Charmed by Dimensional Analysis

I was charmed when as a young student I watched one of my physics professors, the late Harold Daw, work a problem with dimensional analysis. The result appeared as if by magic without the effort of constructing a model, solving a differential equation, or applying boundary conditions. But the inspiration of the moment did not, until many years later, bear fruit. In the meantime my acquaintance with this important tool remained partial and superficial. Dimensional analysis seemed to promise more than it could deliver. (Lemons 2017, ix, emphasis added)

Dimensional analysis has charmed and disappointed others as well…. The problem for teachers and students is that … [t]he mathematics required for its application is quite elementary — of the kind one learns in a good high school course — and its foundational principle is essentially a more precise version of the rule against “adding apples and oranges.” Yet the successful application of dimensional analysis requires physical intuition — an intuition that develops only slowly with the experience of modeling and manipulating physical variables. (Lemons 2017, Preface ix, emphasis added)

A Mistake to Avoid

A model of a state or process incorporates certain idealizations and simplifications. Skill and judgement are required to decide which quantities are needed to describe the state or process and what idealizations and simplifications should be incorporated. Similar skill and judgement are required in dimensional analysis, for the analysis in dimensional analysis is the analysis of a model. And the model we adopt in a dimensional analysis is determined by the dimensional analysis variables and constants we adopt and the dimensions in terms of which they are expressed. (….) While a certain part of dimensional analysis reduces to the algorithmic, no algorithm helps us answer [certain physical questions]. Rather, our answers define the state or process we describe and the model we adopt. We will, on occasion, make mistakes. (Lemons 2017, 11, emphasis original)

Dimensional analysis makes it possible to analyze in a systematic way dimensional relationships between physical quantities defining a model (Higham 2015, 90-91, emphasis added). Dimensional analysis is a clever strategy for extracting knowledge from a remarkably simple idea, nicely stated by Richardson[,] “… that phenomena go their way independently of the units whereby we measure them.” Within its limits, it works excellently, and makes possible astonishing economies in effort. The limits are soon reached, and beyond them it cannot help. In that it is like a specialized tool in carpentry or cooking or agriculture, like the water-driven husking mill … which husks rice elegantly and admirably but cannot do anything else. (Palmer 2015, v, emphasis added)

Physical (material) things have quantitative relationships that are measurable. A dimensional model uses a number of dimensional variables (physical variables) and constants that describe the model. Dimensional analysis is not a straightforward task for it requires skill and judgment — the same kind of skill and judgment needed to construct a model of a physical state or process. Add the complexity of open social systems and this requires even more skill and judgment.

So the legitimate questions arise when confronted with human social systems—such as economics is by its very nature—to what extent can mathematical models capture the true underlying causes of changes in economic behaviour?

When we become charmed by our mathematical tools, and fail to recognize there limitations, there range of validity, we become slaves to our tools rather than masters of them.

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