How math in science is different from math in math
Most science students know that the sciences use a lot of math. After all, all science disciplines require that students take a number of math courses. Research papers in professional journals often contain lots of what is clearly math, including in recent years many biomedical research journals. But often, even after students have successfully passed the required math courses, when they see what's supposed to be the "same" math in their science courses, they find it surprisingly difficult and obscure. Why?
The key reason is that when we do math in science, we are using the math to code information about physical, chemical, or biological variables and how they behave. We are modeling the physical, chemical, or biological system with math. This means:
Our fundamental concern in using math in science is NOT math as a calculational tool, but math as a way to represent information about the physical world.
Here are some specific reasons that math in science looks different from math in math.
- Math in math classes tends to be about numbers. Math in science is not.*
Math in science is about relations among physical quantities that are transformed into numbers by measurement. As a result, quantities in science tend to have dimensionalities and units. These have to be treated differently from ordinary numbers.
- Math in math classes tends to use a small number of symbols in constrained ways.
Math in science tends to use lots of symbols in different ways.
In a typical calculus book, you will find very few equations with more than one or two symbols. Symbols tend to be used following a very predictable convention — $x$, $y$, $z$, and $t$ will be variables; $f$, $g$, and $h$ will be functions; $a$, $b$, and $c$ will be constants. In a typical physics book, you will rarely find an equation with fewer than 3 symbols and you will often find ones with 6 or more. And they won't follow the math conventions.
- The symbols in science classes often carry meaning that changes the way we interpret the quantity.
In science, the symbol we choose for a variable or constant is used to give us a hint as to what kind of quantity we are talking about. As a convention, physical scientists use m for a mass and t for a time -- never the other way round. Even though in pure math it doesn't matter what we call something, in science it does. Here's the reason in point 4.
- In science, different kinds of quantities can't be added or equated.
In pure math, you can typically equate any quantity to any other or to any number. In science, we can't equate a distance to a time or a vector to a scalar. We have to be careful not to write something that makes mathematical sense but is physical nonsense. So it is absolutely necessary to label quantities carefully to remind ourselves what kind of thing they refer to.
- In pure math, symbols tend to stand for either variables or constants. In science, we have lots of different kinds of symbols and they may shift from constant to variable, depending on what we want to do.
A lot of the symbols that we use in science shift meaning. Sometimes we might want to think about them as constants, other times as variables. And we have lots of different kinds of constants — from fixed universal constants like the gas constant, R, to parameters like a mass, m, which could be fixed for a particular set of experiments, but which we could imagine varying to get an optimal result. The same symbol will sometimes be used for an independent variable or a dependent variable, depending on the situation.
- In pure math — at least in introductory classes — we typically use equations to solve for or calculate something. In the natural sciences, we use equations to model a physical system.
Figuring out what equation to use is a key part of the challenge of "math in science". Once we have an equation, we can use it to calculate something, but we can also use it for qualitative and semi-quantitative reasoning. In particular, from the structure of the equations, we can sometimes learn new conceptual ideas about the physical system being described.
- In science, we often have "mathematical homonyms" — where the same symbol has a different meaning when the context changes. As we move through a physics course, for example, we will find that k can be a spring constant (units = force/length), Boltzmann's constant (units = energy/temperature), or a wave number (units = 1/length). The symbol T can stand for a tension (a scalar with units of force), a tension force (a vector with units of force), a period of time (units = time), or a temperature (units = Kelvin). We shouldn't use the same symbol to have different meanings in the same problem, though sometimes it happens. It's like getting used to hearing "there", "their", and "they're" correctly by interpreting the sound in a meaningful context.
All these differences can seem daunting! But if you learn to focus on the physical meaning hiding inside the math, you'll find that the math actually becomes very useful in thinking about what's happening, not just in doing calculations.
* It only sometimes feels like math classes are all about numbers. Especially at more advanced levels, math is about many possible kinds of logical structures, their relationship and implications, both qualitative and quantitative. This more sophisticated perspective sometimes creeps into less advanced math classes; but, unfortunately, not often enough.
Joe Redish and Wolfgang Losert 8/29/12
Last Modified: March 27, 2019