M629 Lecture 1

"Lecture" for Week 1

I intend to start every week with a message like this one. A big advantage of the distance learning setup is that I am not obligated to fill up 150 minutes with talk if I don't have that much to say. Conversely, if I run on a bit long some week, you won't be late to your next class.

Let me 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. (The Vista system (formerly WebCT) has changed since I wrote that in 2003, so I still need to 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. Our current textbook is more oriented to ideas over biography than the on-line materials (Dr. Allen's notes) we were using in 2003. The Links page (when I get it updated) will include the Allen notes and the St. Andrews University site, an excellent source of biographical information.
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. Again, Katz's book is more ambitious than Allen in this regard; since I haven't read it myself yet, I'm eager to see how it works out.
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 week and a half 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.

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