What makes a system of units self-consistent?

What are the rules governing the creation of a self-consistent system of units? To be clear, I'm not asking about making units universal or replicable, I'm asking about the mathematics governing what makes a valid system of units valid.

• I am a bit unclear of what you are asking? Units are not really a topic for mathematics but for physics. Mathematically, it doesn't matter if I measure length in meters or in my-backyard-apple-tree-heights. Physically, though, you might want a unit that can be universally, objectively defined while being measurable. Feb 26, 2019 at 7:36
• @Steeven: units are also a topic of mathematics. Think about the proof of Buckingham’s pi-theorem. Feb 26, 2019 at 7:45
• Perhaps this answer to another post helps? (physics.stackexchange.com/questions/460843/…) Feb 26, 2019 at 7:56
• Not sure how Buckingham $\pi$ theorem fits in here @flaudemus, as that's more about computing dimensionless parameters from given variables. It doesn't say anything about units being valid. Feb 26, 2019 at 11:09
• It essentially shows us that we can do theory without units, but the proof of the theorem shows also, what the conditions are for a consistent system of units. Feb 26, 2019 at 12:12

Your question may be answered within this mathematical note on Buckingham's pi-theorem. It describes the mathematics and consequences of a consistent system of units. The author of this reference provides the following insights that may be relevant to your question :

Suppose you have a set $$\{ F_i \}$$ of fundamental units, where $$i=1\ldots m$$. This could, for example, be the seven fundamental SI-units. Then any physical quantity $$R$$ can be expressed as $$R = \rho\cdot [R],\quad (1)$$ where $$\rho$$ is a number and $$[R] = \prod_{i=1}^m F_i^{a_i}. \quad (2)$$ are the units of $$R$$.

The main formal requirement for the system of units is independence. Independence of the fundamental units can be mathematically stated as $$\mbox{If } \prod_{i=1}^m F_i^{x_i} = 1,\quad\mbox{it follows that }x_1=\ldots=x_m=0.\quad (3)$$

I would argue that a system of units is valid (and self-consistent), if it fulfills the independence criterion (3), and allows us to express all physical quantities relevant to us in the form (1) with units given by (2).

Note also, that Buckingham's pi-theorem actually tells us that we can do theoretical physics without units using only dimensionless quantities and relations between them. It also shows us that experimentalists can always present their data in a meaningful way using dimensionless parameters, i.e., experimental parameters scaled to the natural scales of the physics problem.

A related post on Physics SE is found here.

• Whilst this may theoretically answer the question, it would be preferable to include the essential parts of the answer here, and provide the link for reference. Feb 26, 2019 at 11:04
• @KyleKanos: Thanks for your comment. I found some time to extend my answer. Feb 26, 2019 at 14:51