The Wisdom of Proofs by Contradiction

Do you find yourself grumbling and complaining about how complicated and complex math proofs are? Do you tell yourself there has to be an easier way? Well, reader, there is a solution. Instead of doing needlessly complicated and messy proofs, often involving repetitive casework, you can simply prove that things must exist by proving that they cannot not exist. This is called a proof by contradiction, and I am here, reader, to prove they are best, even better than normal, messy proofs.

First of all, proofs by contradiction are elegant, and often avoid a load of casework. For a simple example, look at the proof that the square root of 2 is an irrational number (a number that cannot be expressed as a common fraction). Without a proof by contradiction, this proof would require calculating every possible combination of numerator and denominator of the fraction that the square root of 2 could be – which is an infinite task. Instead, a far more elegant approach is to first assume that the square root of 2 is a rational number, then go back and prove that if it is rational, there is no way it could exist. Following this line of logic, there is no way the square root of 2 could be rational, which means that the square root of 2 must be irrational.

Second, the logic of proof by contradiction is simply stunning. For starters, let’s explore how this type of proof was created. In summary, a great mathematician got tired of all the casework they were doing, and they decided to think of the problem in a different way. To do this, they thought of the traits of something that’s true; one of the main ones was that something that’s true cannot be false. If a mathematician can prove that it cannot be false, then they will prove that it must be true. Interested in diving deeper? Click here.

This line of reasoning reflects our own lives. Instead of thinking in one direction, of one thing, reader, we should think about the traits of the thing; try to categorize it into its main aspects, and reflect back.

In conclusion, reader, proofs by contradiction are not only an elegant way of simplifying harder proofs, but they are also another way of thinking about life. If you take nothing else away from this article, reader, it is to think about things in a different direction. The best way to do this in math is by proof by contradiction.