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This paper was written as an assignment for Ian Walton's Math G - Math for liberal Arts Students - at Mission College. If you use material from this paper, please acknowledge it.

To explore other such papers go to the Math G Projects Page.
 

The PBS series titled For All Practical Purposes, consists of episodes on different mathematical subjects. I signed up to watch the videos that overviewed computer science and within the subject of computer science, numerical representation specifically.

The overview video starts out by asking the question, what is mathematics? It states that at the heart of the foundation of mathematics is proof and truth. The video then starts to focus mainly on men who have become known today as important pioneers in the field of computer science. Long before the computer science industry was born, these men (mostly mathematicians) developed theories that, unbeknown to them, would be used in the future to lay the initial building blocks of the foundation for computer science. For instance, David Hilbert back in 1900, studied the theorem of proof and truth. He set out to prove two things; mathematics is both consistent and complete. He believed that consistency in mathematics would lead to proof and that proof, in turn would lead to truth. In the 1930âs Kurt Godel proved that David Hilbertâs theory was not correct. The basis of Godelâs scheme was that a statement could be true, while remaining unprovable. This upset the mathematical world because contrary to Hilbertâs theorem, you could have true statements (truth) that were unprovable (proof), which does not lead to consistency, as Hilbert theorized, but rather, inconsistency. Personally, I found these theories to be a bit confusing, but I was able to understand the importance of the ideas. I think that the video presented the material in an easy to understand fashion (or as elementary as it could).

The next important contributor to the foundation of computer science was Alan Turing, who developed the Turing machine. The Turing machine was able to process the ideas of the time by mechanical means. John Von Neumann, who worked with Hilbert on the foundations of mathematics, helped to build the first computer in the 1940âs at Princeton. This "computer" was able to process instructions that could be coded in as numbers and then stored as data in the computers "memory". It was understandably quite amazing at the time, and it received quite a lot of attention. During the excitement Herman Goldstine worked with Von Neumann and together they developed the first electronic computer. This development resulted in faster capabilities and larger memory storage. It is easy to see that with each new discovery, the possibilities of mathematical exploration expanded to lengths far beyond anything that I could have imagined, but these men did have a vision and the intelligence to make that vision a reality.

There are many theories that up until the existence of the computer, would have been impossible for one person (or possibly any amount of people) to prove. Kenneth Appel and Wolfgang Haken were the first two people to use the computer to authenticate this very fact. Their theory concerned the idea of maps and colors. They took Alfred Kempeâs "normal map" idea or "four-color theorem" and used the computer to prove each of his cases.

I do not understand Kempeâs idea fully, but did grasp the concept that when you have a problem that requires a step-by-step analysis; each step builds on the next. If the previous step cannot be "proven to be true", then you have to move on to another step and start the process all over again. This portion of the video also helped me understand that there are mathematical theories that could take beyond a lifetime to prove. Maybe the best example I can give is the ongoing quest for _. Would we have gotten as far as we have without the aid of computers? No, and even with computers how much farther do we have to go? There is not an answer for that question yet. Through the ideas brought to the forefront by past mathematicians, computers were created. With the aid of present day computers, the mathematicians of today and tomorrow can explore theorums that just a short time ago would have been impossible to prove or disprove.

The second episode I viewed explored numerical representation. After having watched the episode, I believe that the title presented at the beginning,"Counting by Twos", was very representative of the concept in this video. The video starts by explaining just what it is that computers do. They receive, process, reproduce, and store data. I learned that in order for computers to do all these things, they have to be "programmed" or told what to do. The way they are told to do these things is by using codes. I like to think of these codes as computer language. Just as the English language has a code of letters that can be put together in an indescribable number of ways to make up many different words, computer language (codes) can be programmed to do the same thing. Each code represents given information. Somewhere deep in my mind, I knew the basic ideas that were presented in this episode of For All Practical Purposes, but what this video did for me was make me think. As Sol Garfunkel explained how nature has a code, a genetic code, and how all the many types of languages have their own codes, I began to think of other things around me, in every day life, that require codes. I began to think about the codes that supermarkets use to keep track of inventory. I thought about the codes my insurance provider uses to keep track of my medical history. Each of these industries put codes into their computers to represent each different piece of information (specific to their industry, of course). The computers then receive the information, process it (according to another code), reproduce it as told by another code, and store it in itâs memory! Of course, this system of codes applies to higher mathematics also.

One of the more interesting parts of the video explains how you can use this concept in music. There are numerous codes (musical notes) that look alike. How does a computer program tell the notes apart, or for that matter how does a musician? It starts with that same code and through the rules of interpretation, the codes take on different sounds. This variance depends on where the note is on the staff. Once a musician learns the given codes, and the rules of interpreting those codes, then they can look at a score and play it the way the author (programmer) intended for it to sound. I learned that in mathematics this same concept is called a "place-value" system, where the value of a symbol is determined by its position.

Before watching this episode, I always thought that the basics of computer language were incredibly complicated. I was surprised to learn that computer language is based on just two symbols, and was developed long ago. This two-symbol code is called a binary system. In the binary system the two symbols used are zero and one. These symbols are called binary digits or bits for short. I have heard of the term bits, but never really understood what it meant before, until now. I found it somewhat difficult to keep track of the rules of binary addition, especially once the "xor" and the "and" gates were introduced. I believe that I did understand the basic concept though. You put in a simple code based on a two-symbol system (binary). You then expand on that system, by adding "gates" to the system (xor and and). These gates are expanded upon again and again, and thatâs basically how a computer is programmed.
 
 

Critique/Evaluation

I donât remember having ever watched any math videos before, so I was curious to see whether or not I would be able to understand the material in the video. I also wondered whether it would be interesting enough to keep me from yawning and getting bored (itâs hard to stay interested in a subject once your completely lost as to what is happening). The first thing that I noticed about the videos was that they had some jazzy music to get them started, which was a good sign, because it made me think that the videos might be exciting and up beat although they were obviously older videos. I could tell they were older because the video looked somewhat faded, not as sharp or clear as the newer video film of today is. While I was trying to focus in on what Mr. Garfunkel was saying, I noticed that there was a steady stream of people walking behind him as he was talking. This was distracting for me, but it did not last long. I thought that the examples used in the videos were well placed and they kept me from being completely lost. One of the most amusing things that I noticed in video number twenty three, was that Sol Garfunkel put a floppy disk into a Commodore computer. I found this amusing because I remember that the first time I used a computer, it happened to be a Commodore!

Overall I would have to say that I came away from watching these videos with increased knowledge about computers, codes, and the history of key contributors in the field of computer science. I was surprised that the videos were as interesting as they were, I was not expecting that. I doubt that I would have been exposed to this information if not for this class, so thanks for the opportunity to learn more about this subject.