Circa 300 before Christ the mathematician Euclid created the 13-book series The Elements. The most influential textbook of all time. Shaping logic as the way we know it today. Making use of definitions, common notions and axioms, which are unproven statements generally accepted to be true. Creating a framework which can be used to prove its propositions. This axiomatic approach has been extremely influential and can now be found in any field relying on logic. In this article I want to reflect on the influence of The Elements and talk about the axiomatic approach to reason that it formally introduced to the world.
The problem with applying logic to prove things is that when we construct a logical argument, we must always start from something that we know is true. However, this is impossible if no axioms are used. Since for us to know that something is logically true it also must be proved, which once again creates the requirement of knowing something to be true. Hence if everything that we see as logically true needs to be proved, we are unable to prove anything. To avoid this problem axioms are used, functioning as the fundamentals of a belief system. Since now we do not create an endless loop of finding truth but rather creates some fundamentals for our beliefs to build on.
While The Elements might have been the worlds’ introduction to an axiomatic approach there are many more systems of axioms that have been created through the years. Raising the question what the requirements for a valid system of axioms are. Since axioms are unproven statements, an axiom may not be derivable from the other axioms. If this was possible it would be a proposition and not an axiom. Hence, we try to keep our list of axioms as small as possible. This makes intuitive sense, since a system of beliefs is more reliable if the list of unproven statements it depends on is as small as possible. Furthermore, a system of axioms cannot be contradictory. While the first requirement is just based on the definition of an axiom, the second requirement is much more important. Since if your axioms result in contradictions then at least one of your axioms must be false.
Up until now I have talked about axioms in a very theoretical way, but without thinking about it we all apply it in our reasoning intuitively. In our everyday beliefs we all have some things we believe not because we know it to be true, but just by assuming it. I think this is an important realization with strong implications in our reasoning.
For example, in a political sense, there are a lot of beliefs on the best policies to rule a country and someone with a certain belief might be just as confident in its policies as someone with completely different beliefs. These differences in beliefs can be so big that it is sometimes hard to believe that people with different views are still both coming from a place of reason. However, with the realization that we all use axioms in our reasoning without even thinking about it, it might just be that the other person has a fundamentally different assumption of “the truth”. If then both persons cannot disprove the other person's axioms, discussion is futile since there is no way to logically determine right from wrong.
But what does it mean to disprove a belief? To answer this question, I will use the second rule of the system of axioms mentioned above. While it might be impossible to directly disprove someone’s beliefs, one might be able to show that their beliefs have logical implications that are not inline with their actual beliefs. Hence showing that one of their beliefs must be false.
I believe that this axiomatic way of approaching a difference in beliefs is very beneficial. Since it allows us to better understand our own viewpoints, but more importantly it lays the focus in truly understanding the other person's belief and instead of debating who is right and who is wrong it creates a search for a difference in a subjective belief.
In short, The Elements of Euclid has been an extremely influential book series, shaping logic as we know it today. The understanding of an axiomatic approach is extremely important in developing a logical way of thinking and helps us see our own axioms that we have created in how we see the world. I hope that realizing helps us all to better understand our own beliefs and those of the people around us.