Technical Reasoning/Introduction to Number Shape

Number and Shape

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

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?

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" is the following proposition:

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, 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 that the parallel postulate can be used to prove the Pythagorean theorem.

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

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 it true" and get a new set of propositions which explain it. But then 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.

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 glance, but from the above consideration, it leads to an infinite regress of reasons.

An Example Argument

The first system of logic in recorded human history, was that of Euclid's book, Elements. In this book, Euclid began from a collection of definitions, and fundamental propositions which we call "axioms". The axioms themselves do not have proofs, but every proposition which is not an axiom has a proof which ultimately derives from the axioms.

But we cannot just move from one proposition to another at random or without reason. There must be principles about how we may infer one proposition from another, and these are called "inference rules".

For example, let's look at the following argument given by Euclid at the beginning of his book:

Theorem: From a Segment, an Equilateral Triangle

Let A and B be any two distinct points, and let ${\overline {AB}}$ denote the line segment between them. Then there is an equilateral triangle with one side equal to ${\overline {AB}}$ .

Proof

Form the circle centered at A running through the point B. Call this circle R.

Form the circle centered at B running through the point A. Call this circle S.

Form a point of intersection of the two circles, call it C.

Form the triangle $\Delta ABC$ .

Both ${\overline {AB}}$ and ${\overline {AC}}$ are radii of the same circle (R) and therefore have the same length.

Both ${\overline {BA}}$ and ${\overline {BC}}$ are radii of the same circle (S) and therefore have the same length.

Therefore all sides of $\Delta ABC$ have the same length, and so it is an equilateral triangle.

$\Box$

I have kept the language of the proof simple, somewhat at the cost of not making the proof as explicit as it could have been.

But what I want the reader to focus on, is the fact that there is some rule, which allows one to say that the circle centered at A and running through B exists.

There is another rule which allows us to infer that the circles have an intersection.

We will investigate these rules in greater detail later, but for now just be aware that there are there rules which allow certain inferences.

Arguments Generally

Notice that arguments are how we demonstrate the necessary truth of some conclusion, from a set of assumptions and accepted inferences rules. We will use the words "argument" and "proof" interchangeably.

In the above proof, we assumed the existence of two points A and B.

With the help of some other assumptions and inference rules, we were allowed to infer the existence of a circle centered at A and running through B.

After a long enough sequence of propositions like these, we eventually ended at the proposition "therefore $\Delta ABC$ is equilateral". This was the conclusion, and it ended the argument for the theorem at the beginning.

In the abstract, this is always the structure of a proof:

A collection of basic assumptions, called the "premises" of the argument.
A sequence of propositions, such that every next one is a premise or inferred from earlier propositions (using an allowed inference rule, of course).
The last proposition in the sequence is the conclusion. The "goal" of the argument is to arrive in this way at the conclusion.