This presentation given today by A.J. Lundgren, after Dr. Barsky's commentary on today in math history, began to generate an overall theme of induction in European mathematics. The topics covered consisted of Dahshour Egypt, Al-Karaji, Levi ben Gerson (1288-1344), and proposition forty-one.
Among these topics, proposition forty-one had intrigued me the most, particularly in the set-up of the cube example. Lundgren started his proof by trying to show that (1+2+3+4+5)^2 is equal to 5^3 + (1+2+3+4)^2. After this set up, he proceeded to break the latter part down to 5 * 5^2 + (1+2+3+4)^2, which then breaks down to 5 * (1+2+3+4) + (1+2+3+4+5). He then proceeded to demonstrate how this applied geometrically to a 5*5*5 cube. He first showed how he went from 5^3 to 5 * 5^2, where the 5^2 equaled the area of the 5*5 squares on the base of the cube, and the remaining five equaled the height of the cube. After this he proceeded to demonstrate where the summations fit in. He drew a square on the board (remembering that the five represents the height), and split it up into sections of one through five and one through four. This demonstrated that five equals the height, (1+2+3+4) equaled sections one through four, and (1+2+3+4+5) equaled the sections one through five. From this one could conclude that you could add extra sections to this and use the same methods, meaning that (1+2+...+n)^2 = n^3 + (1+2+...+(n-1))^2 where n equals any number. I found this to be exciting, however it would be hard to keep track of mentally if n equaled a large number, as Dr. Barsky commented during this lecture. Although we have the notation we do today, the fact still stands that this mathematician had to deal with these summations without modern notation. How did they do it, when large numbers make it so difficult? I believe that they kept track of it some other place than in the brain, like on paper or a counting device, because of the difficulty. Even though medieval times had not been so deprived as primitive times, counting mechanisms seem essential. With a reliable counting system to keep track of the large numbers, one could accomplish a lot without modern notation.
Today's discussion actually began with more on Islamic mathematics. In particular, al-Karaji and an inductive argument that deals with a certain arithmetic sequence. He proved the formula for the sum of the integral cubes. What is interesting about al-Karaji's proof, is that he doesn't state a general result for an arbitrary n. His theorem is stated with 10 and works backwards to 1.
As we moved to medieval Europe, we learned about Levi ben Gerson. He invented the Jacob Staff which was used to measure the angular separation between heavenly bodies. He dealt with combinatorics in a major work of his, The Art of the Calculator. The most important parts of this are the theorems and their proofs.
As we looked at proposition 41, which was the proof of the formula for the sum of the first n integral cubes, we saw that Levi first proves the inductive step, this allows you to move from k to k + 1.
This lecture was wrapped up with two example problems. These helped me a lot because unfortunately I have not been introduced to induction. The proofs were difficult for me to understand, but A J cleared things up through his examples.
Dr. Barsky opened today's discussion by citing a newspaper article which stated that five pyramids built in ancient Egypt were opened to the public this past week. These pyramids include what is believed to be the first true pyramid, and the world's only rhomboidal pyramid. Dr. Barsky then introduced student lecturer A.J. Lundgren.
Mr. Lundgren focused most of his presentation on induction. He started by explaining a proof by al-Karaji that the sum of the first n cubic numbers is equal to the square of the sum of the first n integers. The proof involved constructing a square whose sides were equal to the sum of the first ten integers. The square is divided into a series of gnomons, with the largest being of width ten, the second of width nine, etc.
The first gnomon has two legs, each of which has a width of ten and a length that is equal to the sum of the first nine integers, plus a third part, which is a square with sides of length ten. If we assume that this area is the cube of ten, then it follows by reverse induction that the total area of the original square is the sum of the first ten cubic numbers. So we are left needing to prove only that the first gnomon has area equal to the cube of ten.
Mr. Lundgren presented a proof of this which was done by Levi ben Gerson. Ben Gerson was a philosopher, astronomer, and biblical commentator in the early fourteenth century, and his proof was also by induction. While Mr. Lundgren was giving a further explanation of induction, Dr. Barsky noted the facility of expression created by shorthand notation, and how clumsy it seems to have to use words to describe everything in mathematics.
