6 August 2006 – 18:17 – Sunday - Home
I have been studying Discrete Math for more than 2 hours. Actually, I am already feeling ready for my class. I don’t really worry about it. My enthusiasm can be considered as fun! I enjoy doing math with a book and pieces of papers. I proved a few theorems and I derived a few formulae to calculate number of simple paths, Hamilton Cycles and Euler Paths in a complete graph of Kn. As it happens after each proof, I am amazed with the results of the solutions. For example, the number of simple paths from one vertex to another vertex in a complete graph of Kn equals to exact value of e*(n-2)! This is surprising because number of simple paths is an outcome of a counting process. However, the Euler constant e=2.718… can be used to calculate it. For example, there are 5 simple paths in a K4. It also equals to exact value of 2.718…*(4-2)!=5.437… (By exact value, I mean cutting the digits after the point, not rounding!) This makes me think about the existence of irrational numbers again! Why do we have them? Or do they really exist? Ancient Greeks hid the existence of squared-root of 2 from public because they were not really sure about the existence of the number. How could it be possible to have a non-existent length for a diagonal of a 1 unit square? Beside this, numbers like pi and e are more complicated. For example, in probability theory both numbers are very useful. In Normal distribution, the function has both pi and e as constants. They are useful for modeling and approximating. Many mathematical objects with an optimum excellence use these numbers as their base. When we say circle, we do not talk about individual circles. The circle whose area equals to pi*r^2 is a perfect circle with zero thickness. Then number pi is also a number which belongs to this perfect circle. Ancient Egyptians and Mesopotamians did not know that pi is an irrational number but they used approximations for constructing buildings. It seems they did not have many problems. Even today, except for astronomers, most of the scientists use 3.1416 as pi. Students in high schools still think that 22/7 exactly equals to pi and use it freely. Civil engineers do not need more than 4 digits after the point. It is because we and all the circles we can observe belong to an imperfect universe. We are living in an incomplete universe which continues to evolve and we have no idea whether it will come to a point of perfection or not. But somehow, we have the ability of thinking the perfection. Descartes uses this idea of perfection to prove existence of God. According to Descartes, because we can think of "perfect", then ontologically there must be something or someone perfect. So this should be God. Actually, his reasoning is longer and more complicated but this is what remained in my mind after many years. There is another thing I could remember from Descartes readings: He does not explain why we should deny the existence of pegasus. According to his reasoning, we can think of a horse with wings. There should be a flying horse somewhere in the realm of existence.
Mathematicians create a non-existent perfect universe which has its own entities. These entities are not necessarily useful. Those who discover them do not really worry about their practical purposes. How could Euler imagine that his theorem about Euler Paths or Complex Numbers will be used by the mathematicians, physicists, computer scientists, software developers, system engineers etc of 20th centuries? Euler solved the famous Kongsberg Bridge Problem either for fun or for challenging or both (at the end challenging is fun!). It is amazing that there are people on this earth who are only caring about memorizing more and more digits of pi! Neither pi nor these crazy people give up!
Actually, it is not only mathematicians who create these perfect objects. If we have a quick look at the history of mankind, we can see how the idea of God evolved from “protectors from natural disasters” to “omni-potent, omniscient” God. Isn’t it a similar process? God is perfect and He has everything we don’t or can’t have! We are creating what we need! People needed numbers and they created them. People needed to believe someone who always cares for them and they created God. We all like the idea of perfection. Something imperfect could be considered as incomplete and something incomplete can be considered as “can be completed with a sufficient effort”. I guess, this is one of the ambitions which make us human. This is also how evolution works. If we take our brains as the “best product” of millions of years of evolution process, it would not be surprising that we can easily create perfect objects which do not contradict each other. Actually, Gödel showed that Math itself is not perfect either. Gödel’s Incompleteness Theorem clearly showed that no system –including Logic and Mathematics- can stand on its own feet. This theorem gives more chance to future Mathematicians to create a new system which is complete or at least “more complete” than our present Mathematics.
Now, I will go back to study more Math since I did not finish the chapter yet. There are always amazing things in Math and only finding them with my own effort is enough to make me satisfied with my day. I started to read Graham Greene’s “Quiet American” and I guess I can finish it in a few days.
Today, I also found some other blogs which are kept by Middle Eastern intellectuals. They are living in the countries where there is war going on. For those who are interested in reading things that are usually not told by BBC or CNN, these are good sources.
www.afamilyinbaghdad.blogspot.com
Actually, it is not only mathematicians who create these perfect objects. If we have a quick look at the history of mankind, we can see how the idea of God evolved from “protectors from natural disasters” to “omni-potent, omniscient” God. Isn’t it a similar process? God is perfect and He has everything we don’t or can’t have! We are creating what we need! People needed numbers and they created them. People needed to believe someone who always cares for them and they created God. We all like the idea of perfection. Something imperfect could be considered as incomplete and something incomplete can be considered as “can be completed with a sufficient effort”. I guess, this is one of the ambitions which make us human. This is also how evolution works. If we take our brains as the “best product” of millions of years of evolution process, it would not be surprising that we can easily create perfect objects which do not contradict each other. Actually, Gödel showed that Math itself is not perfect either. Gödel’s Incompleteness Theorem clearly showed that no system –including Logic and Mathematics- can stand on its own feet. This theorem gives more chance to future Mathematicians to create a new system which is complete or at least “more complete” than our present Mathematics.
Now, I will go back to study more Math since I did not finish the chapter yet. There are always amazing things in Math and only finding them with my own effort is enough to make me satisfied with my day. I started to read Graham Greene’s “Quiet American” and I guess I can finish it in a few days.
Today, I also found some other blogs which are kept by Middle Eastern intellectuals. They are living in the countries where there is war going on. For those who are interested in reading things that are usually not told by BBC or CNN, these are good sources.
www.afamilyinbaghdad.blogspot.com
