The main topics of class were about a few of the first Greek mathematicians. First there was Thales of Miletus then Pythagoras of Samos. Thales is considered to be the first recorded Greek mathematician. Although we don't have any material written by him, Proclus wrote about him a little over a thousand years later using a supposed document written in 300 B.C. by Eudemus. Thales is given credit for doing some geometry proofs. He is said to have proven that the vertical angles of two intersecting straight lines are equal and that the base angles of an isosceles triangle are equal. Pythagoras is given credit for forming a way of finding all the Pythagorean triples using squares and gnomons.
I was always under the impression that Pythagoras created the formula a^2 + b^2 = c^2, but it would seem that it might have been the Babylonians who came up with it. From what we learned in class, Pythagoras came up with the way of finding triples that worked in the above mentioned formula. It would seem he did not come up with the formula, but came up with the numbers that worked in it without having to use the formula. His formulas associated the numbers to each other. I found it interesting that both the Babylonians and Pythagoras used squares and/or rectangles to prove and create their formulas. I never would have seen the connection between the Pythagorean theorem and sqaures and rectangles.
Dr. Barsky opened today's discussion by noting that there is currently a debate raging on the math history list arguing different possibilities of how the Egyptians found the volume of a truncated pyramid, ironically the main topic of discussion in the last class period. One writer stated that it's hard to come to any agreement because some 99% of the books written by mathematicians of Euclid's time were actually written closer to the present time than to the time of the author.
After the customary discussion of "this day in math history," Dr. Barsky talked about the beginnings of math in early Greece. Greece was divided into relatively free city-states and had a less centralized form of government than was the norm in the world at the time. This led to less of a strict hierarchy, and, consequently, more free thinking, especially concerning abstract issues. Where the Egyptians were more interested in computations and formulas, the Greeks were more interested in proofs.
The discussion focused momentarily on Thales. Dr. Barsky noted that Thales "proved" that vertical angles are equal, that the base angles of an isoceles[1] triangle are equal, and that the diameter of a circle equally divides the circle. We then discussed the Pythagorean theorum[2] and different "proofs" of it which, although they may not actually be proofs, certainly build a compelling argument. This led to a discussion of square numbers, triangular numbers, and oblong numbers, and the building of a formula for finding Pythagorean triples.
One thing that really struck a chord with me was the type of society the Greeks were. The less hierarchical, more federal system of government would obviously lead to more competition for status and power than, for instance, that of the Egyptians, where the Pharoah[3] was supposed to be descended from the gods. This competition would naturally lead to more use of logical argument and persuasion, which is the basis for mathematical proof.
Katz talks about Aristotle and his interest in logic. Aristotle said that while syllogisms allow new knowledge to be built from old, you need some sort of basis to begin; you need certain axioms, postulates, and definitions as a foundation. Euclid also recognized this, and started each book of Elements with a series of definitions and statements he felt were self-evident. The use of logic, I think, is the most important part of mathematics. Personally, I derive a lot of enjoyment from using logic, which is what draws me to math and computer programming, and I would have been interested in further discussion of the topic.
In this class we discussed a bit about where the Greeks lived and how it affected their society. Then we talked about Thales of Miletus, who was important in that he was the first mathematician to worry about proof. Next came the discussion about Pythagoras. There are many different ways to prove the Pythagorean theorem and we looked at a few of them. We also talked about the ways that the Pythagorans thought. Finally, we closed with the method of finding Pythagorean triples.
The most interesting thing I found with this lecture was the ending discussion about the Pythagoreans. I found it extremely interesting how they worked with numbers, especially how they looked at them visually. I have found that by looking at numbers visually you can see many things that you would not see normally. For example, the perfect squares form a square when you put the objects together. You can also see that any time you add an odd number with another odd number you always get an even number. While this would be difficult to see on paper and pencil, one could place three pennies and five pennies (or any other two odd numbers of pennies) and see that they will always form two rows of pennies of equal length. It is also interesting how the Pythagoreans would "play" with objects and call it math. When I finish the teaching credential program I hope to use this idea of "playing" with numbers to demonstrate mathematical ideas (like the gnomon).
Dr. Barsky began this presentation by mentioning a few comments about A. Cournet, the mathematics economist, and encouraging us to use the internet for Math History class. He then introduced us to the main concept of the importance of proving formulas rather than solving them to the Greeks. The topics of this presentation consisted of Proclus, Eudemus, Thales, Pythagoras, the use of triangles in China, Pythagorean triples, triangular numbers, square numbers, and the Chinese Gnomon.
Among the topics covered, I became most interested in how Pythagoras figured out that the gnomon equaled the difference of two successive square numbers. The Greeks have been known for their philosophical insight, and I believe that they maximized their use of this. According to the presentation, the Pythagoreans found the gnomon to be an odd number, and when using figures to represent numbers, the gnomon appeared in the shape of an L. The Pythagoreans could easily see that when they put a smaller square inside of a larger square, the difference appeared to be a gnomon. So when the gnomon equals a square number, the area of the smaller square added to the area of the gnomon equals the area of the larger square. In my interpretation, the logical sequence of this became the basis for Greek mathematical thought, however I do not believe logic alone could come up with these conclusions. Because the Pythagoreans believed numbers to be the basis for everything, the link between philosophy and mathematics came to life, a significant movement in mathematics history. However because of the Greek link to philosophy, the Pythagoreans came up with their theory of numbers as the basis for everything. Therefore the sequence of proving encompassed both mathematics and philosophy, which brought insight for others to look back upon, even today.
This lecture began with a discussion of the geographical and cultural/political differences between Ancient Greece and the civilizations of Egypt and Babylonia, and how these differences influenced the early Greek mathematicians to take a different approach to the study and development of mathematics. The city-states of early Greece were small and dispersed throughout the Greek islands and surrounding areas. This encouraged trade and the exchange of ideas among themselves and with other cultures. Unlike the Mesopotamian cultures that the Greeks learned from, theirs was not ruled by a strong central government with a priestly class that controlled the knowledge of the civilization. This encouraged free-thinking and the need to "prove" your ideas in order to convince others of their truth. The first recorded mention of an early Greek mathematician using logical arguments to try to prove mathematical ideas was Thales, who lived about -600. He is credited with being the first to prove theorems based on geometry.
The ideas of Pythagoras were discussed in some detail. The Pythagoreans believed all things in nature could represented by whole numbers and ratios of whole numbers. They studied the properties of what we would call the counting numbers, using pebbles formed into various configurations to represent numbers. Using these notions of square numbers, triangular numbers and gnomons, the Pythagoreans were able to develop a method of generating "Pythagorean triples", or whole numbers that represent the sides of a right triangle.
Greek Mathematics was the topic of discussion today. The social and political climate led to issues such as mathematics. The Greeks particularly concerned themselves with proving things.
Thales of Miletus (-600) and Pythagoras of Samos (-500) were discussed. There are four theorems that that were proved by Thales that cause[2] him to be known as the "originator of the deductive organization of geometry." Pythagoras founded a group of disciples that were later known as Pythagoreans. They considered number as the basis of the universe, studied properties of positive numbers and showed interest in the construction of Pythagorean triples. They proved these results from their dot configurations. Although the Babylonians had previous knowledge of what is now called Pythagorean theorem, the uses of these dot configurations and the results made this theorem more useful.
Although the Pythagoreans were a type of cult (kind of), I think it is neat how a group of people can come together and create the types of things that they did. Sure people do work together and develop things quite often, but they don't have the same beliefs, think the same, eat the same foods, etc.