Post #6: In which I talk about the prettiness of proofs

Today another milestone was reached -- assignment 2 is now finished!

For better or for worse, chapter 3 (proofs) is now officially a thing of the past. Well, actually, this is not entirely true. Proofs will reoccur in the next chapter (algorithm analysis) and feature heavily in the rest of my math and computer science career. However, it does feel good knowing that the completion of assignment 2 marks the end of the proofs chapter. There is always comfort in closure.

Assignment 2 was very well-designed in that it required us to tackle a variety of proofs using a variety of different techniques. Going through the process of deciding which technique to use for what proof has helped me recognize the subtle differences in the expressions which make some directions easier to work with than others. Since the scope of the questions covered diverse topics from modulus, inequalities, to the greatest common divisor of a pair of numbers, I found that doing the proofs reinforced my understanding of these often (overlooked and forgotten, but also extremely important) mathematical properties and definitions.

In retrospect, the questions were not too hard -- although my group did spend a considerable amount of time on question #1. Oddly enough, we found this to be the most challenging question. We mainly ran into the trouble of finding a starting point in manipulating one side to look like the other. The proof statement looked easy enough, but the right "step" that sets the momentum of the proof was hard for us to discern immediately. Contrariwise, I personally thought that the algebraic proofs were the easiest, because the intermediate steps were straightforward as long as you saw the end product and tweaked things here and there to achieve what you wanted from the very beginning.

Having been exposed to a formal study of proofs, I now appreciate the creativity and beauty inherent in the process of proving even the simplest of statements. By writing down all the links and connections made in the body of the proof, you are acknowledging the sometimes disparate concepts and ideas from every corner of mathematics, and unifying them in a single context. As the proof-writer, you also play the dual role of a teacher and a student. You are, at each step in the proof, trying to convince both yourself and readers that your line of reasoning is correct. Indeed, until you  actually reach the end of the proof, each intermediate step could either take you to the next vital step, or deter you from reaching the conclusion (in which case you'll have to sit down and  brood and ponder and stare blankly at white walls until the answer comes to you). This is what makes proofs so interesting as well as rewarding; all the insightful "aha!" moments interspersed with the frustration of trying to see where things are leading make you at once vulnerable, and in-control. With this new-found zeal for proofs, I'll move forward in my studies with an open mind.