TITLE: Dimensional Analysis in Mathematics
QUESTION [39 upvotes]: Is there a sensible and useful definition of units in mathematics? In other words, is there a theory of dimensional analysis for mathematics?
In physics, an extremely useful tool is the Buckingham Pi theorem. This allows for surprisingly accurate estimates that can predict on what parameters a quantity depends on. Examples are numerous and can be found in this short reference. One such application (pages 6-7 of the last reference) can derive the dispersion relation exactly for short water ripples in deep water, in terms of surface tension, density and wave number. In this case an exact relation is derived, but in general one expects to be off by a constant. The point is that this gives quick insight into an otherwise complex problem.
My question is: can similar techniques be used in mathematics? 
I envision that one application could be to derive asymptotic results for say, ode's and pde's under certain asymptotic assumptions for the coefficients involved. For any kind of Fourier analysis, dimensions naturally creep up from physics if we think about say, time and frequency. I find myself constantly using basic dimensional analysis just as a sanity check on whether a quantity makes sense in these areas.
Otherwise, let's say I'm working on a problem involving some estimate on a number theoretic function. If I have a free parameter, can I quickly figure out the order of the quantity i'm interested in in terms of my other fixed parameters?

REPLY [3 votes]: To add to the replies above, Terry Tao has a blog entry on this
A mathematical formalisation of dimensional analysis | What's new
https://terrytao.wordpress.com/2012/12/29/a-mathematical-formalisation-of-dimensional-analysis/