"Lecture" for Week 1

Opening remarks

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. (I wrote that in 2003, when we used an online system called WebCT. When I taught the course again in 2006, that had been replaced by something called Vista, so I still needed to discover what is feasible. Now that has been replaced by eCampus, so there is still no need to update the sentence. Don't hesitate to make suggestions.) 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. Stillwell's book is more ambitious than Allen in this regard; since I haven't completely 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 a pplications far removed from the point of origin. For the first two 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.

Very early mathematics

What happened before Egypt? It is hard to know precisely about peoples whose written records were primitive or nonexistent. Perhaps the closest we can come is to look at isolated tribes that are still alive today. Here are three recent anthropological research papers, whose conclusions seem to point in opposite directions:

• P. Gordon, "Numerical Cognition Without Words: Evidence from Amazonia," Science 306 (15 October 2004) 496-499.
• P. Pica, C. Lemer, V. Izard, and S. Dehaene, "Exact and Approximate Arithmetic in an Amazonian Indigene Group," Science 306 (15 October 2004) 499-503.
• S. Dehaene, V. Izard, P. Pica, and E. Spelke, "Core Knowledge of Geometry in an Amazonian Indigene Group," Science 311 (20 January 2006) 381-384.

Clearly we could surf the Web forever, and I don't expect you to read everything I found interesting. But here are some optional threads you might like to pursue (maybe springboards to paper topics):

• I found the Inuit article above while searching for information on the Maya. Maya math is very popular in precollege education right now, so there is a lot on the Web. Type "Maya mathematics" into Google, browse and enjoy. (Don't feel obliged to read all 772,000 hits; even the relevant ones quickly become repetitive.) If you're in too much of a hurry to play the Web game, here are two good sites I noted down (in 2003): Michiel Berger ._. St. Andrews. (The second one has a link to a reference list (things on actual paper!).)
• The development of "culturally appropriate" pedagogy for minority populations is highly controversial. Although there is a huge difference between an isolated village in Alaska and a high school in Harlem, the Inuit article reminded me of this interesting debate:
  • Diane Ravitch, "Multiculturalism," American Scholar 59 (Summer 1990) 337-354;
  • Molefi Kete Asante and D. Ravitch, "Multiculturalism: An Exchange," American Scholar 60 (Spring 1991) 267-276.
• Here's another site that was brought to my attention: Geometry Step-by-Step from the Land of the Incas. Great music and pictures.

More about practical matters

Our textbooks

• Most books, including Allen's and the one by Katz used in 2006, are chronologically organized until they reach fairly modern times, whereas Stillwell starts organizing by field or topic earlier. So, the readings won't always be in synch. I'm advising you to start reading Stillwell's chapters on Greece as soon as you get the book, so that the Greek weeks (3 and 4) won't be overloaded.
• As the title of the book suggests and the preface says, Stillwell's intention is "to give a unified view of undergraduate mathematics by approaching the subject through its history.... Mathematics is the main goal and history only the means of approaching it." This, I hope, somewhat alleviates the temptation to elevate biography or dull facts over mathematical ideas. It also follows, however, that the treatment of history is rather selective (many topics and events are left out) and "whiggish". I didn't like Katz as a textbook for this course, but if you want to get a book on mathematical history for future reference, Katz is an excellent one. Here are some others.

eCampus

• Gradebook (empty so far).
• Discussion forums. So far I have created 3, one for the short bios and ones for discussion of the material of the first 2 weeks. Let's try to keep them organized; that is, please exercise some judgment about whether to post a message as a new one or a reply to a related old one.
• Please don't use the e-mail feature inside eCampus to send me mail. Use my regular e-mail address.

How to submit written assignments

Timing and placement

• Contributions for general consumption (such as your little autobiography) should be posted in the eCampus Discussions right away.
• Weekly homework and major papers (book review, etc.) should be sent to me privately. (Starting with Week 2, the weekly homework should also go to the grader. Homework is due by Monday noon of the week following the week concerned.) Do not post them in eCampus before the due date. On or after the due date they may be posted; should they be? Probably other students are not interested in your routine homework papers, but the book reviews will be of general interest.

Formats

Communicating mathematical symbolism electronically is a scandalous headache. Different students will have different software and expertise available, so I do not aspire to impose uniformity.

1. Paper mail and Fax should be avoided unless necessary. Occasionally you may need to send a hand-written diagram or (less likely in this course) a long hand-written calculation.
2. E-mail attachments will probably be the most common mode, whether you are mailing things to me or installing them in eCampus discussion pages. (Here is some information on how to do that. It refers to Wikis, but I think they work the same as Discussions.) Various formats are possible. I am very much a T_EX partisan, but I understand that many students will need to use something like Microsoft Word.
   • I've been advised that a good method is to use Word or some other word processor with an equation editor, and to export the file in .rtf format (more portable than .doc or whatever the processor's proprietary format is).
   • If you can produce a .pdf file, that is probably the most widely readable, apart from HTML.
   • You can make an HTML file. The mathematical typesetting available within HTML is primitive but often adequate (note the subscript on "TeX" above). There are editors that make the process easier (but I am largely ignorant of them).
   • Sometimes (especially in informal e-mail) it is enough to write plain ASCII text with the mathematics in pidgin TeX.
3. For various reasons, it may sometimes be better to install your paper on a personal Web page and e-mail the URL to me (and, when appropriate, the class).