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?
Best Answer
This may be somewhat obliquely along the lines you are asking about, but I think it's interesting enough that it deserves to be made public.
My friend James Dolan has been developing with a number of other people a big program in which large portions of algebraic geometry are interpreted and explained in terms of concepts from categorical logic. A basic chapter in this program is one he explicitly identifies as "dimensional analysis", which in his rendering is another term for a general theory of line bundles or more general line objects in symmetric monoidal categories -- each line bundle can be considered a "dimension", and quantities of that dimension are sections of that bundle. Dimensions of course multiply (correspondingly, line bundles are tensored), and are the objects of a symmetric monoidal category enriched in a category of vector spaces, which he calls a dimensional category. Jim proposes to study objects in algebraic geometry (schemes, stacks, etc.) in terms of the dimensional categories that are attached to them, and representations of them in other symmetric monoidal categories. These often take the form "the dimensional category attached to (some specified important scheme studied by algebraic geometers) $X$ is the universal dimensional category such that...".
Jim gave a number of very accessible and thought-provoking introductory lectures a few years ago at UC Riverside. Videos of those lectures (as well as a brief written description of his program) can be accessed here. John Baez (one of Jim's collaborators) has also written on this program at the n-Category Café, as for example here and I believe also in week 300 of This Week's Finds.
Not that this program is completely novel, by any means. Right now Jim and I are discussing toric varieties, and I note that Vladimir Arnol'd once remarked on the spiritual kinship between dimensional analysis and toric varieties, in connection with a problem he solved as a young boy, in this article. There was a brief Math Overflow discussion about this, here.