Constants Dimensions Dim One

The dimensionless quantity

A dimensionless quantity is a pure number, one that carries no physical units at all. The speed of light has dimensions of length divided by time; the fine-structure constant, by contrast, is just a number, about 1/137, with no meters, seconds, or kilograms attached. In dimensional analysis, the dimensionless case is the base case: it is the reference point from which every other dimension is defined by adding or subtracting exponents.

The Recognition Science framework's dimensional analysis module formalizes this idea. It defines a dimension as a triple of whole-number exponents for length, time, and mass, written [LᵃTᵇMᶜ]. The declaration dim_one is simply the dimension with all three exponents equal to zero, [L⁰T⁰M⁰]. In plain terms, it is the dimension of a quantity that has no length, time, or mass character: a pure number. The framework's library also defines the individual dimensions for length [L¹], time [T¹], and mass [M¹], and the composite dimensions for the speed of light [L¹T⁻¹], the reduced Planck constant [L²T⁻¹M¹], and the gravitational constant [L³T⁻²M⁻¹]. These are all built from the same triple-of-exponents structure, with dim_one as the zero point.

The declaration itself is a definition, not a theorem. It establishes what the dimensionless dimension is within the framework, and it provides the foundation for checking that physical formulas have consistent units. The framework's documentation uses it to verify, for example, that the Planck length formula √(ħG/c³) has the correct dimension of length, and that the Planck time formula has the correct dimension of time. These checks work by multiplying and dividing dimensioned quantities, with the exponents adding and subtracting, and dim_one is the neutral element in that algebra, the dimension that leaves other dimensions unchanged when combined with them.

What dim_one does not claim is any physical content. It does not say that any particular quantity is dimensionless, only what it would mean for a quantity to be so. It does not derive the value of any constant, and it does not assert that the fine-structure constant or any other specific number is dimensionless. It is a piece of bookkeeping apparatus: a way to keep track of units, not a statement about the world. The framework's larger ambitions, such as deriving the values of physical constants from its axioms, rest on other declarations and theorems, not on this definition.

/-- Dimensionless quantity: [L⁰T⁰M⁰] -/
def dim_one : Dimension := ⟨0, 0, 0⟩

/-- Length dimension: [L¹T⁰M⁰] -/
def dim_L : Dimension := ⟨1, 0, 0⟩

/-- Dimensionless quantity: [L⁰T⁰M⁰] -/
def dim_one : Dimension := ⟨0, 0, 0⟩

What this page does not claim

The declaration dim_one does not assert that any specific physical quantity is dimensionless. The declaration does not derive the value of any physical constant. The declaration does not establish the framework's larger claims about the origin of physical constants.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Constants/Dimensions.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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