The following is a relating of a continuing journey in the discovery of the mystery of what is commonly described as a function. Those that are mathematicians should not be discouraged from proceeding by the simplicity of the question at hand as I am sure that there is a little bit of something for everybody.

1. Background – putting everything in context

2. The 3 amigos – the 3 definitions of a function

3. What are the differences/similarities in the 3

4. System Theory and Complexity

5. Darwinism and Creationism

**Background**

My name is Taurayi Chitaukire. I almost failed my college algebra two or three years ago (I eventually got a C). This happened despite the fact that almost all the material I had covered before. What happened (as happens sometimes when I discover loopholes in something/concepts) was that I developed a psychological block where my (subconscious) mind had picked up on an inconsistency that my conscious mind was yet to understand or “see”.

When I eventually “saw” I realized that the books I was using then (college algebra and trigonometry) had described a function in a way that was the same but different i.e. similar, yet not identical. ASIDE: I have since come to understand/made aware that as one goes higher in mathematics, the definition does become more precise. Thus what I have to say may have been said already but I am confident that some of the perspectives I have are likely to be unique.

**The 3 Amigos**

1. Given 2 sets X and Y of x-values and y-values respectively, a rule/process named f is a function that maps an x-value to a unique y-value.

2. Given a set X and a rule/process named f, then Y is the “image-set” that is created when the rule f is applied to X i.e. applying f to each x-value creates a corresponding y-value.

3. A function is a group/set of one or more ordered pairs of the form (x, y) where all the members of the set (each ordered pair) have the same/identical association (associating entity – rule, process). X is the set of all x-values and Y is the set of all y-values.

# Differences and Similarities

All three definitions have 3 components – 2 sets X and Y and a rule, process or rather, an associating entity that associates a member-value x to a unique member-value y.

Now, to where the loophole(s) began to manifest. Explained in the simplest way, the loophole (inconsistency) in the three has to do with time and/or independent vs. dependant variable i.e. which is the independent and which is the dependant in each scenario.

__1. The Mapping Function__

In the first definition – which coincidentally I named a mapping function (only to find out later that that term is already used) – the sets X and Y are pre-existent before the map that maps each domain-value to a range-value, therefore the independent variables are the members of the sets X and Y and the dependant entity is the map i.e. the rule/process named f.

To appreciate the reasoning, think of when you want travel to a given destination from a given source (departure place). What is known first is the where you start and where you are going. The “map” is then one of possibly many routes to the destination and the chosen one is dependant on the source/destination combination.

Thus, unlike the classical way of looking at the independent/dependant variable as variable as being of the source (domain) or the destination (range), the dependant in this case is the set of “maps” that are resultant of the independent entities X and Y.

__2. The Imaging Function__

In the second definition – which coincidentally I named an imaging function (only to find out later that that term is already used) – the independent entities are the set and the rule/process f and the dependent entity is the set Y.

This is the “classical” understanding although in definitions the rule is not usually referred to as being an independent entity yet it is. If you were to take out x and use G (a set of g-values) the process would still work absent of X.

Therefore, the set Y is an image of the set X, through the process f.

__3. The Triad Definition (a.k.a, The Set Definition) of a Function__

This is the definition I arrived at over time as being the most generic of the three. The finality of the definition was arrived at after taking a Calculus class at with Ms. Houston Community College. She defined it something along the following lines:

*A function is a relation where no two ordered pairs have different first elements (of an ordered pair) and the same second elements*.

In this definition, what is the pre-entity and what is the post-entity is not know (the Divine Mystery?), all that is known is that the three exist at the same time i.e. the x-value, the y-value and the associating “order” (rule/process) that allows the (x, y) combination to be called an ordered pair.

## System Theory and Complexity

Looking at the first two definitions of a function and taking into consideration how to build a hypothetical complex system (where system refers to any generic system without ties to any particular discipline), the question arises, what is better/easier/preferable/feasible, etc., etc., - building a system that has few rules and matter/things that can manipulated in a possibly infinite number of ways by those rules or building a system where you have few items of matter and possibly infinite number of rules that can manipulate the given matter?

If you look at the human gene it is composed of 4 items of “matter” and yet through multitudes of rules (i.e. both the “physics” and “chemistry” rules) you have an emergent set of biological rules that are the complex system called a

## Darwinism vs. Creationism

All of this enters into the whole debate of Darwinism vs. Creationism. Darwinists claim that the complex state of creation can be arrived at through natural evolution (i.e. survival of the fittest) whereas creational evolutionists (evolution just means change) claim that the full complexity of existence cannot be explained by a Darwinist point of view because the “original” complexity that allows everything to exist (including the rules that allow “survival of the fittest” to exist) cannot be arrived at through reductionist science.

Thus, Creationists would argue, there is an “intelligent design” i.e. there is an original architecture that/whose complexity is not known and that is such that it “mandates” a “Creator”. By “mandates a creator” I am referring to the fact that the degree to which everything works and the mechanisms that allow it are so vast that some do not have numbers that can explain them. They are so vast that they allow for the existence of purpose, reason

Thus is the meaning of Creationist – a scientist who believes that the original “science” behind science was created including the rules that allow for change (evolution) to exist.

Darwinist = Natural Evolution

*Creationist = Intelligent Design(er) + Natural Evolution proceeding from that “original” design*

The most common example of a creationist is those that are of Christian faith. Many of believe in Biology, Physics, Economics, etc., and yet at the same time believe in God and the Holy Trinity. And thus we have come full circle, back to the definition of a function and which is “most correct”. The 3^{rd} definition allows for the other two to exist, three including itself.

How it does this is explained by time. For the first two – there is something that exists before (the independent variables a.k.a pre-existent entities) and something that exists after (the dependant variable a.k.a – post-existent entity). In the 3^{rd}, time is no consequence, for when one exists, the others exist.

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