Math 629 -- History of Mathematics

Georg Friedrich Bernhard Riemann
(1826-1866)

Fall 2000

Instructor: G. Donald Allen

Office: Milner 116A
Phone: (979) 845-7950
Office Hours: MW 3:00-4:00PM, or by appointment

Text: Lectures on the History of Mathematics

Book Report Due: October 1

Term Paper Due: December 1 Suggested term paper topics Coming soon!

Readings

1. The Origins of Mathematics - Due 9/1/00
2. Mathematic of the Indians North of Mexico - Due 9/6/00
3. Egyptian Mathematics, Problems 1-11 odd - Due 9/8/00
4. Babylonian Mathematics, Problems 1,2,4,7 - Due 9/11/00
5. Sections 1,2 and 3 of the Greek Mathematics chapter.
6. Read Section 4 of the Greek Mathematics chapter. Do Problems 2,3,4 - Due 9/13.
7. Read Sections 5 and 6 of the Greek Mathematics chapter.
8. Read Sections 7 and 8 of the Greek Mathematics chapter. |Do Problems 5, 6 - Due 9/20
9. Do Problems 7,8,9 - Due 9/22
10. Read Section 9 by 9/23. Archimedes was the greatest mathematician of antiquity. Of that almost everyone agrees. The results included are just a small sampling of the depth and power of this great mathematician.
11. Read Section 10 on the Helenistic Period by 9/27. Note the use of local coordinates by Apollonius. Even though at this time there was no xy-coordinate system overall, mathematicians used horizontal-vertical relationships - the symptoma - to describe the conics. Clearly(?), this anticipated analytic geometry that would take another 1900 years to appear. Note also, the challenge of the mathematical argument is strenthening.
12. Read Section 11 on the modes of algebra at the time.
13. Read Sections 12 and 13.
14. Read Sections 14 and 15. These sections are short and particularly interesting. They both pertain to "how we actually know" history. You will find that much of our information comes from very few sources and that much of it is pieced together. All in all, our knowledge of the history of ancient mathematics is relatively scant. There is ample room for new discoveries. Perhaps the right discoveries could alter fundamentally our perception of ancient mathematics.
15. Read Chapter 5 on Islamic mathematics. At the onset of Islamic power and domination, much of the ancient Greek heritage was ignored or rejected. Within only a century, the Islamic world develop a develoted interest in the great works left behind. They organized massive manuscript searches and tranlations. So effective was their work that it forms an essential link between ancient and modern western thought. Though they were not as committed to geometry as the Greeks, they made several significant contributions. In fact, they discovered another generalization of the Pythagorean theorem.
16. What is important to consider when reading are these topics. They comprise the thrust of the Islamic/Greek essay assignment..
    • Main thrust of mathematical developments
    • Centralization of learning
    • Comprehension of Exact vs Approximate
    • Definite presence of Pure vs Applied
    • Level of rigor
    • Perceived value by society
    • Receptiveness to new concepts
    • Active development of algorithms
    • Wide use of mathematical relationships
    • Importance of axioms and postulates
    • Level of philosophical underpinnings
    • Wide use and understanding of infinity
    • Wide use of sets and symbols
17. Read Chapter 6 on medieval mathematics. Look at the problems being solved. Algebraic, yes. But also problems of mechanics were of interest. Here mathematicians exceeded the great one, Archimedes. Note how mathematicians "dabble" with infinity. There was still a philosophical chasm to cross, and this would take several more centuries. It seems that mankind need time to become habituated to an idea before it can be embraced and then absorbed.
18. Read Chapter 7 on the mathematics of the Renaissance. Note the fundamental and practical need for mathematics had a considerable effect on developments. This meant practical algebra, symbolism, and more. Algebra in this time exceed by far that of the ancients. The general solution of the cubic was determined in the mid-16^th century. Surprisingly, this led to the notion of complex number. Complex numbers, then, arose not a some cerebric invention of theoretical mathematics but as objects needed to "clean up" the cubic formula. Interesting?? As well, the Ptolemaic world was collapsing. This meant better trigonometry and models of the heavens. The three great astronomers of the renaissance, Copernicus, Brahe, and Kepler, lived during this era. At this time the young genius, Galileo was following their work.
