Faulty Proofs

Faulty proofs can come from oversights, logical errors, or missing pieces. There are a few common mistakes, which I'll mention here, but this is not a complete list of every mistake one can make in writing a proof.

Missing Pieces

In order for a proof to work, every statement must have some proof that it holds true. This could be one of many things, but in particular, be careful when using previous lines of a proof to prove later statements.

In induction proofs, a common mistake may be to not create enough base cases. Too many creates a sloppy proof, but too few makes a proof incomplete.

The "horse of a different color" proof is a good example of how this mistake can prove something that is definitely false. We'll discuss this proof in class.

Incomplete Definition/Theorem/Property Considerations

Now, we'll discuss using outside theorems, definitions, and properties. Be sure that the mathematical statement you're using can be applied. The mathematical statement you're using might have some requirement, like the number being positive, or the value being an integer. If the part of your proof does not have this required property, you cannot use the proven property, unless you do something like splitting into cases.

It is unlikely that splitting into cases will help a definition apply. Splitting into too many cases can also make a proof become long and sloppy. Instead, try finding some other definition to use, or use a different proof technique.

Logical Fallacies / Mistakes

Inverse error and converse error were two logical errors we discussed. These errors can occur in proof logic as well.

As a whole, other logical fallacies can make a proof incorrect, but these are less common than inverse and converse errors, since they require some psychological manipulation. However, they can still appear. Be sure that you are sticking to a known proof style, and using mathematically proven techniques and ideas.

There are too many logical fallacies to simply list here, so heed this as a general warning. Don't make assumptions that are not mathematically proven or stated directly in the problem!

Check your Understanding: View the flawed proofs here and try to spot the mistakes before looking at the answers. Were you right?

Check your answer

The answers are in the link!

Visual Proofs

The following video shows that even visual examples are not safe. Watch the video below, and see if you can find the problems with the proofs before the video explains them. This is tough, so don't be upset if you can't find the mistakes!

If you don't have a good knowledge of geometry, this video may be a bit hard to understand. I promise the ideas behind these false proofs translate, so feel free to replay the video after looking up some terms (circumference, isosceles, congruent) if you don't understand the main message. You will likely be completely unable to find the mistake in the last false proof without a good background in the geometry of triangles.

It is okay if you don't fully understand exactly the mistakes that happen in this video. Having a general idea of what went wrong should be enough to warn you that pictures are not proofs.