Considering changing our arbitrary choices

Prerequisites

One of the reasons we use mathematics in science is to make clear communication possible. We want to be able to express physical principles, relationships, and results in ways that are less ambiguous than our everyday speech. Mathematical equations (and standard graphs and schematic pictures) help us do this. If you look at a physics text in a foreign language where you know the physics but do not know the language (even perhaps the characters or alphabet), you may well find that you can make out most of what is going on.

But sometimes we take too much for granted in the universality of our use of mathematics. Many science papers become unreadable to a large fraction of their intended audience because the author incorrectly assumes "everyone uses the same symbols that I do" and fails to define terms clearly.

Units are a particularly dangerous example. Occasionally, scientists or engineers fail to specify the unit system they are working with because they incorrectly assume "of course we're all using the same system." Tragically, NASA lost a $125 million Mars probe because two groups of engineers were using different units and each assumed that everyone was doing the same thing they were so they didn't need to specify.

Dimensional analysis protects against changing our descriptions. We make arbitrary choices to describe something. How does the number we assign change when we change that arbitrary choice? We want the fundamental principle to hold:

Any equation that is supposed to represent an equality in the real world should stay equal even if we change one of our arbitrary choices. The equation should be about the world, not about how we choose to describe it.

Dimensions are about the fact that we can make an arbitrary choice of unit when we define a kind of measurement operationally. Since this choice is arbitrary, we don't want an equation we write that is supposed to represent an equality between physical objects to depend on how we choose to describe the world. ("A difference that makes no difference should make no difference!") This is true for purposes of communication, but it also gives us insight into our description of the structure of reality.

The implications of this is that we need to pay attention to how our quantities change when we change our arbitrary examples. Two critical choices are the dimensions (depend on our arbitrary choice of scale) and vectors (depend on the directions of our coordinate systems). The take-away message is:

Any equation that is supposed to represent an equality in the real world must only add, subtract, or equate expressions that have the same dimensions

and

Any equation that is supposed to represent an equality in the real world can only add, subtract, or equate expressions that have the same vector character (scalars with scalars or vectors with vectors, not mixed).

Joe Redish 7/7/11

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Article 256
Last Modified: August 6, 2018