19. Read Chapter 8 about the Transition period to calculus. By the end of the 16^th century the European algebraists had achieved about as much as possible following the Islamic Tradition. Translating ancient texts took on a priority The desire here, beside the link with the past, was to determine the Greek methods. Among the great mathematicians of this period, we find Vieta, Stevin, Napier, Descartes, Galileo, and Fermat. Vieta produced symbolic algebra; Stevin used limiting processes; Napier invented logarithms; one of the greatest mathematical technologies, Galileo developed an analytical dynamics; Decartes created the philosophy which was to fuel the next century's mathematics; Fermat produced analytic geometry, calculus, and number theory which was to inspire the work of many. In this transition period we see (a) the beginnings of taking limiting process, (b) The emergence of symbolism and its consequences, (g) sophisticated mathematical arguments, and (d) the flourishing of number theory. Hear about this period. (To listen to this you will need the Windows Media Player.)
20. Read Chapter 9, Part 1. This begins a long chapter on the emergence of calculus. In this particular chapter we see the emergence of calculus and probability and the roles played by Blaise Pascal. Note how inventive he was, not just in mathematics, but in physics and technology of his day. You will also see the role played by the Dutch school, particularly Christian Huygens. Following we study the contributions of Pierre Fermat. The calculus he invents is very nearly what we learn and teach. He clearly deals with an infinitesimal, allowing it to cancel out at the correct moment. What is also significant here is the broad band of contributions of others toward calculus. These contributions were not enough to give them standing prominance in the history of the subject, but they impacted the likes of Newton and Leibnitz, who you must understand did not work in a vacuum. Hear an audio introduction to this period. (To listen to this you will need the FREE Windows Media Player. Note this recording uses a different codec than the previous one. So, if you don't hear it let me know and I will rectify the problem.)
21. Read Chapter 9, Parts 2 and 3. One very important idea to note from this chapter is that of indivisibles. Present throughout antiquity, the idea was brought to a mathematical techniques and used with some skill by Bonaventura Cavilier (1598-1642). Cavalieri created the so-called "Cavalieri Principle," still in use today. One question I like to ask students is whether they view the calculus developed in the context of indivisibles, as developed by Cavalieri, could have competed with the calculus developed through infinitesimals that we all know so well. Read also about the contributions of John Wallis and be prepared to compare the depth and scope of Wallis versus Issac Newton. While Wallis may have been the second most important English mathematician of the 17th century, he was far from first. However, and this is significant, not the power of mathematics of this time. Very new and interesting things were going on vis a vis calculus, but there was no coherent theory anywhere. Calculus is a mathematical tool that did not destroy old paradigms like we see so often in physics. It was a brand new tool of immense power, and its "birth pains" were considerable. In these chapters you will once again see the power of Pierre Fermant, where he demonstrates integrals using a "geometric" partition of the line. Very inventive... and so much like what we do today. A final remark is this: Note how power series were used to advance integration. Indeed, mathematicians were using them long before functions in calculus. At first this was surprising, but then understandable because so many other functions we know and use today are defined through their power (now called Taylor) series. So we are still decades from Newton and we see (1) infinite series and products, (2) integrals, (3) derivatives, (4) limits of partitions, (5) rectifications, etc, etc. So, what did Newton do, anyway?
22. Finally, read Chapter 9, Part 4. Here you will see the shear power of Newton and Leibnitz as compared with their contemporaries. You will note that these two mathematicians were eccentric in significant ways. But their abilities were without rival.
23. Begin reading Chapter 12, on infinity. Several levels, from philosophical to purely mathematical, can be encountered when studying infinity. In this chapter pay particular attention to the richness of mathematics that followed clear ideas of infinity. Note also how many millenia and paradigm shifts it took for mankind to finally grapple with, define, and work with infinity.
24. Finish reading the Chapter 12.
25. Read Chapters 10 and 11. In both chapters you will find the mathematical challenge heightened somewhat. However, much information can be gleaned from these pages. In the chapter on the Riemann integral notice how the "fixated" desired to sum Fourier series resulted in more and more complicated sets. In the chapter on algebra, notice how the power of abstraction, first developed for analysis transferred over to mathematical and particularly algebraic structures. Mathematics was never more to be example based. Be sure to read about how mathematics is used in cryptography.
    

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