What is a mathematical paradox? Any statement made mathematically that contradicts another mathematical statement, and therefore falsifies either itself or the other statement. When a mistake in reasoning or assumption that creates a paradoxical situation, the statement is then considered a fallacy. Consider the dialogue:

A: What B will say next will be false.
B: You have spoken truly.

If A is correct, B is false. If B is false, A is false. Because something cannot be both true and false at the same time, the dialogue is paradoxical. Some paradoxical sentences include:

Please ignore this notice.
This sentence is false.

Let's look at a fallacy:

Let f=full
Let e=empty
Half full and half empty are the same.
(1/2)f = (1/2)e
f = e
Full and empty are the same thing.

The fallacy in this proof lies in the fact that it uses mathematical methods on verbal equivalents of equations.