On the Logical Fallacy of "Setting Constants to 1" and the Correct Mechanism of Basis Transformation in Physics

J. Rogers, SE Ohio, 22 Jun 2025, 0801

Abstract:
For nearly a century, the practice of "setting fundamental constants to 1" (c=G=h=k_B=1) has been a standard technique in theoretical physics, prized for its ability to simplify complex equations. While pragmatically useful, we argue that this practice, as commonly employed, constitutes a formal logical fallacy—assuming the consequent—that has obscured the true nature of physical constants and the structure of physical law. This paper demonstrates that the disappearance of constants is not a starting assumption but the result of a well-defined mathematical operation: a basis rotation of our measurement coordinate system into alignment with the universe's natural, orthogonal basis. We present the correct mechanism for this transformation, which involves the identification and application of specific Jacobian operators (Hz_kg = h/c², K_Hz = k_B/h). This rigorous approach not only avoids the logical fallacy but also reveals the constants for what they are: the coordinate transformation coefficients required to patch over our misaligned, human-centric system of units.

1. The Standard Practice: A Pragmatic Handwave

In modern physics, it is commonplace to see introductions to advanced topics begin with the declaration: "We will work in a system of natural units where c=1, h=1, etc." This is done to remove the cumbersome constants from equations like the Einstein Field Equations or the Dirac Equation, making the underlying mathematical structure more transparent.

The pragmatic benefit is undeniable. However, the methodological justification is rarely, if ever, provided. The act is treated as a mere change of units, a re-scaling of variables. We contend that this "handwave" is a profound category error that masks a deep logical flaw.

2. The Logical Error: Assuming the Consequent

The act of setting constants to 1 represents a formal logical fallacy known as affirming the consequent. This fallacy takes the structure:

1. If P, then Q.

2. Q is true.

3. Therefore, P is true.

This is an invalid form of argument. The truth of the consequent (Q) does not guarantee the truth of the antecedent (P).

Let us map this fallacy onto the practice of using natural units:

• The Correct Premise (P → Q): "If we have successfully performed a basis transformation of our measurement system into the universe's natural, orthogonal basis (P), then the numerical values of the coordinate projection artifacts c, G, h will become 1 in this new basis (Q)." This statement is logically sound. The disappearance of the constants is a consequence of being in the right basis.

• The Standard Practice (Asserting Q): The physicist begins by stating, "Let c=1, h=1, etc." In doing so, they are directly asserting the consequent (Q).

• The Flawed Conclusion (Therefore, P): The implicit, unstated conclusion is, "Therefore, my equations are now correctly expressed in the natural, coordinate-free basis (P)."

This is a textbook example of assuming the result. The physicist has jumped to the conclusion without performing the necessary work. The "work" is not the arbitrary re-scaling of numbers; the work is the explicit basis rotation. By skipping this step, the physicist learns nothing about why the constants disappear, what they truly represent, or the geometric structure of our measurement system.

3. The Correct Mechanism: Basis Rotation via Jacobian Operators

The logical error is corrected by replacing the handwave with a rigorous, mechanical procedure. The goal is not to "set constants to 1," but to harmonize our measurement axes.

We have previously established that human-centric SI units (meter, kilogram, second, Kelvin) form a misaligned, non-orthogonal basis for describing reality. The universe's "preferred" basis appears to be one where all physical quantities are ultimately commensurable with a single master axis: Time/Frequency.

To move from our basis to the natural basis, we must perform a series of rotations. These rotations are executed by specific transformation operators—Jacobians—that connect our axes to the master axis of time.

3.1. Identification of the Jacobians

The crucial step, ignored by the old paradigm, is to identify and name these Jacobians. They are not the constants themselves, but specific ratios of them:

• Spacetime Jacobian (c): Scales the Length axis to the Time axis. [L] = c · [T].

• Mass-Frequency Jacobian (Hz_kg): Scales the Mass axis to the Frequency axis. [M] = Hz_kg · [f], where Hz_kg = h/c².

• Temperature-Frequency Jacobian (K_Hz): Scales the Temperature axis to the Frequency axis. [T_temp] = (1/K_Hz) · [f], where K_Hz = k_B/ħ.

These Jacobians are the gears of the transformation. They are not arbitrary numbers but the precise conversion factors needed to make our disparate units commensurable.

3.2. The Correct Procedure

The correct way to transition to a natural basis is as follows:

1. Start with an equation in an arbitrary system (e.g., SI).

2. For each dimensional variable, substitute its equivalent in terms of the natural basis units and the corresponding Jacobian. For example, replace any instance of m (mass) with m' · Hz_kg, where m' is the mass value in the natural basis.

3. Algebraically simplify the entire equation.

4. A Demonstrative Example: E = mc²

Let's demonstrate the flaw and the correction with Einstein's famous equation.

• The Handwave Method: Start with E = mc². "Set c=1." Result: E = m. This is fast, but it teaches nothing. Why did c vanish? The reason is left as a mystery.

• The Correct Basis Transformation Method:

  1. Start with E = mc². The physics is the postulate E ~ M.

  2. We seek to express this in a basis where Energy, Mass, and Frequency are aligned. Let's convert E and m to their frequency equivalents.

  3. We know the basis transformations: E = E' · h and m = m' · Hz_kg = m' · (h/c²), where E' and m' are the values in the frequency-aligned basis.

  4. Substitute these into the original equation: (E' · h) = (m' · h/c²) · c².

  5. Simplify: E' · h = m' · h.

  6. The final result: E' = m'.

Analysis of the Result:
The equation E' = m' states that in the natural, frequency-aligned basis, the numerical value of energy is the numerical value of mass. The constants h and c² did not vanish by decree; they canceled out as a direct and understandable consequence of applying the correct basis transformation rules. We did not assume the result E=m; we proved it by performing the rotation.

5. Conclusion: From Magic Incantation to Rational Mechanism

The long-standing practice of "setting constants to 1" is a logically fallacious shortcut that has, for decades, prevented a deeper understanding of the structure of physical law. It mistakes the destination for the journey, the result for the process. This has led generations of physicists to treat natural units as a kind of mathematical "black box" and to view the fundamental constants as mystical entities that can be wished away, rather than as what they truly are: the evidence of our own misaligned perspective.

The calculus of basis rotation via named Jacobians provides the rigorous, logical, and pedagogically sound alternative. It replaces a magic incantation with a rational mechanism. It respects the rules of logic by refusing to assume the consequent, instead demonstrating that the unity of physical constants and the simplicity of physical law are the earned and inevitable results of seeing the universe not in our own crooked terms, but in its own. The goal is not to force the constants to 1; the goal is to perform the work that leaves them with no other choice.