Axiomatic systems

The intuitive idea of an axiomatic system is very simple. Suppose we want to explain a given assertion A. In so doing, let us say we have to make use of another assertion B. But then why is B true? So we explain B in terms of another assertion C. This means that if we accept the truth of C, we can explain A because C explains B and B explains A. But then the same question returns: why is C true? To answer that, we need to invoke another assertion D, so that now D explains A, and so on. One might ask again why D is true, whereupon the logical regression goes another step. It is clear that there is no end to this regression and a sensible way to handle this is to simply accept D, for example, on faith and say that D is our starting point. In that case, we say D is an axiom.

If it happens that, for all the facts we care to explain, there is a collection of n axioms A1, A2, . . . , An that suffice to serve as our starting point for the explanation of all the things under consideration, then these A1, A2, . . . , An together with all the things they can explain are then said to form an axiomatic system. Of course the formal definition of an axiomatic system is a lot more precise, but precision is not our main concern here in an informal discussion.

Wu, A&E, pp334-335

Euclid did this for Geometry with 5 axioms.

Euclid’s Axioms of Geometry

1. A straight line may be drawn between any two points.
2. Any terminated straight line may be extended indefinitely.
3. A circle may be drawn with any given point as center and any given radius.
4. All right angles are equal.
5. If two straight lines in a plane are met by another line, and if the sum of the internal angles on one side is less than two right angles, then the straight lines will meet if extended sufficiently on the side on which the sum of the angles is less than two right angles.

Discuss: Independence, in particular of the 5th Axiom (“The Parallel Postulate”).

BUT it turns out these five axioms were not quite enough to prove everything. Wu observes “Euclid felt no inhibition about supplementing his axioms with his intuitive beliefs in providing proofs.

Example: In his first geometric theorem, he uses a fact that is “intuitively obvious” but cannot be proven without some deep knowledge of the real numbers.

We can point to another flaw in Euclid’s system, one that is of an entirely different nature. In stating his five axioms, Euclid used the terms “point”, “line”, “plane”, etc., and he added definitions of these terms. For example:

A point is that which has no part.
A line is breadthless length.
A straight line is a line which lies evenly with the points on itself.

Wu, A&E, pp335

Problem: Euclid never provides definitions of “that which has no part”, or “breadthless length”, or “lies evenly with the points on itself.” If he did give such a definition, how would we understand the words and phrases used in the definition? Where does this process stop?

2400 years later, in 1899, Hilbert sorted out this problem (and many others) in his Grundlagen der geometrie (Foundations of Geometry).

This fact notwithstanding, the contribution of the Grundlagen that is directly relevant to the learning of plane geometry lies in its recognition that the terms used in a set of axioms (such as “point”, “line”, etc.) must remain undefined and that their meaning can only be inferred from the axioms themselves. In other words, all we know about points, lines, etc., is completely contained in the axioms, no more and no less.

Wu, A&E, pp336

Axioms and Theorems of Geometry

Wu uses a compromise solution. He starts with eight axioms that provide a reasonable intuitiveness as well as the necessary explanatory power to prove the important facts about geometry. The following pages contain Wu’s 8 axioms of Geometry (L1 – L8), as well a big list of the major theorems of Geometry that are typically addresses in the secondary school curriculum.