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.)

David Corfield(02:40:00) :Your associating Field’s work with torsors raises a couple of issues in my mind. I never had the sense reading Field that he had the slightest interest in reformulating physics without mathematics to produce the kind of rewriting of physics which mathematical physicists such as John Baez see as so important for the next step. If by chance, Field had hit upon a formulation equivalent to the use of torsors I would be amazed.

The philosophical question which seems to relate more closely to the search for a well reformulated mathematical physics, perhaps with torsors or pieces of category theoretic apparatus, is the one I’m always asking. Not ‘Do mathematical entities exist?’, but ‘Can we make any sense of the idea that some mathematical concepts seem essential, are ubiquitous, can’t help but be discovered, while others seem to be gerrymandered, are at best temporarily slightly useful, aren’t continually encountered in different fields?’

Greg(08:55:37) :Thanks for the pointer to torsors — for once, a enlightening bit of mathematics that I actually (kinda) understood.

But it sounded from the Baez piece that, using torsors, there

isa “relation in the language that considers a pair of points defining a distance and a pair of points defining a duration.” (I’m looking at the bit at the bottom of the page where Baez talks about subtracting positions.) But like I said, I only kinda understood this torsor business.David Corfield(04:22:17) :In the same direction as Greg’s comment, were we to find Field’s formulation of physics to be structurally similar to a torsor formulation, wouldn’t those who took the Quinean commitment seriously say that we are committed to torsors? For myself, Field’s exercise struck me largely as the attempt to smuggle the structure of mathematical physics into logic (by no means structureless itself) and an already structured physical universe.

Kenny(17:04:51) :David – I think we would say we were committed to torsors, but they would just be the torsors of space, time, mass, and so on, which just are these things themselves. So nothing abstract or strictly mathematical, in some sense.

Yes, it’s all about getting the structure into the physical universe itself, but that seems like a nice idea for a variety of reasons. (Like not having to explain how abstract things have anything to do with physical space.)

David Corfield(00:56:19) :Kenny wrote:

“we were committed to torsors, but they would just be the torsors of space, time, mass, and so on, which just are these things themselves.”

Let’s see. A torsor is defined thus:

“For any group G we can define a concept of “G-torsor”. For starters, a G-torsor is a set X equipped with an “action” of G…”

and there are then a bunch of conditions.

So a torsor seems to be a thoroughly mathematical kind of entity. John Baez speaks on the same page of energies lying in an R-torsor, to convey that there’s no zero energy point. Would you call this a ‘torsor of energy’? Presumably, there’s nothing special about torsors for you, except if they lend themselves to Fieldian rewriting. Otherwise the situation would be little different to avoiding reference to abstract natural numbers by saying that the quantity of ducks in a flock lies in the natural numbers, and these natural numbers are natural numbers of physical objects, which “just are these things themselves.”

There’s then the additional worry that you’re committed to the whole of the energy torsor, i.e., for all R.

Dan(03:00:26) :Science without numbers? What about the periodic table? Is the question of how many protons a carbon molecule has not a question of science?

Anyway, when we say something, for example Unicorns, are fictional, we mean that we are denying the statement “Unicorns exist in our spacetime continuum”. However, numbers are mathematical objects and have no spatiotemporal properties. Thus the statement “Numbers exist in our spacetime continuum” makes no sense and cannot be denied (or affirmed). So what is the statement that Field is denying? The statement “Numbers exist” makes no sense because it lacks a set or domain of discourse. The statement “Numbers exist in the set of all numbers” is true, so it would be silly to deny that. One is left wondering, what mathematical fictionalism even means. I’ll go further, and assert that it is entirely nonsensical.

Kenny(09:46:05) :Field wants to phrase the question of how many protons a carbon molecule has in a non-numerical way. Instead of saying that six is the number of them, he would say (in the completely formal system) there is a proton, and another proton, and another proton, and another proton, and another proton, and another proton, and no other protons.

And again, I’m not quite sure why mathematical claims need to have a domain of discourse specified, while others don’t. Maybe that’s right, but it would need an argument.

Dan(03:15:10) :Well that’s what we do already in Peano Arithmetic. We say ssssss0. However, to comprehend these terms, ss0, sssss0 etc we need to be able to count. If you can comprehend the difference between sssss0 and ssssss0, you are already counting.