Over at the n-category cafe, John Baez discusses dimensional analysis in physics, as the beginning of a “collection of wisdom on gnarly issues in physics”. (It sounds like “gnarly issues” means “philosophical issues” – so philosophers of physics should definitely go contribute!)
Many quantities in physics (like the speed of light) inherently carry with them certain “dimensions” (in this case, length divided by time). In general, physicists can solve a lot of problems quickly by noting that it only makes sense to add quantities that have the same dimensions, and that in many situations, a single dimensionless constant will be the only relevant factor. For instance, once you know that the only factors relevant to the length of time it takes for an object to fall to the ground are its initial altitude and the acceleration due to gravity, and one knows that the dimensions for these three quantities are time, length, and length divided by time squared, then one can quickly see that when the gravitational acceleration is fixed, the time it takes to fall is proportional to the square root of the initial altitude.
John Baez points out that this sort of reasoning seems to beg a question, in that we have no clear justification for why dimensionless quantities in our theories are always constant. To put it more perspicuously, there are some constants that appear in our theories (like the speed of light and Newton’s universal gravitational constant) that have dimensions associated with them, and the numerical value therefore depends on the choice of units (say, feet versus meters for length, and seconds versus minutes for time). However, other constants (like the “fine structure constant”, which is the square of the charge on the electron, divided by 4π times the permittivity of the vacuum, Planck’s constant, and the speed of light) are dimensionless, and their numerical values (in this case approximately 1/137) don’t depend on our choice of units. Why do physicists find the latter particularly compelling?
At any rate, I found that this stuff all made more conceptual sense after I had read Hartry Field’s Science Without Numbers a few times. In this book, Field argues that numbers don’t really exist – they (like all mathematical abstractions) are just useful fictions that let us extend our physical theories in ways that make calculations much easier. Just as Hilbert showed how to properly axiomatize geometry in terms of betweenness, distance-congruence, and identity of points and lines without talking about real numbers or coordinates or actual distances or the like, Field shows how to add a few more basic relations on spatial points and axiomatize Newtonian gravitational mechanics without any talk of real numbers or coordinates. When thought of like this, it seems to become more clear why the laws have to be phrased in dimensionless terms (or in terms where the dimensions on opposite sides of an equality sign are equal). All the relevant relations that occur in Field’s theories are something like distance-congruence, which is a relation between four points, that are taken to define two distances in pairs. There just is no relation in the language that considers a pair of points defining a distance and a pair of points defining a duration. All the quantities involved are actual physical things (no dimensionless numbers), and the operations only make sense when the quantities are of the same sort.
I don’t know how much this makes sense of what’s going on, but it seems to me to help clarify some things. (I don’t know if it’s evidence that Field’s view is the right way to see things – I’m tempted to think it is, but that may just be my fictionalist bias.) Another way to clarify these things is in the language of torsors, as a commentor on Baez’ post mentions. (I’ve only heard of torsors once before, on a page by John Baez explaining them. But even then, I was struck by how they seem to relate to the way Field thinks of physics and mathematics.)