Faulty Proofs

Proofs can be a hard skill to master, but by examining faulty proofs (or faulty reasoning) we can make note of common mistakes to avoid.

Readings

Howlers (Mathematics)

Howlers (in the realm of mathematics) refer to a "proof" that results in the right answer, but for the wrong reason. As a simple example:

$\frac{16}{64} = \frac{1}{4}$ by cancelling the $6$ on top and bottom of the fraction.

The lesson here is that a correct statement can be proved in a faulty way. A proof can be "proving" something true, but may not have all the steps or logic needed to fully prove the concept.

Moser's Circle Problem

We took advantage of patterns in earlier classes, but it is important to prove these thoroughly. Patterns can be misleading!

Watch the video below for one example.

Think about it: Can you come up with another misleading sequence? Specifically, one that has meaning?

Visual Proofs

How to lie using visual proofs

Now that we've been completing/seeing proofs more regularly, re-watch this video from week 4.

Think about it: Do you understand what the speaker is getting at with proofs?

Spotting Faulty Proofs

These are all mistakes, but they can be hard to see! How can we spot them without being told they're faulty?

First, be thorough. Having a toddler around might be helpful. At every line of the proof, ask the question: why? If there is no good answer as to why this step can be made, the proof needs more work.

On a related note, we also need to be sure that each step being made is correct itself. It may have a good reason, but you may not be able to make the step with valid mathematics! Be aware of statements and theorems being used. They might have some requirements that aren't being met.

Think about it: What may be some other ways to spot faulty proofs?