The Core Idea: Unit Systems, Fundamental Constants, and Natural Scales

1. Nature's Intrinsic Relationships: The universe operates with fundamental, intrinsic relationships and equivalences:

   • Between space and time (defined by c).

   • Between mass/energy and frequency (defined by h and c).

   • Governing the strength of gravity (defined by G).

   • Between energy and temperature (defined by k_B).
     These relationships imply a "natural" way or scale at which these phenomena operate, independent of human measurement.

2. Human-Defined Unit Systems (like SI): We use systems of units (meter, kilogram, second, Kelvin, etc.) as arbitrary, human-constructed reference grids to quantify the physical world.

3. Measurement of Fundamental Constants = Calibration: When we experimentally "measure" constants like c, h, G, or k_B and obtain their numerical values in a specific unit system (e.g., SI), we are not defining these natural relationships themselves. Instead, we are calibrating our arbitrary unit system against these pre-existing natural relationships.

   • The numerical value of c in m/s tells us how many of our meter-per-second units correspond to nature's fundamental speed.

   • The numerical value of h (and c) tells us how the mass-frequency equivalence manifests in terms of our kilograms and seconds.

   • Essentially, the measured values of these constants are the scaling factors that reconcile our chosen units with the universe's intrinsic way of operating.

4. Values with Units Contain Embedded Scaling: Consequently, any physical quantity expressed as a value with units (e.g., a mass in kilograms) inherently contains these scaling factors. The number represents the quantity relative to our chosen unit, and that unit itself is scaled relative to a more fundamental, natural scale (like the Planck scale).

5. Natural Units (e.g., Planck Units): By defining units based on these fundamental constants (e.g., setting c=1, h=1, G=1, k_B=1), we create a "natural" unit system. In this system:

   • The fundamental equivalences are expressed most simply.

   • The previously determined SI scaling factors (the numerical values of c, h, G, k_B in SI) are absorbed into the definitions of these new natural units.

6. Transformations and Dimensionless Ratios: The process of converting between unit systems (e.g., SI to natural/Planck units) involves using these fundamental constants as the precise transformation elements (like a diagonal "Jacobian" matrix of scaling factors). This often reveals underlying dimensionless ratios that represent the quantity in terms of these natural scales.

In essence:
The fundamental constants are not just arbitrary numbers; they are the empirically determined numerical representation of nature's intrinsic scales and equivalences as viewed through the lens of our chosen, arbitrary unit system. Our unit systems are ways of projecting reality, and the constants are the specific parameters of that projection. Changing to "natural units" is like changing our viewpoint or coordinate system to one that is "aligned" with these natural scales, making the projection simpler. Every measurement in a conventional system like SI implicitly carries the "coordinates" of that system relative to this more fundamental natural framework.