Rules of the Jungles
Every proposition contains a set of assertions that are bound to some scope. We will refer to these propositions in a global sense as systems from here on end.
A system can said to be best sound and possibly complete only within itself. No system can be rejected solely by making a reference to any other system for its validity since such action will result in an incompatible verification system without first establishing an absolute system. Therefore, we cannot use a rule from one set and apply it to another set.
Corollary, there must be an absolute universal system where all sub-systems coexist with one another in order to make any verifications among sets. The absolute universal system must not contain any global variables outside itself. All variables then must be contained within the absolute universal system itself.
Each system begins with a base set of laws that holds the integrity of the system. Therefore a system may function throughout its lifetime by being constant or dynamic or both. For there exists governing base laws for a system, all resulting phenomenons within that system is yet bound to the scope of that system. A precise definition of these base laws is not a requirement of the system not only because there may not be any absolute checks for validity but a system may choose to modify its own strucure during its lifetime.
Consider the question:
What is a set?
A common answer to the above question is: “a set is a collection of objects.” Unfortuntely, this answer is far from being satisfactory, for in attempting to define the concept set, it uses the concept collection whose definition has not been given. What the answer has done is essentially changing the question from “What is a set?” to “What is a collection?” So, our question becomes:
What is a collection?
A moment of reflection reveals that to define the concept collection, we have to come up with another concept. Whatever this concept is, we will immediately be confronted with the same problem, that is to give that concept a definition. Therefore, the problem we are facing is that each time we come up with a new concept to define a given concept, we immediately have to find a definition for that new concept. Obviously, this can be repeated indefinitely. So, even before we start our study of Set theory, we are already stuck with a nontermination chain of questions.
Dr. Yung H. Tsin, Theoratical Foundations of Computer Science, p.71, University of Windsor, 1999.
It is impossible to define every concept in our study. We simply cannot develop something out of nothing. We must begin our study with some concepts which need no definition. These concepts are called basic concepts. Every other concept must be defined based on the basic concepts or concepts which have been previously defined. Clearly, each basic concept should be primitive enough so that one will have little doubt as to what their meanings are.
Therefore axioms are simple bits of information that require no further explanation. They are in other words the foundations of any system.
If the starting point of all systems asks us to be in agreement with its statements in order to place a sense of what follows ahead, then what makes one system different (or alike) then any other system is merely a function of appeal and individual's acceptance or perhaps even usefulness . A belief in a system requires only the act of choosing it over another system. Whether this action is a result of a hard personal reasoning or an indifference in a preference or an externally forced decision, the acceptance is still personal and subjective.
An example is due here and so we will take a trivial game that we play in our town. There exists a set of rules for this game which allow us to play correctly only by following them. Since an unrecognized rule would be inconsistent with the nature of this game, we will refrain ourselves from applying them. Let us take a yet another scenario where a variant of this game is played in a neighbouring town. The rules for this game in the neighbouring town has a different set then ours that is understood and applied by their folks. Now lets consider a friendly game between the best players from each town where each player stays within the scope of their own version. Could such a game be properly carried out?
This brings us into a situation where there may be more then one account for the same phenomenon. How can we then approach this to make sense of our reality? Can multiple systems really coexist with one another? Does this lead the mind to have any notion of agency?
Let us consider the story as told in the New Testament of the Bible, the event in which Jesus of Nazareth walks on water demands a point of view to illustrate the complexity of what we are involved in. Given our knowledge and acceptance of the laws that govern this universe, living human beings cannot walk on water. There may exists untapped human abilities in which humans can interact with their environment in such fashion, but, until then, this event stands as a what we call a miracle. A miracle is an event which is not explainable using our current knowledge system. How then can we evaluate the accuracy of this event if it had occurred as claimed? To be more accurate: a belief in which system can best explain this event?
There may be multiple versions to account for the same event. Given the nature in which each event may be explainable within the boundaries of a system of our choice. This begs the question (if, of course, it is the right question to ask): can we and if so how do we make choices? To choose one option over the others clearly makes a mark toward our preference in our life. We don't have to look further then the nature versus nurture debate to explain the whys of our choices. If possible at what probability can we make free choices?
I would like to conclude with my interest in all that I have written so far; I did not intent to handout my version of the partial answers to such unknowns: as such would be a move that I do not wish to face the test of time since new understandings and belief system will arise beyond my reach and comprehension in the future. Instead, my intentions were to solely wrapup my own understanding of the reality in which I believe I live in today and to account for the uncertainty of any given point in the existence I feel that I am part of. I therefore believe that in order to exist (moreso to live) as I claim to, I must make a fair leap of faith toward all systems in order to put them to use in my life wherever I see fit. Hence, within my reasoning, I favour pragmatism.