Today was the first "student lecture" of the semester, but before turning over the podium to today's speaker, Ricardo Gomez, Dr. Barsky opened by telling us that today is the birthday of Bernhard Riemann, a German mathematician who had a great deal to do with the rigorous development of integral calculus. The definite integral in calculus is actually the limit of what is now known as a Riemann sum.
Mr. Gomez began his talk by describing the social and political climate around the beginning of the common era. The Roman empire controlled most of the Mediterranean region and Roman rule brought a great deal of stability to the region. But the Romans felt that mathematical discovery had gone as far as was applicable, and therefore saw no need to pursue it any further.
Mr. Gomez focused most of the rest of his talk on two of the last Greek mathematicians: Nicomachus and Diophantus. Nicomachus wrote several treatises, the most famous of which was an introduction to mathematics. This introduction wasn't written for mathematicians, but rather for people with little or no mathematical background. It focused more on examples than proofs, and was commonly used as a text for teaching. Nicomachus had studied Plato's curriculum, which included music, and also wrote a manual on harmonics.
Mr. Gomez described a series of numerical relationships that Nicomachus found interesting. Nicomachus put labels on certain ratios of unequal numbers. For example, if a=bn for some number n, then a is a "multiple" of b. Conversely, if na=b, a is a "submultiple" of b. Further, if a=b+k, where k is some positive number less than a, then a is a "superpartient" of b, and if a+k=b with 1 ² k < b, then a is a "subsuperpartient" of b.
Nicomachus also separated the numbers into the continuous "magnitude" and the discontinuous "multitude." This was much the same as Aristotle's concepts of magnitude and number, respectively. Mr. Gomez also mentioned that Nicomachus showed that the concept of geometrical numbers (e.g. triangular numbers, square numbers, etc.) can be extended to any n-gon, and that numbers of the form "n cubed" are the sum of n consecutive odd numbers.
The discussion then turned to Diophantus, a Greek mathematician who did a great deal of work defining symbols for powers of numbers. Much of the work of Diophantus parallels what we now refer to as the laws of exponentiation. He also had a particular symbol for the unit, which, when given a coefficient, would be like the constant term in a modern algebraic expression. Then, following a brief discussion of Hypatia, one of the first female mathematicians, Mr. Gomez ended his lecture.
What I find interesting is Katz's discussion of Nicomachus's work in harmonics. Nicomachus wrote about how different ratios determine different harmonies. For example 2:1 produces the octave. In other words, if two chords, such as guitar strings, are made of the same material and have the same thickness and tension, but one is twice the length of the other, then plucking each will produce two notes which are an octave apart. I think it's incredible that 2000 years ago, people were using mathematics to produce music. Most people today don't realize this connection.
At the beginning of class we went through our usual mathematician of the day. Today's mathematician was Georg Friedrich Bernhard Riemann who was famous for the Riemann sum. Then the lecture was turned over to Ricardo Gomez. He started with a map and a bit of the history of what was going on at the time (-31 to 500). At this time, the Romans were in power. The Romans thought that they didn't need any more math. Ricardo then began to discuss mathematicians of the age. He started with Nicomachus who wrote a book of arithmetic for the common man. Next was Diophantus who was one of the first to come up with a notation for squares and cubes and a few other exponents. And last was Hypatia who wrote many comentaries of the work of earlier mathematicians.
The most interesting part of Ricardo's lecture was Diophantus and his notation. I found this interesting because it is hard to believe that there had been no notation for these things earlier. However, there was no notation for the higher powers. So with Diophantus' notation he could go as high as power six. I wondered, though, why he used a triangle to represent a square so I looked in a book and found that the triangle was the Greek letter for "D" (or Delta) which is the first letter in Dynamis (what Diophantus named a square). So I believe that is where the triangle came from.
The presentation given today by Ricardo Gomez, after Dr. Barsky commented on Georg Friedrich Bernhard Riemann whose birthday happened to fall on this same day in 1826, held to the main concept that some of the Greeks made a break away from Geometry and went to abstraction. The topics covered consisted of the Romans, Alexandria, Nicomachus, Diophantus, and Hypatia of Alexandria.
