PHYSICAL DIMENSIONS ARE REAL

If someone tells you the temperature outside is 100, they have not said anything useful yet. Do they mean 100 degrees Celsius (boiling!), or Fahrenheit (a hot summer’s day), or Kelvin (extremely cold)? Quantities like temperature, mass, blood pressure, or GDP are expressed by a number and a unit. We call such quantities ‘dimensionful’. These are different from dimensionless quantities, such as a direct count: if I say that the number of fingers on my right hand is five, I don’t need to add a unit. Contrary to widespread conventionalism, I believe that the dimensionality of quantities has a basis in reality.

In the physical sciences, quantities (such as temperature) are assigned a dimension, which determines the class of units in which it may be measured. For example, length has ‘dimensions of length’, denoted L. This means that length is measured in metres, or centimetres, or inches, et cetera. Likewise, time has ‘dimensions of time’, denoted T: time is measured in seconds, or hours, et cetera. Some quantities have complex dimensions; speed, for example, has dimensions L/T. This means that speed is measured in compound units such as m/s or km/h.

A basic feature of equations in physics is that the terms on either side of the ‘=’ sign must have the same dimensions. This condition ensures that we compare ‘like with like’. It makes sense to compare one distance to another, but not to compare a distance to, say, a temperature. This ‘principle of dimensional homogeneity’ is as important to physics as more famous symmetry principles.

As dimensions play such a foundational role in physics, it seems that they must track some real feature of quantities (the realist view). For example, it seems that speed having dimensions of L/T reveals something about the way speed relates to distance and duration. The prominent view of dimensions, however, is anti-realist: it is merely a convention to assign quantities particular dimensions. The physicist Henry Langhaar ([1951], p. 51) describes this consensus well:

The general conclusion that emerges from the discussions is that the concept of dimensions is of little importance to philosophy. On the other hand, dimensions serve a mathematical purpose. They are a code for telling us how the numerical value of a quantity changes when the basic units of measurement are subjected to prescribed changes. This is the only characteristic of dimensions to which we need to ascribe significance in the development of dimensional analysis.

The Bureau International des Poids et Mesures states it more pithily in the ninth edition of the SI brochure ([2019], p. 24; emphasis mine): ‘Physical quantities can be organized in a system of dimensions, where the system used is decided by convention’. In my BJPS article, I argue against such conventionalism: some dimensional systems are closer to the truth than others. The SI brochure has it wrong.

The main case for conventionalism about dimensions is that it is in fact possible to construct alternative dimensional systems. This is a typical example of underdetermination of theory by data. We could, for example, assign dimensions of time (T) to length. This amounts to a commitment to measure distances in temporal units. We could stipulate that a length of one second is equal to the distance covered by a photon in one second. There is no empirical difference between this dimensional system and the standard one: the same equations of physics continue to hold. But if there is no empirical difference, the conventionalist claims, we are merely faced with a choice as to which dimensional system is most simple or useful—not which one is ‘most real’.

There are three ways to change a dimensional system. First, it is possible to reduce the number of dimensions by equating the dimensions of two or more quantities, as in the earlier example of length and time. Conversely, it is possible to increase the number of dimensions by assigning a base dimension to a quantity that previously had complex dimensions. We could assign speed a new primitive dimension, S, and then introduce a new ‘conversion constant’ that determines how S relates to L and T. Finally, it is possible to leave the number of dimensions the same, but change which quantities have ‘base’ dimensions. I noted that length and time have base dimensions, L and T, and that speed has complex dimensions, L/T. But we could equally stipulate that length and speed have base dimensions, S and L, and that time has complex dimensions, L/S. In this case, the total number of dimensions stays the same but the base dimensions vary.

Against the conventionalist consensus, I defend realism about dimensions. Since I cannot appeal to any empirical evidence, I argue that dimensional realism is more explanatory than conventionalism. This is similar to how scientific realism employs an inference to the best explanation to show that our best theories are approximately true.

