Overview: Modeling with mathematics

Physics uses mathematics as a critical component of how it looks at the world, perhaps more so than any other science. As a result, it's a great place to learn about the use of math in science.

Math lets us create long chains of reasonably tight, logical arguments — longer chains than we can typically hold in our head at one time. And it often helps us learn things that we could not easily have worked out without it.

By now you've had lots of experience with math in lots of math classes. But in all those math classes they might not have told you this:

     Math in science is not the same as math in math.

In math classes, you manipulate symbols and numbers, but they don't represent anything physical. In science, we use the tools of math embedded in a real world context.  That turns out to be very different.

Representing a physical system with math

Interpreting the math in the context of a physical system adds two key steps that crucially distinguish "math in science" from "math in math":  Modeling the physical system with equations and interpreting the mathematical results. The diagram below shows some of the important components of using math in science that make it different from math in math.

We start with some physical system on the lower left. Here are the steps in creating a mathematical model of that system:

• Modeling — We identify some quantity in our physical system that we want to model and find a way to assign a measurement (or set of measurements) to it. We then assign that measurement to a symbol and treat the symbol as a kind of mathematical quantity.
• Processing — Whatever mathematical structure we've assigned, we can then use mathematical procedures: ways of building new quantities and ways of solving equations. These let us create long chains of arguments and see results that we'd otherwise have a lot of trouble figuring out.
• Interpreting — Once we've solved something using math, we have to get a physical meaning back out of it. This means we have to figure out not just the answer, but what the answer tells us about the physical system we are talking about.
• Evaluating — The last step is to decide whether, in fact, the result correctly represents the physical system. The fun (and useful) thing about using math is that it often leads us to results that we didn't expect and that surprise us. A lot of times, they are true! But sometimes, they are not. Then we have to figure out whether we made a mistake or if we left something important out of our model and have to change it.

In the process of using math in science, we'll use all these steps, sometimes jumping back and forth across the diagram, sometimes following it in order.

You may think that biological systems are too complex for these simple-sounding ideas to work. But we use math lots in our everyday lives and, even in the simplest cases, the issues in the diagram apply — and we know how to use them.

A simple example

Consider pieces of chalk for writing on a blackboard. If the physical system we are considering is ordering chalk for a school, a reasonable model is to model individual pieces of chalk by representing them mathematically by the system of integers. A single piece is represented by the number 1 in our model.

From the math of integers, we know how a lot of processes: addition, subtraction, multiplication, and division. We can then represent a box of a dozen pieces of chalk by the number 12, a carton of 50 boxes by the number  $50 \times 12 = 600$, etc. Interpreting these numbers as pieces of chalk works fine as long as we are just buying and distributing them.

But now consider a wider physical context. Once teachers start using the chalk, the pieces get shorter. If someone drops a piece on the floor and it breaks in half do they now have 2 pieces? Maybe. Twice as many people could use it, but not for as long as a whole piece. Or should we consider it to be 2 x 1/2 pieces? Should we allow not just integers but fractions and make a new model? But if I drop the chalk and it shatters into 1000 pieces, none of them can be used to write on the board anymore. We would just throw them out. So the process from using fractions, in which 1000 x (1/1000) = 1 doesn't work for chalk in the situation of use.

The math of integers works very well for keeping track of how much real world chalk I have — but it breaks down once the situation we are considering is expanded, and there is not a simple model with rational or real numbers that is particularly useful.

In the chalk example, as in any example of math in science, modeling the world with math gives you a lot of power — but only if you understand not just the math but are also able to connect the math to a physical system, when it works and how to use it. You must just "do the math" but connect it to the physical system you are describing.

Learning the process of mathematical modeling

Note that although in our diagram we have separated the analysis into 4 steps, when you actually use math in modeling you will eventually learn to integrate the steps  to think about the physics and the math as two sides of the same coin — and to blend together your physical and mathematical ideas.

Because we add physical modeling to math in science and blend mathematical and physical intuitions, a lot of things look different when you use math in science. Sometimes, students have trouble in a science class that uses a lot of math — even if they have studied the math successfully in a math class! Working through the following sections can help

• How math in science is different from math in math — a section with several follow-on pages that elaborate on how math is used in science
• Mathematics recap — a section with pages that review the math needed for this class with a spin that emphasizes how to think about it in useful ways for science
• Building your mathematical toolbelt — a section that identifies specific methods for blending math and physics that we'll come back to through the class.

Joe Redish and Wolfgang Losert 8/19/12