A project is a short research paper. The project descriptions (below) are just hints to look at some significant individual or incident. You are to research the topic, using several references (library work will be necessary) and write it up (with a bibliography) making sure that you put the topic in its historical context, i.e., if the topic is about something Archimedes did, you should report on it as well as talk about Archimedes and his place in history. Remember that all topics have antecedents (things that led up to the subject) and consequences (things that the subject influenced).
A project is defined by two parameters, time period and theme. There are four time periods, I - before 1478, II - 1478 to 1800, III - 1800 to 1900 and IV - 1900 to June 16, 1947. There are several themes which are listed in the table below. To determine a project you need to select a time period and a theme, subject to the restrictions that your projects must be chosen from different time periods and you may not repeat a theme. After making your selections, look up, in the table below, the appropriate project numbers and then go to the project descriptions to choose a specific project. Throughout the semester I will be adding projects to this table. You will be able to view them on the webpage.
|
Theme\Time |
I |
II |
III |
IV |
|---|---|---|---|---|
|
Women in Mathematics |
1 |
2 |
3,4,5 |
6 |
|
Non-western Mathematics |
7,8,11-16 |
9,17,18,19 |
|
10 |
|
Mathematics and War |
20 |
21 |
22 |
23,24,25 |
|
Mathematical Institutions |
26,27,28 |
29 |
30 |
31,32 |
|
Symbols and Notation |
33,34 |
35,36 |
37,38 |
39 |
|
Who gets the credit? |
40 |
41 |
42 |
43 |
|
The great mistakes |
|
44,45 |
46-50 |
51 |
|
Paradigm Shifts |
52 |
53 |
54 |
55 |
|
Potpourri |
56,57 |
58 |
59 |
60,61 |
|
|
|
|
|
|
Project Chart
The Themes
One of the reasons to study history is to gain insight on issues that we have to deal with in the here and now. The historical data points shed light on how these issues came to be and how they were dealt with in the past; valuable information as we move forward. In order to avoid a "scatter-gun" approach to the study of the history of mathematics, I've selected a few "issues", which I call themes, to concentrate on. My selection is fairly idiosyncratic, so I should say a few things about what they mean to me.
Women in Mathematics: A cursory glance at history, as taught in the west, would lead one to believe that we study only DWM's (Dead White Males), and this seems to be especially true in the history of mathematics. To counteract this DWM syndrome, we need to look at the lives of women who have contributed to mathematics and the difficulties that they overcame to do so.
Non-western Mathematics: Ditto the above with respect to "White". Mathematics is a discipline of the mind and as such does not respect the boundaries of gender or race. Great mathematics can come from anywhere and we need to recognize it without the filters of western civilization.
Mathematics and War: Mathematics does not exist in a vacuum. It is a human activity which impinges upon and is impacted by the culture which surrounds it. In this theme we investigate this interaction with humankind's most destructive activity.
Mathematical Institutions: Mathematicians are human and, as any group with special interests, tend to congregate with others who share their interests. These social groupings are often formalized in various ways and they permit complex interactions to be played out. Looking at some of these institutions gives a better picture of the social interactions which play an important, but often overlooked, role in the creation of mathematics.
Symbols and Notation: Mathematician's have developed a specialized language to discuss their work with each other. This language did not develop overnight, it has a history of its own which can give insight into the thinking process of those that developed it.
Who gets the credit?: Mathematics is rife with cases of people getting credit for something they did not do. How credit is assigned is part of the social aspect of the subject and, as these examples should show, it is a process which is far from infallible.
The great mistakes: The impression one gets from reading mathematical texts is that the subject grows steadily by building on past work. This is a fallacy engendered by the way the subject is presented. In doing "real" mathematics, wrong turns, gaps in reasoning and mistakes are the norm and not the exception. History can shed light on this process and correct our thinking about it.
Paradigm Shifts: As in the history of science, there are key points in the history of mathematics when a new way of thinking about a problem comes in and replaces the old thought patterns. This can occur rapidly, in which case we refer to it as a "revolution", or more gradually. Examining such shifts will give us a better idea of how mathematics grows in reality.
Potpourri: This isn't really a theme. I put this catch-all category in the list to reflect the fact that there is a great deal of mathematical history which can not be fit into neat boxes and yet still has great interest.
Project Descriptions
Hypatia (370-415 A.D.)
Maria Agnesi (1718-1799, Italian)
Sophie Germain (1776-1831, French)
Ada Augusta Lovelace (1815-1852, English)
Sonya Kovalevskaya (1850-1891, Russian)
Emmy Noether (1882-1935, German)
Horner's method for finding approximate solutions of polynomial equations as found in Ch'in Chiu-shao's book "Su-shu Chiu-chang" (1247).
