What is Math?

The late Bill Thurston asked me and a classroom full of mathematicians (mainly graduate students) "What is Mathematics?". There were nearly as many different responses as there were people in the room. Regardless of your relationship with mathematics, I'm sure you as well can think of a few interesting answers to the question. Here I explore one possible answer.

There and Back Again

Below we follow the course of solving a basic math problem:

Choose a Problem

2 + 3

5 units

Take a problem and abstract away the real-world content, leaving only the underlying structure. This typically involves a mathematical model, and so is only as correct a step as the model is.
Solve the resulting math problem.
Put the real-world content back in using our original model.

Notice that steps 1. and 3. are potentially imprecise, but step 2. is precise. We'll see some examples below where converting to a math problem introduces a definite amount of imprecision. One way to visualize this is as pushing the imprecision to the beginning and end of the problem-solving process:

1. Convert to a mathematical model
(Introduces Error)

2. Solve the corresponding math problem
(Doesn't Introduce Error)

3. Interpret your solution using the model
(Introduces Error)

As such, my definition for mathematics, at least, the one I'm proposing here, is as follows:

Mathematics is the precise, error-free study of abstract structures, relationships, patterns, and quantities, such as those which appear in models of the real world.

Choose-your-own Mathventure

The pattern above of converting, solving, and converting back shows up within mathematics itself:

If you were tasked with solving the problem of adding the roman numerals II + III, you would probably first convert to arabic numerals (2+3), solve the problem there (5), and then convert back to get an answer of V.
Many geometry problems can be solved by converting them to algebra problems, solving, and then converting back.
My Ph.D. Thesis involved converting a problem from logic to a problem about finite automata, solving it there, and then converting back.
The successful proof of Fermat's Last Theorem (a problem of number theory) involved converting it to a problem involving complex curves and then solving that problem.

The interactive below will walk you through many ways of solving some real-world problems by converting them between different branches of mathematics. Can you think of another way of solving one of these problems?

Hint: Hover your mouse over the buttons to get more information on that mathematical step.

Choose a Problem

Various Gravity Problems

An object thrown upwards at 16 ft/s experiences downward acceleration of 32 ft/s².
When does it reach its maximum height?

Maximize
-16x²+16x

Solve (-16x²+16x)' = 0
and test

This box is a ghost and shouldn't be seen

The length plus width
of a rectangle is 1.
What is its maximum
possible area?

Maximize x-x²

Solve (x-x²)' = 0
and test

Find the vertex
of the vertical parabola
passing through
(0,0),(1,0),(2,-2)

Maximize
-(x-½)² + ¼

Maximize
-(x-½)²

Solve -32x+16=0

Minimize
(x-½)²

Solve
1-2x=0

Solve
x - ½ = 0

Answer

x=½