What is Math?: The Beginnings of the End

Again, I find myself using this blog! Last time, I was an aspiring teacher, bright-eyed and ready for my junior year. Now I'm a senior, a semester away from BEING that aspiring teacher (how exciting!) and ready to finish learning about mathematics! So, in introduction to developing a relationship between every single mathematical concept I've covered in my four years at Grand Valley, I was asked: What is math?

What is Math? 

I would define math as using given information and prior knowledge to draw conclusions. Take the following logic question for example: Suppose there is a basket full of 14 apples and we take 3 apples from it. How many apple do we have? Of course, we have to use our assumptions and given information to determine the answer to this. We know that we definitely have at least 3 apples, but was the basket in our possession as well? In this case, we have at most 14 apples. So, the answer is that we have between 3 and 14 apples. In this way, we were able to draw conclusions from given information and knowledge we previously had.
Let's extend this definition to a real-life problem-solving situation. Suppose you and your friends want to go out to dinner. You have the choice between a little mom and pop diner you have never been to before, or Outback Steakhouse, which you know you like, but had a bad experience with the previous time you went. You know this: You like steak, but you have never tried the diner. You know you enjoy experiences at Outback, but your previous experience was very negative. How can you use what you know and the given information to draw conclusions? Well, there are several conclusions you can draw: You can try something new and go to the little mom and pop diner. You can ignore the previously bad experience you had at Outback, remembering the positive experiences you had, and decide to go there. Or you could forego steak completely, either deciding on a different restaurant or even cooking dinner at home. This type of thinking relates more to permutations and combinations, since it deals with choices and logic about how to make a decision and how many ways a conclusion can be reached.
Now, let's discuss a more "mathematical" example. In Euclidean Geometry, we were able to use given information and our prior knowledge (in this case, the theorems we had proven in class and the axioms we were given) to prove the Pythagorean Theorem. The theorem is as follows: "If a right triangle has legs of length a and b and hypotenuse of length c, then a^2+b^2=c^2."
As a brief outline, the proof begins as follows: "We are given a triangle, which we denote as ∆AEF, such that L(AF)=a, L(AE)=b,  and L(EF)=c. We will prove that a^2+b^2=c ^2.  We construct line AF, and we copy AE onto AF with F between A and D such that FD is congruent to AE. This is possible by Theorem 31. Then, by Axiom 1, L(FD)=b." This will continue, using previous knowledge and given information, until we are able to construct the following figure:

At this point, we have proven what the formulas for the areas of triangles and squares area. We see that the shaded area (or the areas of the four triangles) forms the area of the interior square subtracted from the area of the exterior square. So, we know that 4(ab/2)=(a+b)^2-c^2. That is, 2ab=(a^2+2ab+b^2)-c^2, which reduces to 0=a^2+b^2-c^2. Finally, by addition, this forms a^2+b^2=c^2. Thus, this a real-life mathematical example of how to apply my definition of math.

How Has Math Made an Impact? 

While my knowledge on the history of mathematics (and thus the real impact it has had on the historical realm), I do know some major strides have been important. Bear with me, though, as my knowledge is incredibly limited and I may not get all the facts completely right. (Feel free to correct me in the comments! I love learning new things.) 

Development of Number Systems 

I'm not sure where the development of number systems really got their root. Part of me thinks that it came with the proposal of adding "zero" to the numbers, since that allowed for people to be a lot more flexible in their mathematical reasoning. Nevertheless, the development of number systems was pretty awesome because it led to the explanation for numerical behaviors we are familiar with today. For example, why are we "allowed" to add 2 and 2 to create 4? Does this work in all number systems? In what areas does division work, and where can we not divide? With the separation of numbers into categories (which leads into "ring theory"), we can learn more about why numbers behave the way that they do. To put it into perspective: we know that subtraction is closed in the real numbers; that is, we can subtract one real number from another real number an obtain a real number. For example, 2-3=-1, and -1 is in the real numbers. However, subtraction is not closed in the natural numbers. As a counterexample, 2-3=-1, but -1 is NOT in the natural numbers. This development of number systems not only helps to explain the overall "behavior" of numbers, but also helps to make proofs much easier, since we can use the knowledge we have of number systems to rationalize why a certain property (such as addition) holds.

The Parallel Postulate 

While (I believe) this has yet to be proven, its existence is what makes Euclidean geometry possible. The parallel postulate states this: "For every line l and every point P not on l, there is at most one line containing P that is parallel to l." Let's suppose this property did not hold. This would create several structures of "lines" that, although parallel to l, were hyperbolic, conic, and three-dimensional, generally not Euclidean in nature. The parallel postulate allows us to construct and measure squares and rectangles, and to find their respective areas. It is the backbone of a lot of geometric mathematics that we do, and thus it has made a huge impact on our mathematical world today.

René Descartes 

I know that Descartes is responsible for so much more in the realm of mathematics, and I hope to get a chance to learn more about these accomplishments this year, but I think his development of the Cartesian plane made the expression of two-dimensional functions much easier. This eventually led to the expression of derivatives, integrals, and even a three-dimensional plane for hyperbolic structures, among many other strides. The Cartesian plane is something students learn to draw early on in their education, and something they continue to use throughout college and beyond. 

The Takeaway 

I guess the takeaway here is that, while I have learned a lot about math in my years of school (and to think I came into college not even liking math!), I still have quite a bit to learn. I know very little about the history of mathematics, and what I do know revolves around theory rather than the people behind these theories. Additionally, I would like to be able to extend my definition of mathematics to beyond the scope of doing math with numbers and theorems, and actually being able to relate it to all real-world scenarios. I guess my ultimate goal is to be able to answer the question: "Why do we have to know this? When will we use this in real life?" 

I look forward to learning even more about mathematics this semester, and in being able to post my findings online again!