The implications of finding Pascal's triangle in the 1303 text of Chu Shih-chieh, "Szu-yuen Yu-chien" (The Precious Mirror of the Four Elements).
The effect on Chinese mathematics of the publication in China of Ricci and Hsu's translation of Euclid's Elements ("Chi-ho Yuan-pen" [1607]).
The effect of the People's Cultural Revolution on Chinese mathematics.
Brahmagupta and the "School" of Ujjain.
The Golden Age of Bagdad during the reign of the first three caliphs.
The life and works of Mohammed ibn Musa al-Khwarmizmi
The solution of the cubic equation by Omar Khayyam.
Fibonnacci's role in the introduction of arabic numerals to the west.
The role of astronomy in the development of Indian mathematics.
The general nature of the literary activity in Mexico in the 16th Century with special reference to the need which produced the work of Juan Diez.
The introduction of Western mathematics into the East in the 17th Century.
The work, the general standing, and the influence of the Japanese scholars in the 17th Century.
The death of Archimedes and his role in the protection of Syracuse.
Tartaglia and his role in artillery science.
Napolean hobnobbed with the a number of the great mathematicians of France, who were they and what role did they play in his campaigns?
William Friedman spearheaded the effort to break the Japanese diplomatic code "purple".
Stanislaw Ulam and the Hydrogen Bomb.
Why did Norbert Weiner stop doing applied mathematics?
The Pythagorean Brotherhood.
The Library at Alexandria.
The earliest universities were established at Oxford, Paris and Bologna. What was the role of mathematics in their curricula?
Why was the ecole Polytechnique founded and who were the mathematicians who taught there?
Little effort was made to encourage the study of modern mathematics in the United States until Johns Hopkins University hired Prof. J.J. Sylvester.
Why was the University of Gottingen considered the mathematical center of the world for the first third of the twentieth century?
Trace the rise to prominence of the mathematics department at the University of Chicago.
What is the derivation of the name of the trigonometric function sine?
How were the Hindu-Arabic numerals introduced in the west?
William Oughtred used over 150 mathematical symbols in his writings, many of his own invention. How many of these have survived in modern times?
The invention of the symbolism for modern decimal fractions is usually attributed to the Belgian Simon Stevin.
Who advocated the use of the solidus ("/") to write fractions (as in 5/8)?
For many centuries there has been a conflict between individual judgments, on the use of mathematical symbols. On the one side are those who, in geometry for instance, would employ hardly any mathematical symbols; on the other side are those who insist on the use of ideographs and pictographs almost to the exclusion of ordinary writing. Trace this conflict by examining various editions of Euclid's Elements that are written in English.
What is a googol?
Who discovered the Platonic solids?
Who formulated L'Hospital's rule in calculus?
Who invented Steiner Triple Systems?
What did Pasch have to do with Pasch's Axiom in geometry?
Saccheri's “proof” of Euclid's Parallel Postulate.
Newton's classification of cubic curves.
Legendre's several "proofs" of the Parallel Postulate in the various editions of his Elements of Geometry.
Joseph Fourier's work on Fourier Series has been called "a classic example of physical insight leading to the right answer in spite of flagrantly wrong reasoning."
Cauchy's result on convergence of series that was fixed by Abel.
Kempe's “proof” of the 4 Color Theorem.
Riemann's incorrect statement of what he called "Dirichlet's Principle."
Wile's first "proof" of Fermat's Last Theorem.
Why is Thales called "the Father of Mathematics"?
Descarte's introduction of analytic geometry.
Weierstrass and the crisis in the foundations of analysis.
Godel's Incompleteness Theorems.
Who bought Archimedes' palmiset?
What is Plimpton 322?
Where did Newton get his data on the tides for the Principia?
Compare the constructions of the natural numbers by Peano, Frege and Dedekind.
What is the status of Hilbert's list of 23 problems?
- What was Edgar Allen Poe's contribution to mathematics?
These projects are meant to be interesting and enjoyable assignments, not chores, so choose your topics with care. Your report should satisfy the following constraints:
It must be a paper on the history of mathematics; it cannot be all history nor all mathematics.
It must be written in an expository style, with enough background material so that it can be read and enjoyed by itself. Test your paper for readability by asking a friend to read it. Form is important!
You should use many different research materials, from articles to books to the internet. Pay attention to your sources, especially to their reliability. Verify everything, document everything.
The paper should be done on a word processor, preferably one that can handle mathematics (Latex or if you must - Microsoft Word can). It is worth investing the time to learn how to do this.
Note:The source of this document is due to Bill Cherowitzo.