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.

When I think of math, I naturally think of all of the math courses that I have taken at Grand Valley.  Throughout all of these different math courses, I have seen that math can look vastly different and has just about an infinite amount of applications.  Math, for me, has ranged from how to teach it to children using manipulatives all the way to proving that zero is, in fact, less than one.  Throughout the duration of my journey from colored blocks to more abstract applications of mathematics I have found that the best way to describe math is that it is helpful.  That's what math is.  Given that there are so many different variations of math, I find that it would be unhelpful to list what math is in all of these circumstances.  I do, however, find it helpful to talk about the fact that, in light of there being so many different mathematics applications, math is helpful.  Why are there so many different types of math? Because there are so many different purposes for it.  Math can help me figure out how many more apples I have than Jane, it can help me to calculate the number of years it will take me to pay off my students loans, or it can even help me to figure out the best way to invest my (hypothetical) lottery winnings.  So the most basic and fulfilling ways to describe what math is, in my opinion, simply to say that it is helpful.  It helps me, it helps you, it helps everyone.

As far as the biggest achievements in math go, I would say that they are:

1. The development of the pythagorean theorem. Well done Pythagoras. (What can't that be used for?!)
2. Calculus.  What on earth would we do if we couldn't estimate the volume of objects by slicing them into tiny discs?
3. Matrices can literally be used for more things that I can think of.  I'm glad that those are around.
4.  Multiplication and division.  How else would we divide 35 camels up for some strangers!?
5. Addition and subtraction.  We have to pay for things, sometimes at the same time.  Without addition and subtraction we would be left wondering how much the cone costs.