That dimensional realism is more explanatory can be shown with the technique of dimensional analysis. Briefly, this is a technique that determines the dependence of one quantity on other quantities purely on the basis of their dimensions. Suppose we know that the period of a pendulum, P, depends on the gravitational acceleration, g, and the length of the pendulum, l. But we do not yet know whether the period is proportional to l, or to the square of l, or something else. Period P has dimensions of T, l has dimensions of L, and g has complex dimensions of L/T². Recall that the principle of homogeneity states that the terms on either side of an equation must have the same dimensions. If we want to find an equation of the form P = f(g, l), then since P has dimensions of T, the function f(g, l) must have dimensions of T as well. It turns out that the only quantity that can be constructed from g and l that has dimensions of T is √(l/g), so we correctly deduce that P = α √(l/g), where α is a dimensionless constant of proportionality. Dimensional analysis is incredibly powerful: We have derived the formula for the period of a pendulum without any equations of motion! And, as Lange ([2009]) effectively argues, dimensional analysis is not only powerful but also genuinely explanatory.

The explanatory power of dimensional analysis depends on the system of dimensions. This is formalized in Buckingham’s pi-theorem, which states that the number of ‘dimensionless products’, m, that we can form is equal to the number of quantities, N, minus the number of base dimensions, n, that is, m = N – n. The smaller m is, the more determinate the results of dimensional analysis are (for a philosophical analysis, see Jalloh [forthcoming]). Therefore, if we increase n (the number of dimensions) without a simultaneous increase in N (the number of quantities), we thereby increase the effectiveness of dimensional analysis. For example, in the case of the pendulum there are three quantities (P, l, and g) and two base dimensions (L and T), so N = 3 and n = 2. Thus m = 1. This means that there is a unique product of l and g with the same dimensions as P. We have seen that this is √(l/g).

But now consider a system of dimensions in which length has dimension T. In this system, g has dimension T^-1. There are still three quantities (N = 3), but there is only one dimension (n = 1), so m = 2. This means that there is no unique product of l and g with the same dimensions as P. Therefore, the principle of dimensional homogeneity does not yield the result that P = α √(l/g) anymore! From this example it may seem as if it is always better to increase the number of dimensions, n, in order to decrease the number of dimensionless products, m. But this only works up to a point: after this point, we have to introduce additional conversion constants to our equations, so that the number of quantities, N, increases as well. There is thus a ‘Goldilocks’ system of dimensions with just enough dimensions, but not too many constants. The situation is illustrated well by Dingle ([1942], p. 338):

The net result of these considerations, then, is that if we wish to give the greatest scope of usefulness to the principle of dimensional homogeneity, we must choose sufficient fundamental magnitudes to prevent two or more magnitudes from having the same dimensions, but not so many that indeterminable ‘constants of nature’ are forced into our equations.

Dingle was an operationalist, so he did not attach any metaphysical significance to this result. But I argue that these ‘Goldilocks’ dimensions are more explanatory because they are closer to the truth than any other system. Put differently, if the choice of a dimensional system were merely conventional, then the success of only some of them is a miracle. Dimensional analysis is most explanatory for these dimensional systems because it is these dimensional systems that best describe reality. What that reality is—what dimensions metaphysically are—is a question that deserves further exploration (for one suggestions, see Skow [2017]).

There is, however, a complication. The optimal system is not unique. I have shown that the number of base dimensions is fixed by explanatory considerations; but the choice of base dimensions is not. Dimensional analysis is equally explanatory when a different set of dimensions is chosen as basic. There are two possible responses. The first is that there is a true dimensional system, but it is unknowable to us. All we can know are the dimensional relations between quantities. This is akin to epistemic structural realism, which likewise says that we can know the structure but not the nature of the physical world. The other option is that the structure of a dimensional system is all there is. This is similar to ontic structural realism. The latter position has the virtue of knowability, but the problem is how to make sense of dimensional relations without any dimensions. Which of these options is preferable is the question that realism about dimensions has to answer.