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Sunday, March 6, 2011

About proofs in ( studying ) mathematics

Exercise, show that for $n>1$: $$\sum_{d/n} \mu(d)=0,$$ where $\mu$ is the Möbius function.

There has been a time when I did not like proofs. I suppose that was when I got first introduced to the theorem - proof - protocol. My goal was to study ( just ) enough so that I could do the exercises, or better ( read: worse ) questions from exams of the previous years. I must have complained about a lack of examples, I am sure. Going through all the compulsory proofs must have been hard because I never studied doing proofs as such. The idea that proofs can be beautiful was known to me but I thought I would never understand its meaning.

I liked mathematics a lot because I liked doing exercises. Think of the satisfaction the completion of a Sudoku puzzle can give you. Every completed exercise gave me a similar rush. It is as with Sudoku's: you know you will complete the game because, in essence, they are all the same ( at the same level anyway ). At a certain moment you begin to realize that doing exercises doesn't add anything to your mathematical abilities. You begin to search for ever harder exercises, preferably those that require a lot of study before you can complete them. And thus you can be sure of increasing math skills.

Reading ( read: thinking ) about new mathematics is interesting but does not let the subject sink in, digest into your system, make it part of your toolbox. - Doing exercises does. I suppose this is the principle of the books that present the theory in the form of ( almost just ) exercises. The theory is summarized, for example, in the form of a short list of theorems and then slowly, exercise by exercise you construct all the proofs yourself. If the solutions are included then books like that are priceless. It is, in fact, the method the Open University uses to present mathematics. The mathematics books of the Open University are not just like any other mathematics book.

Anyway, I made it full circle today... I know it is a wide open door: Doing mathematics is the best way to learn mathematics. But... use a consistent exercise set. If you notice you are cherry picking exercises that seem interesting you have fallen into the trap of an inconsistent ( or not hard enough ) set. In the protocol of mathematics a sequence of exercises for the reader contains a single message.

Exercises in a text generally have two functions: to reinforce the reader's grasp of the material and to provide puzzles whose solutions give a certain pleasure. Here, the exercises have a third function: to enable the reader to discover important facts, examples, and counterexamples. The serious reader should attempt all the exercises (many are not difficult), for subsequent proofs may depend on them; the casual reader should regard the exercises as part of the text proper.
(J. Rotman - An introduction to the theory of groups )

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