Technical Reasoning/Examples and Motivations

Mathematical Puzzles

Let us start from a collection of puzzles, which will motivate the need for a system of logic.

Divisibility

Every even number seems divisible by two. Is that always true? What about zero? Is zero an even number, anyway? And is it divisible by two, anyway? Negatives?

0.99...

Here is a classic topic that has spun pages, even tomes of internet writing. Some people even get very emotional about the question: Is 0.999... (with infinitely many 9s) equal to 1?

Imaginary Numbers

You may have heard of the imaginary number, $i={\sqrt {-1}}$. But you may have also heard that it is not possible to take the square-root of a negative number, and that i is actually "imaginary". How does any of this make sense? Are we always free to make up numbers?

The Parallel Postulate

The "parallel postulate" is the following proposition:

Animation of the objects referred to in Euclid's parallel postulate.

Consider any two distinct lines l and m. Let A be a point on l but not m. Let B be a point on m but not l. Call s the segment from A to B. If the two angles formed on one side of s sum to less than 180°, then l and m must meet on that side.

If one assumes the parallel postulate, it is possible to then prove that the interior angles of a triangle sum to 180°.

On the other hand, the exact reverse is also true! If one assumes that the sum of the interior angles of a triangle is 180° then it is possible to prove the parallel postulate.

Even more striking is the fact that the Pythagorean theorem can be used to prove the parallel postulate; and the parallel postulate can be used to prove the Pythagorean theorem.

So in fact we have three propositions, and all three of them imply the other two!

One then wonders: which one of these principles should we take to be foundational, and which of them should be regarded as a consequence?

Well-foundedness

Take any proposition, like "the sum of interior angle measures of a triangle equals 180°", or anything else that you like.

We can ask "why is that proposition true" and perhaps get an explanation in response.

But then the explanation is itself a collection of propositions. For each of those propositions, we can again ask "why is that true?"

And of course the reader knows what comes next: We get more propositions, and more "why?" questions, and so on, in an infinite cycle.

This leaves us with a dilemma: Do we require that every statement has a reason, or must we accept some statements on the basis of no reasons? The former perhaps sounds better at first. But from the above consideration, it leads to an infinite regress of reasons, which is maybe even worse than having propositions with no explanation.

Motivation

In each example above, we see some suggestion that an analysis of logic could be helpful. Either by organizing our existing body of knowledge, or by making uniform what we mean by certain terms, or by helping us to carefully inspect each step of reasoning in an argument -- logic will allow us to shed light on the puzzles above.