Class Content I

2.

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.

One of the main things that math does is it 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, but embedded in a real world context.  That turns out to be very different. 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 in using math in science that makes 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 it to a symbol and treat it as a kind of mathematical quantity.
• Processing -- Whatever mathematical structure we've assigned, from math we inherit procedures, ways of building new quantities, 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 simply-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.

As a simple example, consider pieces of chalk for writing on a blackboard. If I am ordering chalk for a school, a reasonable model is to map pieces of chalk into an integer. A single piece maps onto the number 1 in our model. From the math of integers, we inherit addition, subtraction, multiplication, and division. We can then map a box of chalk into the number 12, a carton of 50 boxes onto 50x12 = 600, etc. But once we start using the chalk, the pieces get shorter. If I drop a piece on the floor and it breaks in half do I now have 2 pieces? Or 2 x 1/2 pieces? Should I use fractions now and make a new model? But if I drop the chalk and it shatters into 100 pieces, none of them can be used to write on the board anymore. I would just throw them out. So the model using fractions, in which 100 x (1/100) = 1 doesn't work for chalk. The math of integers works very well for keeping track of how much real world chalk I have -- but only up to a point.

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.

Note that although we have separated the analysis into 4 steps, when you actually use math in modeling you will eventually need to learn to integrate the steps -- to be thinking about the physics and the math as two sides of the same coin -- and be blending together your physical and mathematical ideas.

Because of adding modeling to math in science, blending mathematical and physical intuitions, a lot of things look different when math is used in science. The Follow-Ons link to discussions of issues that can give you trouble in a science class that uses a lot of math — even if you have studied the math successfully in a math class!

Follow-ons:

• Using math in science
• Math recap
• Building your mathematical toolbelt 

Joe Redish 7/5/11

Wolfgang Losert 8/19/12