Mathematics gives us tools to do a lot of different things. Physicists use calculus to explain the motion of objects. Electrical engineers use differential equations to process signals. Psychologists and sociologists use statistics to analyze and understand human behavior. Even computer scientists use mathematical logic to understand the limits of programming.
I intend to use geometry to provide insight into worldviews. While this probably seems like a ridiculous claim, it will make sense by the end of this article.
In geometry, there are two types of terms and two types of statements you can use. The terms are undefined words and defined words. The existence of undefined words is viewed as an unfortunate necessity to avoid circular definitions. If you try to define everything, you’ll run into a loop (circular definition, see definition: circular). Alongside are axioms (statements that are assumed to be true without proof) and theorems (statements that are proven to be true from other theorems and/or axioms). Axioms are another unfortunate necessity, as your first statement cannot have a reference to determine whether it is true or not.
In geometry, the three main undefined terms are:
Common axioms are:
- “For any two distinct points, there is exactly one line that passes through them”
- “Three points not all on the same line define a plane”
- “For any line L and a point A not on the line, there is exactly one line parallel to L that goes through A in the plane determined by A and L.”
These should be familiar to you from your high school geometry course (if you remember it). These correspond to the idea of a plane being “an infinite sheet of paper”, a point being “an infinitely small dot”, and a line being “an infinitely long straight line.” Note: these are notions, not definitions!
However, that third axiom could be replaced with a different axiom and provide a perfectly legitimate, but quite different, geometry. One option is: “For any line L and a point A not on the line, there are no lines parallel to L that goes through A in the plane determined by A and L.” Another is “For any line L and a point A not on the line, there are infinitely many lines parallel to L that goes through A in the plane determined by A and L.”
The first corresponds to what is called “spherical geometry“, where a point is the two “points” on opposite sides of a sphere, and a “line” is a great circle around it, and the “plane” is the sphere. The second corresponds to “hyperbolic geometry“.
When constructing a set of axioms, there are two possibilities: either they lead to a contradiction, or they do not. If they lead to a contradiction, that means you can prove some statement and the opposite of it. If you can do that, then the axioms are called “inconsistent”, which means you can prove anything, and they are meaningless. To demonstrate a set of axioms are consistent, we find a model that satisfies them. A model is a representation of the terms (defined and undefined) that satisfies all the axioms. By extension, it must also satisfy all the theorems derived from the axioms.
The Connection To Worldviews
Our worldviews work in a very similar manner to geometry. We have beliefs that we accept as true without trying to prove them, and others that are conclusions from those beliefs. Granted, we can usually point to experiences that support one belief or another, but at some point, we are stuck with beliefs that we just take as a given. However, that doesn’t mean we can’t analyze those beliefs for problems. Just like geometry, these axioms can lead to consistency or contradiction.
Consider a trivial example: “There are no absolute truths.” There are people who accept this statement as an axiom of their worldview. Unfortunately, analysis of this axiom reveals problems. Let’s start with whether it is absolutely true. If it is not, then it has exceptions, which means there is at least one absolute truth. The alternative is that it is an absolute truth, which means there is at least one absolute truth. Regardless, it becomes clear that this belief is self-refuting. Without considering any other specific statement, it falls apart.
Clearly, this is a simplistic example, but it illustrates the point: if your set of fundamental beliefs (axioms) leads to a contradiction, then you either have to abandon one of them, or admit that your worldview is illogical. Regardless, other people won’t have a reason to accept your beliefs as true.
Additionally, for a belief system to be useful in everyday life, it needs to conform to reality as a model. For example, you can believe (as an axiom or theorem) that the Earth is flat. You can defend that position based on a variety of arguments. However, it does not conform to objective reality. Even if your beliefs are internally consistent, if reality is not a model for them, any conclusions you draw have no relevance to the real world.
For every idea we present, and every argument we encounter, it will be important to state our assumptions. If there appears to be a contradiction, we either need to demonstrate how it isn’t really there, or accept that at least one of our assumptions is false and needs to be abandoned. If an assumption is cherished, but is the source of the contradiction, we may have to accept something very unpleasant as true.
Additionally, our assumptions and their conclusions must conform to reality. If they violate what we can observe about reality, then we must either resolve the apparent discrepancy, or admit that our belief system has nothing to do with reality and abandon it.
When it comes to this blog, that is our standard for ourselves, but also for those challenging our ideas. If we find that your assumptions lead to contradictions, we will challenge you to drop one or more of them. Regardless, until you can make your challenge internally consistent, or show our worldview is internally inconsistent, we will not need to abandon our position. Further, any worldview presented needs to conform to reality as a model for it. Otherwise, what is being described is irrelevant to any practical discussion.
It also means we can have contradicting worldviews with no way to show who is right. Further, if we cannot clearly show that either worldview contradicts observable reality, then we can logically agree to disagree.