"Lecture" for Week 1

This is my first time to teach a history course, and my first time to teach a Web-based course, so please be patient if I am occasionally slow to make decisions. I should start by declaring some of my prejudices.

• Graduate students -- "distant" ones especially -- should be largely self-directed. You will sometimes need to interpret and adapt instructions to your personal situation. A large amount of the assignments will be labeled "optional" or stated in an open-ended way. No one student could possibly do all the things suggested. On the other hand, I don't think that a student who always does just the bare minimum required should expect a grade of A.

• A Web-based course should include significant electronic communication among the students, thereby both taking advantage of the technology and compensating for the lack of personal contact in a classroom. The precise modalities of this communication will be made clearer later, as I discover what is feasible. It follows from the previous point (about optional and imprecise assignments) that not all students will be doing the same things, and I encourage you to make your productions available for the rest of the class to read.

• Books versus the Web The WWW has some obvious advantages: Almost by definition, it is available to every distance-learning student, whereas not everyone has easy access to equally good libraries. Search engines make finding information (when it exists) very easy. Documents can be kept up to date (although this is not always done) when new information becomes available or mistakes are discovered.

  On the other hand, the information available on the Web is still only a small part of what is available in print. Documents online are hard to reference properly and even tend to disappear without warning. And since almost anybody can publish on the Web, without filtering by editors, etc., there is a problem of quality control. The reliability quotient is lower than for archival printed material. Of course, you can't believe everything you read in a book, either. One should be particularly skeptical, and diligent to separate fact from opinion, while reading about such an emotionally and politically charged subject as the early scientific discoveries of non-Western cultures, the topic of our first week of readings.

  One of the most productive uses of the Web is getting references to books and journal articles that one can then find in a library if they are not, themselves, available online.

• The history of mathematics is more important than the history of mathematicians. Biography can be interesting and sometimes instructive, but the development of ideas is central. Writings on the history of mathematics often tend to lapse into an obsession with "genius" that can be discouraging to students.

• Recent (e.g., 19th century) mathematics history is important and interesting, but hard to cover in a historical survey course because of the higher technical sophistication required to discuss it. I'm going to try to leave some room for it; this means going through some of the early chapters slightly faster than Prof. Allen's syllabus did.

• One of the most important things about mathematics and its history is the constant interplay between applications and math-for-its-own sake, or (something just slightly different) between the abstract and the concrete. A new mathematical concept almost always arises because it is needed to solve a particular problem. (The problem could be one in pure math or one in a practical or scientific application.) Later (with luck) the concept proves to be useful in a variety of applications far removed from the point of origin. For the first few weeks we will be studying applied mathematics, because that is all there was. Later the focus will shift, but we should always keep the technical developments framed in the broader scope of ideas and human life.

You will also write a book review on another book. That one should not be a general history of mathematics, but something more focused. The only requirement is that it should contain both history and mathematics at a nontrivial level. (More on this later.)

The bottom of the link page lists some places where you might find suitable books.

Homework: For the first week, the "problems" in Prof. Allen's text may make good discussion questions, but they don't strike me as very suitable for homework papers. Instead, in addition to a get-acquainted essay, I have assigned a more mathematical problem concerning magic squares. The first part is easy (essentially a writing exercise). The rest is harder.

In preparation for this exercise I need to make some commments on the text material about magic squares. It contains a number of misprints and omissions.

1. Most obviously, in the example on page 5, n = 5 (not 7) and q = n - 1 = 4.
2. The main formula in Theorem 1 is missing a term "+ 1" (clearly visible in the Maple code).
3. The theorem and the Maple code are both correct, but to make them precisely consistent one should let i and j in the theorem range from 1 to n instead of from 0 to n - 1, and delete the "+ 1" from the subscripts on the left of the main formula. The result of this change is just to move the first row and column to be the last row and column. [Why?].
4. If you have sharp eyes, you noted that the Chinese magic square on page 3 and Dürer's square on page 4 have the extra property that each diagonal also sums to the magic number! The algorithm in Theorem 1 does not have this property; it is understood that we are not including it in our definition of a magic square.