After going through ben Gerson's proof, Mr. Lundgren showed by way of an inductive proof that the principle is true for any positive integer. Dr. Barsky commented that these mathematicians seemed to be showing that properties held for given numbers that had no special significance, with the assumption that the properties would then hold for any other numbers. Mr. Lundgren wrapped up his discussion by explaining other examples of proof by induction.
Katz talks about several other properties proven by Levi, including the associative and commutative properties of multiplication. Most of these proofs are done by induction, a method which is beautiful in its simplicity. I would assume that the reason induction wasn't introduced before is that most of the previous mathematicians put a geometric flavor on everything. They were dealing with what they referred to as continuous magnitudes, and, of course, when dealing with continuous sets, induction isn't an option.
A.J. talked about al-Karaji, his use of induction with numbers from 10-1, and his work with the sum of cubes. The mathematician and astronomer Levi ben Gerson's work with induction was also covered.
What I found most interesting was how this class and the Math 350 class have tied into each other with the discussion on induction.
Levi ben Gerson is yet another mathematician that has made contributions to astronomy. Levi ben Gerson is known for his invention of the Jacob Staff, which was used for hundreds of years to measure the angles between heavenly bodies.
A. J. showed us two different ways a form of mathematical induction was used. The first was by al-Karaji, an Islamic mathematician. He did a form of induction, but went backward from 10. He showed how if you have (1+2+2+3+...+10)^2 it was equal to 1^3+2^3+3^3+...+10^3. He did this by first showing that (1+2+3++10)^2 = (1+2+3+...+9)^2+10^3. From there he was able to prove that each time you took out the highest number in the square you could add it to the other side as its cube plus the square up to the number before, plus the cube of any number higher up to ten. In other words (1+2+...+10)^2 = (1+2+3+...+8)^2+9^3+10^3 = (1+2+3+...+7)^2+8^3+9^3+10^3 = ... = 1^3+2^3+3^3+...+10^3. Al-Karaji's form of induction works backwards from a fixed number, instead of forwards up to infinity.
The medieval European that dealt with mathematical induction was Levi ben Gerson. He lived from 1288 to 1344. He was not only a mathematician, but also an astronomer, philosopher and biblical commentator. He created the Jacob Staff, which measured the angular separation between heavenly bodies. He wrote Maasei Hoshev, which was one part theory and one part calculations. It was in this book that Gerson proved (1+2+3+...+n)^2 = n^3 + (1+2+3+...+(n-1))^2. He proved it by showing how the equation was true when n=5. The part of the equation that used induction was showing that n^2 = (1+2+3+...+n) + (1+2+3+(n-1)). With that knowledge when you are splitting up n^3 into n*n^2, you can substitute in the previous equation for n^2. Then you get an equation in the form x^2+2kx+k^2, which can be factored. I was wondering if Levi ben Gerson actually did the induction of n^2 like we did or if he showed it with numbers like he did in his proof of problem 41?
This lecture focused on the development of mathematical induction as a means to prove properties of integers. The work of two mathematicians was featured: one from the Islamic world and the other from from slightly later in medieval France.
The Islamic mathematician al-Karaji used a form of induction combined with the familiar geometric proof to show that the sum of integral cubes is equal to the sum of the integers squared. Unlike modern induction, al-Karaji did not state the case for an arbitrary integer n but showed it was true for the number ten, with the argument clearly being extendible to any other integer.
The next mathematician discussed was Levi ben Gerson, whose work dates from the early 1300's. Levi presumably learned from Arabic texts and would thus have been familiar with work of the earlier Islamic mathematicians. Levi's text the "The Art of the Calculator", from which A.J.'s example was taken, gave many important results in the field of combinatorics and featured the use of induction in many of the proofs (Katz). The proof presented was that the square of the sum from 1 to n was equal to n cubed plus the square of the sum from 1 to n-1. The presentation was a bit unorganized and hard to follow. I think he needed to explain the principle of induction a little more clearly for those not yet familiar with it and present his work a bit slower, but I know that's a lot easier said than done. It was also a little unclear how modern induction differs from the methods used by Levi and which was which during the presentation.