I just took a look around my work area to see what the most useful thing has been over the past week. It has been a rather long week since I’ve been studying for an exam. There is nothing I hate worse than exams. They are the least practical way to evaluate real knowledge that I can think of, but that is another story. What I found to be most useful is How to Prove It: A Structured Approach, a book that does a good job of presenting a very important skill for anyone working toward a degree in Computer Science (or math I imagine). I really wish I had known about this when I was getting my Bachelor’s degree. It would have helped a great deal with the “introductory” courses in discrete math, computer science theory, and graph theory.
I have found that writing proofs, much like writing code, is kind of an art. Most either get it or they don’t. I have been lucky since I have always just got it when it comes to code. Unfortunately I’ve had to be able to write proofs as well. Proofs I did not get. Why is another story but I think it has a lot to do with the cultural innumeracy described in Innumeracy: Mathematical Illiteracy and Its Consequences.
How To Prove It is a book for those of us that did not come to proofs naturally, and I must say Daniel J. Velleman has captured on paper easily understandable incites that allowed me to build the new skill of writing proofs atop the well trodden skill of writing code. Basically if you grok code and the process of creating it all you need is to study this book in order to achieve the same status for proofs. For me this has opened up a very important area for potential application to my research. Having a real understanding of proofs has allowed me to begin reading math (and understanding instead of glossing over proofs), and math like it or not is the foundation of computer science. These of course have been the long term benefits of having purchased and read the book several months ago. What made it the most reached for item over the last week has been the four page summary of proof techniques conveniently listed in the back of the book. If you are looking for a great guide to mathematical proofs and by extension reading and writing math, this is it. If you’re trying to win an argument with your wife keep looking, these are not those kind of proofs (I can’t prove it but I believe those kind of proofs don’t exist).
For the last week being an excellent reference and most used item I’d rate this book 5 beers, but overall I’d have to say 4 beers. It is eye opening for those with limited proof skills, but the reference value could have been raised significantly by printing the proof techniques on the insides of the cover, or in some other easy to reach location.