Turtles All the Way Down

A student asked his teacher: "Master, what holds up the world?"
The teacher said: "The world is held up by four giant elephants."
The student asked: "So what holds up the elephants?"
That teacher said: "The elephants are held by a giant turtle"
The student asked: "So what holds up the giant turtle?"
"Ah!" proclaimed the teacher, "It's turtles all the way down!"

To begin understanding Epistemology, start with any statement you believe to be true. If I ask you to justify your belief, you will need to come up with a different statement, a more basic statement you believe to be true, but then again I can keep asking you for justifications. This process seems infinite, and indeed this is a problem first recognized by Aristotle, called infinite regress. Aristotle believed that infinite regress is an invalid argument. Instead when you regress into more and more basic statements you will inevitably reach a statement so basic that requires no justification, something that is so obviously true it is self-evident. These statements are called axioms.

But this only raises more questions: How can we tell if we have reached an axiom? Is there a complete finite list of axioms? How can we derive such a list? Will such a list allow us to derive everything that is true? Will such a list allow us to evaluate each statement to resolve its truthness? In order to answer these questions we need to develop a language, a set of principles, that are themselves independent of the axioms.

Another way to think about this is by imagining a dictionary in a foreign language. Searching for a definition in such a dictionary will only yield more words that are by themselves foreign and require definitions. It seems we have reached another infinite regress, not of arguments this time, but of definitions. The only way to resolve this is by using a bilingual dictionary. In other words: In order to describe a system, one must use terms from outside of that system.

So we need an axiomatic system, and we need a framework in which to discuss the axiomatic system, but how can we justify the truthness of any such framework? We would have to use an axiomatic system for that, otherwise we will reach another infinite regress. This creates what Douglas Hofstadter termed a strange loop, a cyclic structure with no definite beginning, each part necessitating a prerequisite, which in turn necessitates another prerequisite until you reach the exact point in which you have started.

So how could we possibly hope to untangle such a mess?