(Last modified Dec 13, 2002.)
The nature of mathematical objects has been disputed since antiquity. For example, how do we reconcile the apparent continuity of points on a line, which suggests that a line can be subdivided 'forever', with the seeming indivisibility of points on a line? Zeno's paradoxes are an attempt to express this puzzle in a way that exposes the problems that result.
Viewed as a question about the material world, the problem is susceptible to experiment, though of course scientific theories continue to evolve as new data emerge.
Viewed as a question about mathematical idealizations, we have no recourse to experiment to resolve the issues. The axiomatic method is an ingenious solution to this lack, in that it sidesteps the question of the ultimate nature of the objects we are studying, in favor of the study of their properties.
That is, we posit certain properties as axioms, and then show that other properties necessarily follow. It then no longer matters whether we agree on the nature of the mathematical objects we study: so long as we agree on certain of their properties, we will also agree on others. Further, we are able to isolate the properties which give rise to differences.
There is an unexpected benefit to this method. As new mathematical structures are discovered, if they have, for example, the properties which define a group, then they also have all the other properties which pertain to groups. In this way, we can build complex structures and study them without having to return to first principles at each step of the way.
The price we pay is that we must find reliable ways of deducing the consequences of our axioms, rather than relying on the intuitive understanding of one particular example at a time. This turns out to be more difficult and subtle than first supposed, but 19th and 20th century mathematicians made good progress on the problem.
This course, then, is an introduction to the methods of modern mathematics. When we have finished, one beneficial side effect will be a much greater understanding of some of the most familiar mathematical objects such as the rationals and the reals.
| Start | Topic | |
| ----- | -------- | |
| Sept 3 | Logic | Chapter 1 |
| Sept 17 | Sets | Chapter 2 |
| Oct 1 | Test 1 | |
| Oct 3 | Relations | Chapter 3 |
| Oct 10 | Integers | supplement |
| Oct 17 | Rationals | supplement |
| Oct 24 | Functions | Chapter 4 |
| Oct 31 | Test 2 | |
| Nov 5 | Cardinality | Chapter 5 |
| Nov 19 | Real Numbers | Chapter 7 |
| ------- | ----------- | |
| Dec 17 | FINAL EXAM | 25 State Hall |
First, READ THE BOOK . You should read each section before we talk about it in class, then again after class, before doing the homework for the section. If you have any trouble understanding it, read it several times, first, quickly for an overall idea what the section is about, then in detail, working out the examples the book uses to make sure you know why each statement is true. Only after this should you start the homework. You may be pleasantly surprised how much easier the homework is with this sort of preparation. You will certainly understand the material and retain more of it, if you study in this way.
Second, READ OTHER BOOKS . There are many books in the library which deal with the topics we will cover. You should get in the habit of reading as widely as possible on any topic you expect to understand. Other books will not apprach the issues in exactly the same way as our text does, and this is the point. You will see things from other perspectives, and will have to sort out the relations between the ideas, thereby making the ideas your own rather than a rote repetition of ideas others have told you. (Here are some recommendations.)
Third, discuss, and even argue about the material with your friends and classmates. Present your proofs to them to see if they are convinced by them, before trying them on me. Be merciless in your criticism of your classmates' proofs when they try them on you. If there is a claim, or a step, you do not understand, insist on explanation. If you think they are wrong, see if you can produce an example to show that their argument doesn't work. Apply these habits to yourself: that is, read your proofs with a critical eye and try to find any possible errors in them before giving them to me.
Fourth, come to class. Attendance will not be formally counted as part of your grade, but I can reliably predict that if you do not attend, you will will have a very small chance of success. Further, there will be material and discussion in class that is not in the text. You will be responsible for this material.
Your grade will be determined by your scores on your ten best homework assignments, worth 100 points total, on 2 in-class tests, worth 100 points each, and a comprehensive final exam, worth 200 points, for a total of 500 points possible.
| 10 best homework sets | 100 |
| 2 In-class exams | 200 |
| Final | 200 |
| ------ | --- |
| Total | 500 |
Makeup exams will not be given, since no one item contributes a commanding portion of the grade. If you have a legitimate excuse for missing an exam, it will omitted from the calculation, so that your grade will be based on a smaller possible total. If you do not have a legitimate excuse, your grade will be 0. In general, a legitimate excuse is one over which you do not have control and which you could not reasonably anticipate. For example, a late bus, or other exams the same day, can be anticipated, and would not be considered legitimate.
Paul Halmos, Naive Set Theory.
Thomas Körner, The Pleasure of Counting.