Monday, June 30, 2014

Nature of Mathematics (Exemplar)

Nature of Mathematics

There are two questions that I aim to clarify, in light of my own understanding, through this post. The first question is, what is an axiom?  An axiom is essentially to math, what a universal truth is to life.  For instance, one of these universal truths, though not often thought about or commonly used, is that the truth is always the truth, even if no one believes it.  So what this is saying is that, in life, if something happened and is true, even to the knowledge of only a single person, that true thing is, in fact, what actually happened.  This holds true whether everyone or no one else thinks that this "true thing" is true.  It is something that can be applied to any situation and must be accepted as real or fact by the person to whom it is true.  This expands to all of life, in fact.

What this looks like in mathematics is an axiom.  So an axiom is a truth about math, that we hold to be true regardless of the situation.  In mathematical terms, this truth would in actuality be reminiscent of a proof.   Though a proof must be proven to be accepted as true, an axiom (a statement about an aspect of math) is accepted as true without having to prove it....what the heck math!? Anyhow, I'm not sure who, but somewhere along the line some mathematician, who definitely was not bizarre in at least one way, decided that these statements didn't need to be proven.  Who'd have thought?

So, what in the world do we use these axioms for? Why would we need something that is accepted as true, without having been proven? Axioms are used for proving other mathematical statements which could not otherwise be proven.  That is, for many mathematical statements you come to a point in the proof of said statement where you have to accept at least something as true.  If you didn't then you would have to continue breaking down the statement forever...but seriously it would get frustrating or impossible to prove some things if you didn't accept some truth as true.  That truth is an axiom.  They are essentially the most basic statement that you can use in a proof that doesn't have to be proven itself. An example would be that if we could not assume that all right angles were congruent, we could not logically apply the pythagorean theorem to any triangle.  This would be in light of the fact that, if not all right angles are 90 degrees, then they could be other degrees.  Smaller or larger. A change in angle measure means a change in segment length on a triangle, which would be impossible to calculate with the pythagorean theorem, because the pythagorean theorem only works if there is a right angle.

That seems confusing. But a simple way to think of an axiom is this.  "I would not be at Grand Valley right now if I didn't apply, I wouldn't have applied if I didn't know it existed, I wouldn't have known it existed if...if...if....if...if....................axiom."  An axiom is the last in the line of the "ifs"...if you will.

Exemplar Additive:

For the exemplar of this weekly post I started off sort of unsure what I should do.  It seems that I pretty thoroughly covered what an axiom is and why it would be important to mathematics.  I even explained it in non-mathematical terms.  So as I began to think more about what I should do with this idea of axiom, I found myself starting to think of where axioms even came from.  As I researched this idea further, I found, as it turns out, that axioms came from ancient Greek findings in mathematics.  They basically decided that to deduce anything about math, there had to be a starting point.  These starting points came from logic.  If something was entirely logical, then there would be no way to dispute it.  For instance, if we look at the associative property of addition we see that it doesn't matter if we have 3 rocks in one and 4 in the other, in total there are 7 rocks.  It doesn't make a difference if you count the 3 or the 4 first, it still adds up to 7.  This is the kind of reasoning behind any axiom.  

After spending some time with this idea, I began to think about the concept of creating a new axiom.  Could it be done? If it could be done, would there be any point to it?  I began searching around for any literature on the subject and the best that I came up with was this

There is a lot of filler talk in this article, but it does offer some good viewpoints on the matter that helped me to better form my opinion. I would recommend this read to anyone who has interest in thinking more deeply about the need or the logistics of creating new axioms.  The biggest question that I came into, as noted in the article as well, which axioms would need to be created and for which purposes?  Definitely food for thought.  

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