In any deep and complex endeavor, it is easy to make mistakes. Even after intense scrutiny, errors in logic and reasoning can be difficult to catch. However, when properly harnessed, errors can lead to new and interesting ideas.
Recently, a bright junior colleague made a mistake in a calculation, and I am sure that he was down on himself over the whole thing; so, I wrote him the following email (which I have modified slightly for you). After rereading it, I thought it would be good general advice to all students who are struggling with work at the boundaries of the unknown where mistakes are common.
Dear Student,
Do not be hard on yourself about such errors. We all make them. When
you have been around long enough to have made all the simple errors,
you move on to the more complex ones. As you mature, you will make
these transitions over and over again. The two of us are similar in the sense that we
perhaps make more errors than the average physicist. This can become a
strength if you use it to your advantage.
I believe that the tendency to make mistakes is associated with creativity. Mistakes take us into new territories that others may never imagine. Often, it's a mistake that leads me in new directions that brings me into uncharted domains. When my colleagues question me about how I ever even
thought of doing X, I can't explain it. Now that I think about it, I
should answer that it was a string of errors that led me to X. These random meanderings avoid the huge walls that block the straighter paths.
The key is to work hard and tirelessly in the pursuit of entertaining lots of new ideas, be slow to
publish so that you can catch your mistakes before they become public,
and allow yourself times of unrestrained creativity but temper those times with
disciplined thought. And most importantly, do not get caught up in
mathematics without thinking about the physical consequences. It is
the physics that is the most fulfilling and the best guide through your
intellectual hardships.
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