Among the mathematicians covered, Nicomachus interested me the most, primarily in his work "Introduction to Harmonics," covered primarily in the Katz text corresponding to this presentation. I am intrigued at how he analyzed a link between mathematics and music. As I recall, the Pythagoreans considered numbers as the basis for everything, and Nicomachus used the term harmonic with a type of proportion. I would figure that he would apply the term to the analysis of intervals rather than proportions, e.g., an octave equals eight units of pitch higher than a previous octave, and a fourth equals three units of pitch higher than one, etc. He did do as I expected with the use of the sixth, fourth, and the third which are the most common harmonies. He then set up the ratio six:four equal to three:two which gives us the fifth. I would think that the Greek musicians had studied music in the form of mathematical analysis before actually picking up the instrument and playing it. Because they thought in terms of abstraction, I believe they would sit around and contemplate their actions first before doing them which usually involved some type of mathematical thought. When they had finished their mathematical analysis of it, they would then try to play the instrument to confirm their analysis of the harmonies of pitch.
Today's lecture was about three mathematicians who lived during the reign of the Roman Empire (-31 to +500). The first mathematician was Nicomachus (-129 to -60), who worked on Euclid's Algorithm for Greatest Common Divisors (GCDs) and in other areas of math. The second was Diophantus, who worked with squares and cubes (and higher powers) of numbers. Lastly was Hypatia, the daughter of a mathematician and whom she later came to be regarded as even more intelligent than in the lore of mathematics.
What interested me the most was Nicomachus' ways to write GCDs. His ten ways of "classifying" GCDs was very interesting because I am currently working with GCDs in two other classes. It must have taken much time and thought to come up with the different ways to write A and B in terms of each of those five different ways. I say five since each form of A= had a B= counterpart.
The mathematician of the day was Georg Friedrich Berhard Riemann. The remaining information on the history of Greek mathematics was discussed and how Alexandria was the safe haven for mathematicians in a time that did not value higher learning to the same extent as in the past. Nicomachus and Diophantus were highlighted as well as Hypatia, one of the first great female mathematicians.
The Greeks were lucky to have Nicomachus write a basic text of arithmetic, the other authors could then build from there. Although I don't understand the nature of naming the unequal ratio's, I read in Katz that it may have something to do with music which does shine a small light on the subject.
The Student Lecturer (Ricardo Gomez) made an excellent presentation concerning final period of Greek mathematics. The lecture began by illustrating the setting in which the events to be detailed took place. This was, of course, one of Roman dominance, in which theoretical mathematics took a back seat to practical applications. The practical applications were more like calculating values for building roads, canals, etc., and the kind of intellectual approach found in the writings of Euclid are rare in this period(-31-+500) indeed. Gomez identified two areas that supported the furthering of mathematics (but I only wrote down one in my notes). One of these areas was Alexandria, which, of course, supported an impressive library that tended to attract scholars from all over. After setting this stage, Gomez elaborated mainly on three mathematicians of the times: Nicomachus, Diophantus, and Hypatia.
Nicomachus (~60-+~120) was the first of the mathematicians that Gomez mentioned. He is given credit for writing Egsagoge (Introduction to Arithmetic) which contained theorems but no proofs and was intended as a textbook. Nicomachus also acknowledged the ideas of "magnitude" and "multitude", roughly equivalent to "continuous" and "discontinuous" respectively. Nicomachus stated that "magnitude" dealt with "numbers that can be continuously divided indefinitely", while "multitude" dealt with the counting numbers. Finally, Gomez indicated that Nicomachus had noted that cubes are the sum of odd numbers.
The second subject of Gomez's lecture was Diophantus (c250), whom Gomez credits with coming up with a convention of notation to deal with numbers that essentially amounted to the use of variables. This notation system dealt with exponents, as well as variables and constants. Also discussed briefly was Hypatia (?-+415) who wrote "A Commentary on the Arithmetica of Diophantus," and "A Commentary on the Conics of Apollonius."
To me, one of the more interesting things found in this lecture is Diophantus and his use of variables , especially the fact that he considered numbers in powers higher than three. Of course, squares and cubes seem natural, almost, to human thinking. But what, in the human experience, points to a fourth dimension? If one thinks of powers geometrically, (as was common to the Greek way of thinking), then how does one consider an idea like cubing a cube? I know that it seems very simple for us to think in terms of fourth, fifth, even 125th powers, without considering their geometric equivalent, because our learning of these concepts doesn't involve a geometric treatment.