Philosophy and History of Dimensional Analysis

My dissertation comprises a philosophical examination and exploitation of the fundamental principles of dimensional analysis and a partial history of its development. Dimensional analysis is a logical method largely, though not exclusively, used by physicists to derive functional equations and estimate quantity values. This method is based on the fact that our physical equations are quantitative and so specify equivalences between dimensional quantities. Roughly, we can consider quantity dimensions as kinds of quantities, with only quantities of like dimension being commensurable, e.g. mass, length, charge.

In “The Π-Theorem as a Guide to Symmetries and the Argument Against Absolutism” I use a fundamental result of dimensional analysis, the Π-theorem, to define a class of quantity transformations which are both empirical and dynamical symmetries. As such, they license an argument against absolutism, a commitment the fundamentality of intrinsic quantity magnitudes relative to quantity relations (ratios). Earlier symmetry arguments on the basis of quantity transformations like global mass doublings are party to counterexamples. These single quantity dimension transformations cannot be both empirical and dynamical symmetries. I show that dimensional analysis diagnoses this problem in a general way and provides an algorithm for generated true quantity symmetries. Generally such symmetries involve the transformation of multiple quantity dimensions, which is usually encapsulated in the transformation of some physical constant, in this case G. This raises a question regarding the nomological status of the values of the physical constants: Is it a law of nature that the gravitational constant has the magnitude it does? I do not settle the question, but I do set out some of the positions and show how they interact with the absolutism-comparativism debate.

Another paper from the dissertation “Metaphysics and Methodology in Dimensional Analysis 1914-1917” chronicles a debate in the methodological foundations of dimensional analysis with a particular focus on the positions of Richard Tolman and Percy Bridgman. The methodological debate yields a metaphysical one: Are quantity dimensions natural kinds (as I glossed above)? Or are they mere conventional devices meant to guide use in unit transformations? After analyzing the dialectic between quantity dimension realism and quantity dimension conventionalism I put forward a third position which seems to survive their mutual critique: quantity dimension functionalism. Under the functionalist position, there is a minimum number of basic quantity dimensions needed for an adequate model of kinds of physical phenomena: three for mechanical phenomena and one additional dimension each for thermal and electromagnetic phenomena. Empiricism tells against postulating unnecessary addition basic dimensions (e.g. treating forces and masses as basic). However, conventionalism still reigns when it comes to the identities of the basic quantity dimensions. Either force or mass may be treated as basic and the other as derived—This makes no difference to mechanics. A vector space representation is used to illustrate quantity dimension functionalism.

A paper recently published in Theoria, “The Nature of the Physical and the Meaning of Physicalism” leverages some of my work on dimensional analysis to provide a quantitative account of the physical that avoids “Hempel’s dilemma”. On this view, a physical object is any object which can be described by dimensional quantities. This account makes physicalism into an empirical claim that will or will not be borne out by experience.