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# Copy of Modeling with Mathematics (2013)

last edited by 6 years, 7 months ago

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 is useful because it lets us create long chains of tight logical arguments -- longer chains than we can reasonably hold in our head at one time. And it often helps us learn things that we could not have easily thought out.

By now, you've studied much math in many math classes. But there's one thing that they might not have told you in all those math classes:

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

When using math in physics (and in other sciences) we use the same tools of math, but they're now embedded in the context of a real system.  Interpreting the math in this context 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. Here's a diagram that shows some of the elements about what goes on.

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

1. Modeling -- We identify some quantity in our physical system that we think is interesting 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. We'll think about how other quantities in the system might relate to our original quantity, and assign symbols as needed. From this, we might have an equation, a set of equations, or some other mathematical structure.
2. Processing -- Whatever mathematical structure we've assigned, from math we inherit procedures (i..e. taking derivatives), ways of building new quantities, ways of solving equations, making reasonable approximations, and incorporating known physical laws. These let us create long chains of arguments and see results that we'd otherwise have a lot of trouble figuring out.
3. 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.
4. 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. It's often the job of the experimentalist to determine whether the model was 'correct.' Did it capture what was going on in the real system? If not, start over! (On the other hand, even incorrect models often point experimentalists in interesting directions they might not have expected to take.)

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 a piece of chalk to an integer - how about the number 1? So a single piece is represented by 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 twelve pieces of chalk into the number 12, a carton of fifty boxes onto 50 x 12 = 600, etc. The model works.

But once we start using the chalk, the pieces get shorter. If I drop a piece on the floor and it breaks in 1/2 do I now have 2 pieces? Or should I use fractions now and make a new model? On the other hand, if I drop it 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 even incorporating fractions, the model wouldn't totally work for keeping track of how much chalk the school has. The math of integers works very well for keeping track of how much real world chalk I order, but then fails once we start using it. A better model is desired, if I care about modeling chalk inventory.

Learning to model 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.

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 the modeling nature of math, a lot of things look different when math is used in science. Here's a page that links to discussions of a lot of that sort of issue, many of which 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!

Here are a set of reviews of some of the basic issues you have learned in math, but possibly looked at from a new and useful angle.

Joe Redish 7/5/11

Wolfgang Losert 8/19/`12

Kerstin Nordstrom 9/3/13