Levison Sam 000296128 October 19, 2008 Math 331 Project #1
Since humans in the 21st century where born they have always been taught the fundamentals of mathematics. As young children before any organized education we have been able to tell who has more candy by the first concept of quantity. As Pre-School began we have been able to count down the months till our birthdays by addition or subtraction. As our knowledge of the of numbers and theory we where taught simple theorems such as an odd plus an odd will always be even, and an even plus an odd will always be odd. The rules and formulas we have followed never stopped in middle school we learned the Pythagorean theorem where the squares of the side of a right triangle will always equal the square of the hypotenuse. In High School the theorems grew more advanced and complex such as the root of any prime number will always be irrational. At some point in our education, most of the times far down the road, either by force or curiosity, we where forced to ask ourselves why these statements do work, what makes them true, and how is it that they are. As history tends to do we find ourselves asking the same questions as earlier civilizations, much earlier in this case. Way back in the Ancient Hellenistic (Greek) times is where we find good sources of men asking the philosophical questions, why, what, and how. Why are the base angles of an isosceles triangle equalWhy does the diameter of a circle bisect the circle?(Eves, 72) From such questions ways of providing proof to
explain these formulas were introduced. Proofs back then might have been different from the proofs we are use to in modern mathematics. As mathematics has grown in different directions and become more complex so have the proofs. The basic proof or the direct proof method cant explain or prove all the statements and answer all of the whys and hows of mathematics. We now have many ways of proving and answering these questions for the plethora of mathematical problems that are out there such as proof by mathematical induction, proof by contradiction, proof by exhaustion, elementary proof, two-column proof and the list keeps going. As mathematics has grown more intricate and advanced even more technological proofs have been a way of demonstrating a mathematical statement to be true. As mentioned earlier proofs can be found in a number of different formats for a variety of problems but they all are for the same goal, to prove a mathematical statement, through a series of broken-down, basic, to the bone concepts. A successful proof will take the statement to a theorem which can even be used to in another proof to prove another more complex statement. For one of the more famous statements a2 + b2 = c2. The Pythagorean theorem we are able to prove this statements through a proof using geometry. By taking this equation and putting it into shapes it is clear that the statement is true.
By literally putting the squares on the sides of the right triangle according to there length you can see that the area of the a and b equals the area of c. proving these statements has allowed the ancient Greeks to put the theories and numbers to work in architecture. However proofs are not always going to be accurate. They might overlook simple details of knowledge. For example the parallel postulate by Euclid has come to be quite a contradiction and a great example of how evolution and the understanding of the universe can hander the accuracy of a proof. When you look at postulate five you can see that there is only one parallel line passing through a point outside of another line that is parallel to the first. This could be true as far as Euclids understanding of the universe but when you put it into modern contexts with the understanding of the universe and the earth as a sphere there are never two parallel lines with great circles. Great circles around the surface of the sphere will always intersect other great circles at exactly two points. This simple contradiction sh...