Problem-solving

First I would like to discuss some quotes from Feynman. Here is the first:

Know how to solve every problem that has been solved.

What is Feynman trying to say? He means understand the solution to every problem that has be solved so far. Well, to most problems that have been solved. The way I see things so far, physics can be seen in some ways as a collection of canonical problems in different disguises. For example, the linearisation of non-linear equations of motion. Some of them admit solutions that involve harmonic oscillations about a certain point. In this case, the harmonic oscillator is the canonical problem. Even though this approach does not gives us the complete equations of motion, it still gives us some information about this small oscillations. And of course, the importance of the (classical and quantum) harmonic oscillator cannot be overstressed.

I do not think I have the capabilities of ever learning how to solve every problem that has ever been solved. But I guess it is a nice goal to set in life, unrealistic as it is. Is it worth learning all that stuff? Well, one can imagine that if we adopt this task, it will be eternal; new problems are solved everyday. In fact, in mathematics it is customary to re-solve problems with different approaches until the most elegant and simple way of getting the results have been reach. But life is all about learning. One will never know enough. In my opinion this is true, only arrogant will believe that they will know enough one day that they will simply stop learning.

Another quote from Feynman goes like:

The worthwhile problems are the ones you can really solve or help solve, the ones you can really contribute something to.

Now Feynman is trying to tell us that some problems are more worthwhile than others, based on their resolvability. I think this is a comment address to individuals. In my opinion it says something like "Know when you have more than you can bite". That is, everybody has their talents and it does not make much sense to approach problems where you will just stare at it silently for long hours with no idea at all. Now I am not talking about classroom-type problems, but in general physical questions. For example it could refer to some sort of experiment that cannot be carried out for some particular reason, or simply a calculation that is really out of ones league.

I am not saying that one should not try something hard and just comfort on the easy problems, whatever those are. The thing is this too is individual. Some people find some problems easy and others hard and some other people do the opposite, so in the end it also depends on interest. What I am trying to say can be explain better through an example. Say I am a theorist, and I want to calculate something that involves an application of a branch of mathematics I really do not know much about. In this case learning the new math will come as part of the knowledge package that comes during the process of solving the problem. But if I am really bad with that particular branch of mathematics, then depending on the degree of badness one can try and try and someday solve the problem or just be stuck forever on something that simply is not your stuff.

Maybe, just maybe, that quote was also referring to Feynman's view on string theory or quantum gravity in general. Maybe Feynman wants to tell us that there are some questions that really do not matter. Personally I do not think that the problem of quantum gravity does not matter just because it might not be pertinent in life. The truth is that there are also other open questions in modern physics that are more "basic", like understanding turbulence. It is "basic" in the sense that the problem is of classical origin, and has been around for a long time. I recall reading something along the lines of "There are other exiting problems in physics other then quantum gravity, like understanding the flow of water through a pipe". This is true. But I believe that turbulence is not an easy problem so in my case it is not worthwhile ;-).

In the end there are no such things as easy or hard problems; all problems require a consideration of certain ideas and situation that sometimes happen to be the most familiar to the person trying to solve them. For example, in my Quantum Mechanics final exam I could not answer one question about eigenfunctions of the quantum harmonic oscillator. Of course as an undergraduate student I went through all the derivation of the solutions (which apparently I forgot). Was the problem hard? No, I just happened to forgot a particular set of information, namely a set of symbols that represent the energy eigenfunctions for the quantum harmonic oscillator in an arbitrary energy level. It was not just a set of symbols, but I am still bitter about it ;-).

Some time ago I started another blog, The problemsheet. My intentions to start that blog were (1) to share with others solutions to some of the problems I encountered as a graduate student, and (2) to practice my typesetting with LaTeX.

In the back of my head I had some other intentions, like pursuing my own version of Feynman's goal and trying to show "someone" that I can solve problems. Maybe I wanted to be a smart ass like the people who display the solution to famous textbook problems online. I guess most of them do not want to show-off, they are just trying to help other students. That is particularly my first intention, but I know that in the back of my head I also want to show-off the grandness that I sincerely do not have. Nevertheless I comfort myself by thinking that by typing this solutions I am actually going through the problem again from the start, following it and looking for mistakes.

Another point that is also related to teaching is the fact that sometimes it is not easy to present one's results in a clear way without omitting some important steps. When I write down my solutions to assigned problem sets I try to be as clear and explicit as possible. Maybe it is the mathematician in me (who does not help me that much, that bastard...), or maybe it was the product of my undergraduate professors. I notice that fellow graduate students are not as explicit as I am in their solutions. I guess I want to be as clear as possible that (I think) I know the stuff I am talking about and am not fooling anybody. Nevertheless, I have been able to fool myself a couple of times.

Do people really expect students to solve every problem that is thrown at them? Now that I look back, I think my professors did. And I wonder why I was not able to solve every problem from the problem sets. Most of them were easy, in the sense that they involve some application of the stuff that was discussed in class. I do not think I encountered any evil problem during this two semesters, evil in the way that it requires some amount of genius to realize some not-so-obvious pattern or what-have-you. But as a professional, am I supposed to solve all the problems I encounter in life? I am afraid that a negative answer might imply some relying on others or something like that to avoid responsibilities. But after all, are we not working on a community? Well, I guess I tried being independent during this past semesters... but I suppose most people discuss their problems with somebody else.

I would like to mention something attributed to Gell-mann about Feynman's problem-solving algorithm:

1. write down the problem;
2. think very hard;
3. write down the answer.

I am going to adopt this mantra from now on.

One last thing I would like to address. Solving textbook problems does not make me great. After all, their purpose is to teach and get some ideas across. All this problems have been solved countless times by many others. So me and my little blog about problems does not makes me any better than anybody else. I am eager to start working on research problems, the type of problems that nobody has solved yet. This is what science is about in the first place, to understand the unknown.