ m (in SI)
[M] = m_P = √(hc/G) = 5.456×10⁻⁸ kg (in SI)
[T] = t_P = √(hG/c⁵) = 1.351×10⁻⁴³ s (in SI)

These are specific numerical values, not abstract categories.

Every dimensional formula [M^α L^β T^γ] is arithmetic:

[M^α L^β T^γ] = m_P^α × l_P^β × t_P^γ

Dimensional analysis is Planck-scale computation using h, c, and G.

8.2 What Planck Actually Discovered in 1899

Max Planck's 1899 natural units are not "a convenient unit system."

They are the bridge from measurement coordinates back to Newton's dimensionless ratios.

The Planck values l_P, m_P, t_P (derived from h, c, G) are:

• The Jacobian inverse from arbitrary unit charts to substrate reality
• The numerical values of dimensions in each unit chart
• The conversion factors that remove coordinate pollution

When you "divide by the Planck scale," you're removing dimensional contamination to reveal the pure substrate ratios that Newton worked with.

8.3 The Pedagogical Fraud

Physics education commits systematic deception:

They teach: "Dimensions are abstract labels with no numerical values."

The math proves: Dimensions equal specific Planck values in every equation.

They claim: "Dimensional analysis is symbolic manipulation."

The reality: Dimensional analysis is arithmetic with Planck-derived values from h, c, and G.

This is not innocent simplification—it's obfuscation that prevents students from understanding:

• Constants are coordinate artifacts
• Dimensions are Planck-scale values
• Natural units reveal substrate reality
• Newton's ratio-based physics was architecturally correct

8.4 The Implications

For physics education:

• Dimensional analysis textbooks must be rewritten
• The "abstraction" framing must be abandoned
• Students deserve to know dimensions have specific numerical values derived from h, c, and G

For theoretical physics:

• Stop treating constants as "fundamental mysteries"
• Recognize them as Jacobian transformation coefficients
• Understand Planck's 1899 work as revealing the substrate bridge
• Return to Newton's ratio-based approach as architecturally primary

For metrology:

• The 2019 SI redefinition operationally acknowledged constants are design choices
• The formalism presented here provides the missing conceptual foundation
• Future unit systems should explicitly align to the Planck scale (mantissa = 1)

8.5 The Final Irony

For 120+ years, physics has taught:

"Set h=c=G=1 as a convenient trick for advanced work."

The truth they missed:

Setting h=c=G=1 isn't a trick—it's removing the coordinate system entirely and working directly in Newton's dimensionless substrate reality.

The dimensions [L], [M], [T] are the Planck values l_P, m_P, t_P.

Dimensional analysis is Planck arithmetic.

"Abstract dimensions" is a pedagogical lie to avoid teaching this truth.

The emperor has no clothes.

The "abstraction" was always fiction. The math was always arithmetic. The dimensions always had specific values derived from h, c, and G.

We just taught generations of physicists to not look at what the equations actually say.

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Appendix: Historical Note on Planck's 1899 Paper

Max Planck introduced natural units in his 1899 paper using h (not ℏ), along with c and G. His formulas were:

l_P = √(hG/c³)
m_P = √(hc/G)
t_P = √(hG/c⁵)

These are the original, non-reduced Planck units.

The reduced constant ℏ = h/(2π) was introduced later by Dirac as notational convenience in quantum mechanics. The "reduced Planck units" using ℏ instead of h are a 20th-century derivative, not Planck's original discovery.

Planck was right in 1899: The natural units derived from h, c, and G form the unique bridge from measurement coordinates to dimensionless substrate reality.

When students ask "Why these specific combinations?", the answer is:

These are the only combinations that invert the Jacobian transformation from Newton's ratio-based physics to our arbitrary coordinate system.

Planck found the map home. We've spent 120+ years pretending it was just a convenient shortcut.

It wasn't. It was the revelation that dimensions are not abstract—they are specific numerical scales defined by h, c, and G in any given unit chart.

This paper simply makes explicit what the mathematics has said all along.