Chapter 2
Postulates and the Basic Theorems

What is a Postulate?
The core of this book is dependent on a system of postulates and theorems, as in Euclidean geometry. It presents an example of the logic system inherent within God's Creation. As stated in the previous chapter, the words are not the reality. If I did not put something into words, nothing could be expressed. Some kind of symbol system has to be developed to get the subject across.

To explain how this chapter works, I need to define what is a postulate. The postulate in geometry is only an observation. Somebody makes an observation, and it cannot be proved or disproved. It just is. In fact, as mathematics grew up, a procedure developed for the establishment of postulates. Somebody comes up with an observation. After a couple hundred years as an axiom -- if nobody can find any exceptions to it -- the axiom becomes a postulate. If one exception is found, it ceases to be an axiom.

A postulate is readily verifiable, although it cannot be proved. Verification of a postulate is dependent on a set of conditions that apply to the postulate. The postulate of the geometry of parallel lines and intersecting lines was used as an earlier example. These conditions are: the parallel lines (the operator sets up the parallel lines) and the intersecting lines (the operator sets up the intersecting lines). However, if the conditions are not present, it is impossible to verify the postulate. Therefore, a person cannot verify the postulate if they do not have two parallel lines. You need to have the conditions first before the postulate is observed.

What's a theorem?

My handy-dandy dictionary states that a theorem is a proposition that is provable on the basis of explicit assumptions.¹ Theorems can be proved or disproved. This is because they are not an observation per se, but a way of thinking about an observation. For example, the Pythagorean theorem is one:² (a) squared, plus (b) squared equals (c) squared (a² + b² = c²). One proof was made by breaking the sides of a [3, 4, 5] or a [6, 8, 10] triangle into squares showing that this was true. Theorems are subject to proofs or disproofs; postulates are not.

Because the elements of the theorems are?

A theorem is composed of elements of logical thinking. A theorem has an end-point/conclusion. The theorem is an end product of some form of logical thinking. The elements of the logical thinking can either be postulates or previously proven theorems. Theorems are places and conclusions that an individual comes to within that logical order system. Through one form of the logic system, a theorem is derived. But, the logic system (like math), being a true logic system -- when folded back upon the theorem -- proves the theorem. There is a proof involved one way or another with a theorem, whereas postulates do not have a proof.

The ideal being: through a logic system, a conclusion is made. The logic system being holistic, a proof occurs when the individual comes to the same conclusion another way.

So whether you're inducing or deducing... It doesn't make any difference. The conclusion should be the same.

Right. Being lazy, I am going to leave the proofs of these theorems up to the reader.