What is dimensional consistency, mathematically?

Question

When going through symbolic calculations involving physical measurements, it's common to check that the final result is dimensionally consistent. (If I'm calculating a frequency, I'd better get something with units 1/s in the end.) Mathematically, where does the requirement of dimensional consistency come from?

To make this more concrete, say I have some ugly PDE describing a physical problem. After we write it down, it's just a pile of symbols; there's no longer anything indicating that $x$ is a length and $t$ is a time. How can we know a priori that dimensionally inconsistent expressions will never be valid solutions? How can I prove, in general, that the correct solution has the "right mix" of dimensional quantities?

I understand the physical reasons, and I'm familiar with using dimensional analysis, but I'm looking for a more mathematical explanation.

Answer 1

Buckingham Pi Theorem. See e.g. chapter 1 of Bluman and Anco, Symmetry and Integration Methods for Differential Equations.

Answer 2

You can check dimensional consistency all along. $\sin x$ or $\exp x$ are only defined if $x$ is dimensionless. If you take the sine or exp of some combination, it has to be dimensionless. This is a very effective way to catch errors in a derivation. Even if you have a dimensionless problem, it has to stay correct if you attach dimensions. If you have the standard quadratic, $ax^2+bx+c=0$ you can just assign $x$ dimensions of length. Then $a$ must be length$^{-2}$ and so on. If your final result doesn't respect this, you have made an error.