Axioms

I think the following quote is my own, but maybe I heard it, forgot it, and fooled myself into thinking I created it. It goes like this, “The easiest person to fool is yourself “. Ironic.

The clashes between cultures and religions stir up a lot of dust in the world. Bombs and armies will do that. Through the dust of war, I realized that my understanding and visibility into the problem was limited as to why some cultures and behaviors differed so much from my own. For example, why do some fundamentalist cultures do things that are non-sensical to me? The easiest explanation is that they’re crazy and I shouldn’t bother to make sense of it. For a while I did exactly that, but a gnawing feeling persisted that I hadn’t looked deeply enough. What follows are my thoughts pertaining to why we have so much strife in the world and how we might begin the mountainous task of reducing it.

First, a few boring details; an axiom is a starting assumption, postulate, or belief taken to be “true” — a “self-evident” truth. Starting from axioms, organized systems of thought can be developed using logic, additional beliefs, and assumptions. Systems of thought are the basis of all social organizations such as political, religious, and scientific. Systems of thought are usually internally consistent — in other words, they often “make sense” to those immersed in them. But when viewed from outside of that system of thought, the behavior and thinking might not be sensible to the uninitiated observer. A slight change to any one, or more, of the foundational axioms can result in a radically new, but yet internally consistent, system of thought.

Second, mathematics is a field of systematic inquiry based on axioms. One starts from a set of axioms and derives a system of relations using logic. Euclidean geometry starts with five axioms and from these are derived the rich set of planar geometric rules we learned about in high school. Remember the shortest distance between two points is a straight line?

I want to make a point through allegory that illustrates how easy it is to find systems of thought that even though are based on nearly identical axioms, and where both systems are governed by pure logic reasoning, one can arrive at a system of thought so foreign to the other that one can imagine hostilities breaking out between these systems based on nearly identical axioms.

A little history on the matter might provide some context. Euclid was this Greek scholar who played around with math, because he didn’t have anything better to do, apparently. He developed a system of geometry that is still in wide-spread use today. I know this because in the 9th grade, I got a D in it. Euclid was not my hero and because of his system of geometry and my lack of learning it, I almost lost my snowmobile racing privileges granted to me by my parents. But I digress. Euclid based his system of geometry on five axioms and it is Euclid’s fifth axiom has been the subject of study amongst mathematicians. The fifth axiom is often called the parallel postulate because it deals with the idea, overly simplified here, with lines and a point. Essentially, it results in two parallel lines never intersecting. These lines can go out to infinity and never cross each other – I tried to convince my parents that this was the root cause of my D; I had tried to prove the 5th postulate by traveling along these parallel lines to infinity and that takes more than a day or two. They were not amused and wondered out loud if one could be grounded for infinity – good move, I did not see that coming.

Riemann, among others, believed that one could alter the fifth postulate and in changing this one axiom, a new set of geometric rules were derived. Riemannian geometry can be more easily visualized as the surface of a sphere where “lines” are those comprising great circles. The great circle flight paths of modern aircraft provide proof that the shortest distance between two points is not a straight line — if the geometry is curved. These non-Euclidean geometries helped Einstein formulate new theories about the universe and didn’t help me impress the “ladies” in the 9th grade. I didn’t see that coming either – talk about unsettled dust.

Now for the allegory. There is a nation of Euclids living in a land aptly named Euclidia. That reminds me, one could write a tasteless limerick … but again I digress.

These residents of Euclidia are well versed in the flawless logic of Euclidean geometry but strangely ignorant of Riemann’s different assumption. The answer to the age-old question, “What is the shortest distance between two points” is easy-peasy for Euclidians. It’s a straight line between those two points.

Living next to Euclidia, in Riemannia, are the Riemannians. They are well-educated in the flawless logic of Riemann’s geometry, but sadly ignorant of Euclid’s version of the 5th postulate and curved geometry. The answer to the age-old question, “What is the shortest distance between two points on a sphere” is easy-peasy for Riemannians. It’s a great circle.

One day, over an overpriced coffee at some cutesy coffeeshop, idle chit-chat turns to this question. The Euclidian says to the Riemannian, pass the cream and sugar and can you believe that little Johnny didn’t know the answer to the shortest distance problem. They both chuckle to each other knowingly. The Euclidian and the Riemannian simultaneously state emphatically, “Straight line” and “Great Circle”, respectively. What transpires is not respectful at all. There is no way to even begin to rationally discuss this because a great circle is not a straight line. (I know this is difficult because I tried to tell my parents that this is also why I got a D because I was traveling in Riemannian circles.) Because of their frustration with each other hostilities broke out between them and the coffeeshop lost money and could no longer play depressing music. Sad, but true.

This miniature allegory about the axiomatic process serves as an example of how two systems based on mostly the same axioms, except for one, can result in two systems with internally consistent views, but with a view of the world unrecognizable to the other. Worse yet, mathematics is a highly disciplined field where logic and reason prevail. Imagine how much more complex the situation is when the axioms of two systems are much less similar and less logic and more emotions are used in the derivation of the system of thought. It’s no wonder we have cross culture problems and religious intolerance today. I think that all human systems are based on axioms, some stated clearly and others not so clearly stated or understood, and yet these systems are internally consistent to those living in them – they make “common” sense. Yet differences in the founding axioms prevent one system from understanding another system’s viewpoints and there really is no way to logically and rationally understand the other system without first starting with its axioms. If we educate ourselves in axioms and understand the basic assumptions behind a system of thought, perhaps we can avoid some hostilities. And, if this allegory doesn’t help produce world peace, then it may someday help explain why you got a D in 9th grade geometry.