The historical context and additional material will come from Katz.
In the process, we will learn something of the actual history of mathematics. One studies the history of mathematics for a number of reasons. For us, the primary reason will be to learn about the nature of mathematics and to learn to appreciate why it is in its present form. The nature of mathematics is an elusive topic, because mathematics may be described in various ways, such as:
By studying historical documents, we will observe how mathematicians have studied a variety of topics, created new language and notation, developed new concepts and structures, and employed various means of argumentation. By the end of the course, it is hoped that you will have developed a reasoned opinion on the nature of the mathematical endeavor and will be able to clearly discuss and rationally defend your opinion, based on what you’ve read.
| Number | Reading | Response Essay Topic |
| 1 | Katz, pp. 1 - 8. | Define the terms "grouping", "multiplicative", "ciphered", and "base", in the context of numeration systems (i.e., techniques for recording numbers) and give examples of systems that did and did not use each of these four principles. |
| 2 | Katz, pp. 8 - 14. | The choice of numeration system necessarily has an impact on the algorithms one uses to perform arithmetic operations. Illustrate this principle by showing how one would add and multiply in two different numeration systems. |
| 3 | Katz, pp. 14 - 39. | Ancient civilizations seemed willing to accept answers that were not exactly correct, if the results were reasonably close. Briefly describe four problems from the text where approximate answers are given and compare the ancient results with the exact answers. |
| 4 | Katz, pp. 46 - 96. | Compare and contrast Greek mathematics with that of the ancient civilizations with respect to the type of problems they studied (i.e., the goals of mathematical investigation) and the manner in which they approached their solutions. |
| 5 | Katz, Chap. 1 and 2. | One is limited in the types of problems and solutions one can consider by the complexity of the mathematical objects one is allowed to employ. In particular, both the Greeks and earlier, ancient civilizations were hindered in their mathematical investigations by their limited understanding of the "number" concept. Give two examples from each Chapter illustrating how people were forced to resort to various circumlocutions because of their limited conception of "number". |
| 6 | Katz, Chap. 3 and 4. | Compare and contrast the types of problems and methods of solution in Chapters 3 and 4 (of Archimedes, Apollonius, Ptolemy, etc.) with those in Chapter 2 (of Euclid, et. al.). What might this indicate about their views on the goals of mathematical investigation? |
| 7 | Katz, Chap. 5. | Finish the following sentence and give a brief justification for your answer: "Diophantus would have been most comfortable working with the Greek mathematicians from: a) Chap. 2 or b) Chap. 3 and 4." |
| 8 | Katz, pp. 192 - 210. | Describe two types of problems that the ancient Chinese could solve that the Greeks apparently could not. Conjecture on why the Greeks and Chinese seemed to approach mathematics differently. |
| 9 | Katz, pp. 210 - 232. | Describe what you consider to be the two most important advances in Indian mathematics, in terms of their superiority to Greek mathematics and their importance for mathematics' future development. |
| 10 | Katz, pp. 223 - 228. | The invention of the Hindu-Arabic numeration system was one of the most crucial advances in mathematical history. Describe at least three ways in which this system empowered mathematicians in their work. |
| 11 | Katz, Chap. 7. | Because of the Greeks' limited number concept, they could not pursue algebraic investigations in the same manner as we do in modern times. Describe how the Arabic concept of number was closer to our own and describe three of their most important algebraic advances. |
| 12 | Katz, Chap. 8. | Summarize the ways in which mathematical knowledge was preserved and transmitted from ancient societies to medieval Europe. |
| 13 | Katz, pp. Chap. 9. | Pinpoint the invention of as many modern, algebraic symbols and notational conventions as you can find in this chapter. |
| 14 | Katz, pp. Chap. 10. | Describe the sociological forces propelling mathematics during the Renaissance. |
| 15 | Katz, pp. Chap. 12. | Newton is often credited with the invention of differential and integral calculus. Argue against this proposition by citing historical examples, and identify three of Newton's key contributions to Calculus. |
| 16 | Katz, pp. Chap. 1 - 12. | Mathematics has flourished in "modern" times (since 1700) with mathematical knowledge growing exponentially. Based on your reading, give at least five reasons why mathematicians in modern times are better equipped than the ancient Greeks to make mathematical advances. |
| 17 | Katz, pp. Chap. 13 - 18. | Both the types of mathematical objects being studied and the manner in which investigations are carried out have expanded greatly in modern times. Describe two ways in which the collection of objects for geometrical consideration has grown. Likewise, describe two advances in the modern "number" concept. Finally, describe two ways in which we approach mathematical questions in a fundamentally different manner in the 20th century than in earler times. |
| Essays | 50 % |
| Papers (2) | 15 % |
| Final Essay | 20 % |