Did you know that 1 > 0? Neither did I until my professor proved it in the second week of my first college mathematics class, Honors Mathematics. It is unlike any other class I have ever taken. What makes this class so different from other mathematics classes is its emphasis on proofs and the actual logical process that every branch of mathematics, including calculus, derives from. Rather than learning how to find the volume of a circle rotated around the x-axis or how to integrate specific expressions, we started with the basic self-evident statements that all of mathematics is founded on. Day by day, we have been working our way through the basic statements toward more complexity, rigorously proving every step of the way. However, I’m not writing this article to teach you mathematics. Rather, I will demonstrate that this same mathematical structure of logic, when compared to our everyday beliefs and values, can teach us a lot about ourselves and what we know about the world.
The foundation of all of mathematics is a group of statements, called axioms, that are believed to be self-evidently true. These propositions are not only considered unnecessary to prove, but they are not even possible to prove! We reason with these supposedly self-evident statements to then prove further statements, called theorems, which are not necessarily self-evident, but can be derived by the axioms. These theorems then go on to prove even more theorems, and so on, building more and more complex statements that all rest upon a simple foundation of axioms. So, strange as it may seem, all of mathematics is built upon statements that are not provable by the mathematical process. Of course, this does not mean that what we know about mathematics is wrong. I am simply using this mathematical example to demonstrate how axioms underlie every field of inquiry and science including mathematics: the most logically-founded field of inquiry that we have. The axioms of mathematics have been carefully chosen to create a system of mathematics that describes the physical world and have been tested through scientific applications. Unfortunately, we have not yet established any kind of self-evident truths outside of mathematics that everyone can agree upon. It may not even be possible.
Now, imagine what this says about everything that we know, or think we know, about the world. Looking at our systems of belief in a completely logical structure, every truth that we hold should be proven by other truths, which need to be proven by other truths, and so it tumbles until inevitably hitting a solid floor of axioms. Axioms that we can not possibly prove and underlie everything that we believe about the world. Where we get these axioms from is outside of my scope of knowledge. Some will say these supposedly self-evident beliefs are emotionally instilled in our brains as a result of their advantages in the process of evolution. Others claim that our unexplainable values prove the existence of an absolute morality and the existence of God. Either way, every belief we hold is based on basic propositions that we can not logically prove – truths that we simply take for granted. And as there is certainly a conflict of beliefs in the world, either we are irrational creatures, unable to logically reason our way to truth from these foundations, or we are not building upon the same axioms.
This is what open-mindedness really is: understanding that every philosophical, cultural, and even mathematical belief that we have can be broken down into smaller and smaller components until all that remains is a pile of axioms that can not be broken down any further; axioms that will forever remain unproved. It is realizing that we are all in the same, taking what we believe to be true, and doing all we can to spread this truth around the world. Is it depressing or is it beautiful? It leaves us as minions in a war of beliefs where every man and woman in our ranks is born into an ideological army, equipped with the weaponry of voice and muscle to do everything in our power to spread the unprovable set of beliefs that we just happened to be born into. So, never stop preaching your values, but never lose touch with this openness, this idea that maybe you don’t stand for some kind of absolute truth. Because your time in this world is an opportunity to explore and stand up for your own self-evident truths, and leave the world a different place than you found it. Then someday, even though the values that you fought for were ultimately not provable, you will have done your part in the story of humanity.