This course is one of several first-year seminars in the mathematics department. You are probably taking it for distribution credit, and this may be the last formal mathematics you will ever see. It is a chance to be a little bit philosophical.
Since this course is not a prerequisite for anything else, there is no necessity to "cover" any specific body of material. The result, somewhat paradoxically, may be to give a truer look at what mathematics is really like than a "real" mathematics course would do. We will have plenty of chances to reflect on the nature of mathematics as we actually DO some mathematics. This is quite a different activity from reading a textbook or listening to a lecture. It is more like getting hunches and trying to prove them; discovering a counterexample (so the hunch was WRONG!) -- and consolidating the result of experimentation in a new concept or definition. The course will consist almost entirely of mathematics we can actually DO, without a text. This makes it important, if the course is to mean anything to you, that you participate and stay engaged. I have therefore devised an unorthodox grading system to encourage this. If you look at it, you will see that it is straightforward to get a good grade in this course: just stay engaged and participate in each exercise. Isn't that what you expect to do anyway? Warning: this system does not permit you to put things off, then do them all at once at the end. The automatic consequence of that is a low grade. But I think you will see that the benefits of this system more than make up for its strictness.
Ancient mathematics was often very sophisticated, and a number of issues arose very early, so we will have a look at some of them in their original setting. In the modern period (i.e., since the Renaissance) those old questions gave rise to modern mathematics. We will look at work of Fermat, Euler, Gauss, Cantor, and others. It would be good timing if we could end with a famous theorem of Sophie Germain, which is relevant to the most spectacular number theoretical result of the last few years, the proof of "Fermat's Last Theorem" by Andrew Wiles.