Today's discussion was a continuation of the last discussion on mathematics in ancient Greece. Dr. Barsky started off with a general overview of what was going on in the world during the fifth century B.C.E. He spoke briefly of the Greek Persian Wars and how they established Greece as a major political power in the world. Dr. Barsky also talked about the Peloponnesian wars, which pitted Athens against Sparta, the assassination of Philip of Macedonia, and the rise to power of Alexander the Great. Alexander was a student of Aristotle, who was, in turn, a student of Plato.
We then spent a little time talking about Aristotle. Aristotle, being a logician more than anything, stressed that mathematicians are more interested not in what is known, but rather how it is known.
Next on the agenda was the Pythagoreans. Dr. Barsky told us that the Pythagoreans found incommensurability a crisis. They felt that, when measuring things, there should exist a single unit that could be used in such a way that anything being measured could be divided evenly by this unit. They didn't deal with irrational numbers. The Pythagoreans differentiated between number and magnitude. Numbers were theoretical; magnitudes were geometric. Dr. Barsky then did a proof of the statement "the square root of two is irrational."
We then returned to the Pythagorean theorem, and a proof of it based on similarity of triangles. What intrigued me was how similar Euclid's proof was to Dr. Barsky's, yet at the same time how different the proofs were. They were both based on drawing a perpendicular from the side of the square on the hypotenuse to the vertex of the right triangle, and then showing that each rectangle created by this perpendicular had area equal to that of the square on one of the legs. But from that point, the proofs diverged, with Dr. Barsky's proof being very simple and Euclid's getting very involved and complex. The reason Euclid used such a difficult method was to avoid using the similarity of triangles. He had to do this because of the differentiation between number and magnitude.
Also because of this differentiation, the Greeks couldn't really merge algebra and geometry. But in modern times, algebra plays a great role in geometry, and vice versa. I think the accomplishments of many of the ancient mathematicians were incredible, considering what they had to work with. But I wonder how much more they could have accomplished had they been willing to deal with negative and irrational numbers.
Dr. Barsky opened his presentation with some commentary on James Joseph Sylvester, the mathematician born on this day, after which he proceeded to introduce an overall theme of Greek rationality in mathematics. Within the scope of his presentation, the topics consisted of James Joseph Sylvester, Pythagoras, the Greek-Persian Wars, the Battle of Marathon, the Peloponnesian War, the First Mathematical Crisis of Incommensurability, and a similarity based proof of the Pythagorean Theorem.
Among the topics covered, I became especially interested in Incommensurability. How could the Greeks label an irrational number as incommensurable? According to Dr. Barsky, in the case of the triangle with a hypotenuse equal to the square root of the value of two and each side equal to the value of one, the[2] Greeks determined that the side and the diagonal of the square obtained are incommensurable. This means that they could not write both segments as fractions due to the incommensurability of the hypotenuse. Dr. Barsky found that by setting the value of two equal to the value of the square of the ratio of the diagonal to one of the segments -- written as a fraction squared, i.e., (p/q)^2. He found that p and q had a common factor of two, which makes the contradiction that the square root of two must not be irrational. The Greeks wanted to use similarity because they could easily rationalize this, however they did not completely trust similar triangles after they found incommensurability. When I interpret this, I see that the Greeks did not have the mathematical knowledge that we have today, however they did have rationality. The Greeks had the rationality that the Egyptians and the Babylonians did not, and therefore the Greeks brought mathematics to a different level. This rationality probably facilitated the strength of the Greek empire, and the fact that we study the Greeks, both philosophically and mathematically, tells us that we might not have the same mathematical ideas today if the Greeks never existed. Each generation expands on the knowledge of previous generations. I couldn't imagine how difficult it would be to learn about triangles without the Pythagorean Theorem, and this theorem came from a generation that existed before the birth of Christ!
The main topics of the day were a brief political history of the times, the incommensurability crisis that faced the Pythagoreans, and a bit of Euclid's geometry. Right after Pythagoras died there were many wars involving the Greeks and many changes in rulers. During all this[3] the basic mathematical theory shifted from the Pythagoreans' number theory to Euclid's geometry. It is unclear as to what cause this shift, but it might have possibly been the incommensurability crisis. One good example of this crisis was the Pythagorean triple of 1,1, sqrt(2). Since irrational numbers couldn't exist in the number theory, they were just ignored.
It was Aristotle who came up with a way of dealing with irrational numbers. In class you said that magnitude and numbers were separated into two categories. I found this a hard concept to grasp at first. Upon reading Katz, I saw things more clearly. First, I discovered how the Pythagoreans defined numbers. That was truly interesting. Katz said their definition of a number was "a 'multitude composed of units'". They saw a unit as something that couldn't be divided or made smaller. Thus they didn't view 1 as a number, because they couldn't write it as a multitude of some smaller unit. This made it much easier for me to see how irrational numbers could be so hard for them to understand. Then I read about Aristotle's definition of number and magnitude. He used the concept of infinity without truly getting the whole idea. He could see how a line could be divided "infinite" number of times, it just depended on when a person chose to stop. Yet he didn't connect the idea of dividing numbers an infinite number of times.
Like last Thursday, Dr. Barsky gave a brief recollection of some of Greece's history, some of its rulers, and the famous mathematicians who tutored them. The majority of this lecture was about the crisis that arose when Pythagoreans, who were working with numbers, basically positive integers, encountered irrational numbers.
I have seen the "square root of two" proof before in other math classes and still I am unsure about it. How does it follow that p and q turn out to have 2 as a factor disproves that the square root of two is rational. I understand how proofs work, but the assumption that p and q have no common factors doesn't seem to me to have anything to do with the square root of two being rational or not. It seems irrelevant to the initial assumption. On the other hand, the part about the Pythagorean Theorem was very comprehensible because I enjoy Geometry the most of all mathematical subjects.
In this lecture, we learned that the first crisis in mathematics was incommensurability. The Pythagoreans wanted to understand rational numbers, not irrational numbers and to determine whether or not a number was rational or irrational; proof by contradiction was used. We also learned that Euclid proved the Pythagorean theorem but he did not use similar triangles because that was based on commensurability. We looked at the proof of the Pythagoren theorem using similarity. This concept was interesting to me because I have a difficult time seeing things geometrically. After it was discussed in class, I was able to see things a little clearer. The proof of the Pythagoren theorem had been shown already and Euclid insisted on proving it again. Did this type of thing happen very often? I'm well aware of the fact that a group of scholars may take a general concept from others and expand on an idea, but why prove the same thing?
Today we were introduced to James Sylvester; he researched Algebra and matrix theory. Greek conquest of the Mediteranean area afforded the building of a center for learning in Alexandria. Many mathematicians were attracted to Egypt. Euclid studied there. Even earlier, Thales and Pythagoras had visited Egypt.
Pythagoras spent time in Egypt and Babylonia. He established a"cult" of followers that were known as the Pythagoreans; the group was religious and philosophical in nature. The Pythagoreans were rationalist. They believed that the sides and diagonal of square were commensurable. A crisis of incommensurability resulted. The Pythagoreans believed the numbers are the basis for everything, including the basis for organization of the universe.