The Role of Mathematics in Physics

[Chapter 1 Objectives]

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It took a long time in the history of humankind before it occurred to anyone that mathematics is useful - even vital - in the understanding of nature. Western thought was dominated from antiquity to the Renaissance, turn by turn, by Plato and Aristotle. Plato taught that reality consists of idealized "forms", and our world was a flawed, inadequate shadow of reality - hardly worthy of passing notice, let alone study. Aristotle thought that the intricacies of nature could never be described by the abstract simplicity of mathematics.1 Galileo recognized and used the power of mathematics in his study of nature, and with his discovery modern science was born.

In the text (section 1.2 Mathematics - The Language of Science, page 1), Professor Hewitt discusses some of the roles that mathematics plays in science, particularly physics - as well as why mathematics is not emphasized in this particular physics course.

A couple of points about the discussion in the book:

  1. While it is true that most scientists would agree with Prof. Hewitt's claim that "when the ideas of science are expressed mathematical terms, they are unambiguous" (page 1), some would object that a mathematical statement can't be more precise than the (verbal) concepts and definitions that it came from.
  2. You should understand that while the statement, "When the findings in nature are expressed mathematically, they are easier to verify or disprove by experiment" (also page 1) is certainly true, Prof. Hewitt is not saying that it is easy to "verify or disprove" something expressed mathematically by experiment - because it isn't (as you will see)! Most students' experience with experiment involves following a fill-in-the-blanks lab book, and then writing the conclusion "It didn't work because of experimental error," turning it in, and forgetting about it - or some such baloney. In reality, it requires a great deal of thought, skill, and perseverance to get even reasonable experimental results.

Mathematics as Abbreviation:

A role that mathematics plays in physics not mentioned in the text is that mathematics is a really great way to get a very concise statement that would take a lot of words in English. For example, Newton's Second Law can be stated as follows:

The magnitude of the acceleration of an object is directly proportional to the net force applied to the object, and inversely proportional to the object's mass. The direction of the acceleration is the same as the direction of the net force.

Exactly what all of this means is not important (at the moment) - what is important is that the statement above can be expressed mathematically as:

a equals Fnet over m

The point is that to a physicist, both statements say exactly the same thing. The symbolism of mathematics can replace a lot of words with just a few symbols.

Mathematics as Concept Map:

Many beginning physicists get the notion that equations in physics are just something to "plug the numbers into and get the answer" - which is one reason that numerical calculation is not emphasized in this physics course. Physicists think differently - equations tell them how concepts are linked together.

For instance, this equation arises in the study of kinematics:

average velocity equation

The symbol on the left side of the equation represents the concept "average velocity". Since there are two symbols (forgetting the division sign, and the delta x counts as one symbol) on the right side, to a physicist, the equation says (among other things) that the average velocity of an object depends on two (and only two) other concepts - the object's displacement (delta_x), and the time it has been moving (t). Thus equations tell scientists how concepts are related to one another.

Mathematics as Mechanized Thinking:

Once an idea is expressed in mathematical form, you can use the rules (axioms, theorems, etc.) of mathematics to change it into other statements. If the original statement is correct, and you follow the rules faithfully, your final statement will also be correct. This is what you do when you "solve" a mathematics problem.

From a scientific point of view, however, if you start with one statement about nature, and end up with another statement about nature, what you have been doing is thinking about nature. Mathematics mechanizes thinking. That's why you use it to solve problems! You could (possibly) figure it out without the help of mathematics, but mathematics makes it so much easier because all you have to do is follow the rules!

As a very simple example, suppose you start with the equation above, which is often considered to be the definition of average velocity (in mathematical form, of course):

It is a perfectly acceptable mathematical operation to multiply both sides of an equation by a variable, so multiply both sides of this equation by "t". You get:

average velocity equation 2

On the right side, the rules of algebra say that t/t = 1, so it must be true that:

average velocity equation 3

And the commutative property of algebra says that this is the same as:

average velocity equation 4

This is a new statement about nature (equivalent to the familiar "distance equals speed times time") - derived using the rules of mathematics. Using mathematics, physicists can discover new relationships among physical quantities - mathematics mechanizes thinking.

1 see Arnold B. Arons, Teaching Introductory Physics, 1997, p. 48
[Chapter 1 Objectives]

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last update August 27, 2009 by JL Stanbrough