Math 300: Fundamental Concepts of Mathematics - Spring 2005 - Philosophical Remarks

Instructor: Farshid Hajir                     Office: LGRT 1118                     Phone: 545-6015 (office)     

This page contains philosophical remarks. For more practical course information, go to the main page.

Goals for Math 300

I. Get some understanding and perspective on the general philosophy of mathematics from a mathematician's point of view.

II. Learn techniques of proof and the logic behind them.

III. Learn basic material for more advanced classes.

IV. Get some practice in speaking mathematics and in giving proofs in front of class and learn to work together in groups.

Some Words To the 300 Club

Is Mathematics an Art or a Science? It is neither and it is both. When was the last time you made your own mathematical discovery? You probably did more of this on your own as a kid than you have in years. (Think of the first time you convinced yourself "There is no 'biggest' number..."). In this course, you will have what is likely to be your closest contact so far with mathematics in the raw: Not the pre-packaged mathematics of typical high school Math and Calculus classes, but the mathematics which is the Science/Art of discovering and comprehending hidden patterns and structures. In a sense, this is a language class: the language of rigourous reasoning. The structure of the course, is what I like to think of as Challenging/Supportive. The intensity will be somewhat similar to that of a language class: methods, technical tools, and new ideas will be flying fast and furious during the large lectures and no detail will be too gory for inclusion. On the other hand, you will meet in a very small group for an hour a week with extremely talented and friendly undergraduate TAs who were in your shoes not so long ago and who are very eager to help you to understand and appreciate the material. The TAs and I will also be available on a fairly flexible timetable so there will be people you can talk to one-on-one about the material on an informal basis.

To use another analogy, think of this as a beginning class on flying an airplane. If doing mathematics is like flying an airplane, then in typical Calculus classes, you learn how to enter the cockpit, enter the coordinates for your destination in the computer and engage the Automatic Pilot, who then takes over and does everything while you sip your Diet soda. My main goal for this course is to take you through enough manual take-offs and landings that by the end, you have some kind of permit of basic skills competency for flying on your own. This permit will hopefully also serve as your ticket for success in future Mathematics classes such as Math 411, 461, and 523. At times, you might become so engrossed in the details of how to fly a plane that you will forget to look at the beautiful view. My secondary job will be to remind you to take your eyes off the instruments and look out the window once in a while. For me, and hopefully for you, too, mastering the techniques, which is not always easy, is well worth the fantastic vistas we will have from on high.

Some Maxims We Will Encounter

It's good to be confused.

It's easy to make bad mistakes. It's very hard to make good mistakes.

To learn to fly a plane, it's important to crash it a few times.

When we figure this out, we're gonna feel pretty stupid.

If you build the right definitions, theorems will come. Pay homage to the Theorems, but Worship the Definitions.

Sometimes when a question is too hard, you need to ask a harder one to see what to do.

As the course goes on, hopefully you'll contribute some of your own maxims.

When working on any type of problem, it's good to keep in mind Polya's Method:

Understand the problem --> Dream up a plan --> Carry it out --> Look Back.

page last updated 22 January 2005