J. Rogers, SE Ohio, 16 July 2025
Abstract
This paper exposes a pervasive logical fallacy in modern physics: the claim that physical constants are somehow exempt from basic arithmetic operations. Using the example of Planck's constant h, we demonstrate that the argument "h/c² is different from 20/4 because physical constants work differently" is mathematical nonsense that violates fundamental principles of arithmetic. We show that this "special number" fallacy has infected physics education and prevented recognition of obvious mathematical relationships.
1. Introduction: The Arithmetic Denialism Argument
Consider this argument commonly made by physics educators:
"The analogy with 20/4 = 5 doesn't actually capture what's happening with h/c². Physical constants work differently than pure numbers. Physical constants like h have specific empirical meanings tied to experimental observations... The fact that we can write h/c² doesn't mean h is 'composed of' something involving c²."
This argument is mathematically illiterate. It claims that physical constants are somehow immune to basic arithmetic operations that apply to all other numbers.
2. The Fundamental Error: Numbers Are Numbers
2.1 The Universal Nature of Arithmetic
Arithmetic operations work identically on ALL numbers, regardless of their origin or meaning:
- 20/4 = 5 (abstract numbers)
- $20/4 = $5 (currency)
- 20 apples/4 baskets = 5 apples per basket
- h/c² = Hz_kg (physical constants)
The mathematical operation of division works identically in all cases. There is no special arithmetic for "physical constants."
2.2 The Composition Principle
When we write 20/4 = 5, we are stating that:
- 20 = 4 × 5 (20 is composed of 4 groups of 5)
- 5 = 20/4 (5 is the quotient when 20 is divided by 4)
This is identical to what happens with h/c²:
- h = c² × Hz_kg (h is composed of c² groups of Hz_kg)
- Hz_kg = h/c² (Hz_kg is the quotient when h is divided by c²)
There is no mathematical difference between these operations.
3. The "Special Number" Fallacy
3.1 The Mystical Thinking
The argument that "physical constants work differently" is pure mysticism. It treats h as if it were a magical number that doesn't follow normal mathematical rules.
This is equivalent to claiming:
- "20 is a special number because it represents twenty things"
- "Therefore 20/4 doesn't really equal 5"
- "The 5 isn't really 'derived from' the 20"
This is obvious nonsense when applied to regular numbers, and it's equally nonsensical when applied to physical constants.
3.2 The Measurement Irrelevance
The fact that h was "discovered through blackbody radiation experiments" is completely irrelevant to its arithmetic properties.
Consider: The number 20 might be discovered by counting apples, but this doesn't make 20/4 = 5 any less true or meaningful.
How you discover a number has nothing to do with how arithmetic works on that number.
3.3 Scientific Numerology
Physicists resist the conclusion that h/c² reveals real compositional structure because they’ve embraced a form of numerology. They treat certain constants—like h, c, and G—not as numbers governed by arithmetic, but as mystical entities immune to basic operations like division.
How a number is discovered has no bearing on how arithmetic works on it. If you can write h / c², then h = c² × (h/c²). That’s not interpretation. That’s arithmetic.
This isn’t abstract philosophy. It’s basic math. And denying it isn’t deep—it’s denialism.
4. The Empirical Evidence: Unit Scaling
4.1 The Mathematical Proof
The unit scaling code proves that h/c² functions as a fundamental quantity:
Hz_kg = h / c**2
# Hz_kg serves as the basic mass scaling coordinate
{"symbol": "kg", "factor": Hz_kg/t_P}
This is empirical proof that the division h/c² extracts a fundamental component, exactly like 20/4 extracts 5.
4.2 The Scaling Relationships
At the Planck scale, all units reduce to time coordinates through operations involving Hz_kg. This proves that Hz_kg is a fundamental scaling unit extracted from h by dividing out c².
This is basic arithmetic revealing physical structure.
5. The Logical Absurdity of the Counter-Argument
5.1 The Inconsistency
The argument claims that h/c² "doesn't mean h is composed of something involving c²" while simultaneously using this very composition in calculations.
This is like saying:
- "20/4 doesn't mean 20 is composed of 4 groups of 5"
- Then using exactly this composition to solve problems
You cannot use arithmetic relationships while denying they exist.
5.2 The Practical Contradiction
Physics routinely uses h/c² in calculations, proving that h IS composed of c² × Hz_kg. The argument against this is purely philosophical denial of obvious mathematical facts.
If h weren't composed of c² × Hz_kg, then h/c² would be meaningless. But it's not meaningless—it's a fundamental scaling quantity.
6. The Educational Damage
6.1 Teaching Mathematical Incoherence
This fallacy teaches students that:
- Arithmetic works differently for different types of numbers
- Physical constants are somehow exempt from mathematical analysis
- Mathematical relationships don't reveal physical structure
This is mathematical illiteracy disguised as sophisticated thinking.
6.2 The Anti-Scientific Stance
The argument prevents students from recognizing obvious mathematical patterns in physics by declaring them "ontologically neutral."
This is equivalent to teaching that 20/4 = 5 tells us nothing about the relationship between 20, 4, and 5.
7. The Correct Understanding
7.1 Arithmetic Universality
All numbers follow the same arithmetic rules:
- If you can divide h by c², then h = c² × (h/c²)
- The quotient h/c² is a real quantity (Hz_kg)
- This relationship is as real as 20 = 4 × 5
7.2 Mathematical Realism
Mathematical relationships in physics are discoveries about reality, not arbitrary manipulations. When arithmetic reveals that h factors as c² × Hz_kg, this is a real discovery about the structure of h.
8. Specific Refutations
8.1 "Physical Constants Are Different"
FALSE. Physical constants are numbers. Numbers follow arithmetic rules. There are no special arithmetic rules for "physical" numbers.
8.2 "Mathematical Operations Don't Imply Ontological Priority"
FALSE. Mathematical operations reveal compositional structure. 20/4 = 5 reveals that 20 is composed of 4 groups of 5. h/c² = Hz_kg reveals that h is composed of c² groups of Hz_kg.
8.3 "Historical Discovery Determines Meaning"
FALSE. How you discover a number is irrelevant to its arithmetic properties. The number 20 has the same arithmetic properties whether discovered by counting apples or measuring distances.
9. The Broader Implications
9.1 Scientific Progress
Science advances by recognizing mathematical relationships. The argument against mathematical reductionism would prevent all scientific progress by declaring that no mathematical relationship reveals anything about reality.
9.2 The Unity of Mathematics
Mathematics is unified. The same arithmetic that applies to counting apples applies to physical constants. There is no separate "physics arithmetic" with different rules.
10. Conclusion
The argument that "h/c² is different from 20/4 because physical constants work differently" is mathematical nonsense that violates basic principles of arithmetic.
h/c² is exactly the same mathematical operation as 20/4. Both operations extract a quotient from a dividend by factoring out a divisor. Both reveal compositional structure. Both follow identical arithmetic rules.
The claim that physical constants are somehow exempt from normal arithmetic is mathematical illiteracy that prevents recognition of obvious relationships.
Anyone who makes this argument has forgotten that numbers are numbers, regardless of their physical interpretation. Arithmetic doesn't change based on what the numbers represent.
The unit scaling evidence proves that h/c² = Hz_kg is a fundamental physical relationship revealed through basic arithmetic. Denying this is equivalent to denying that 20/4 = 5 tells us anything about the relationship between 20, 4, and 5.
This is not sophisticated thinking—it's arithmetic denialism.
No comments:
Post a Comment