Class Content I  > Modeling with mathematics > Using math in science > Dimensions and units

2.1.3.4

One reason we use mathematics in science is 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 will 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.

Our dimensional analysis is about protecting 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 the equations 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 the structure of reality.

Follow ons:

• Scaling and functional dependence

Joe Redish 7/7/11