Nothing limits you like not knowing your limitations.
We spend too much time showing how well things work, and not nearly enough time figuring out where they don’t. Progress is there to be made by systematically exploring the limits of applicability for methods rather than show them off on cream-puff problems. Despite this, the reward system is hopelessly tilted toward the demonstration of methods through relatively trivial tests.
I mean that gods do not limit men. Men limit men.
Exposition in science is quite success focused much to the detriment of everyone. Failures are not tolerated much less expressed in many structured ways. Numerical methods are no different. In writing, people show demonstrations of their methods working, but rarely how the methods fail. This is quite unfortunate because it is limiting progress. For a numerical method people will typically simply solve a few standard test problems show results and call it a day. Rarely does anyone discuss how things fall apart or describe what the limitations on a methodology are.
Only those who dare to fail greatly can ever achieve greatly.
I have always found that the best way to make methods better is to continually break them. I will routinely torture methods to the breaking point fully knowing that “breaking” is itself complex. Methods break in many ways starting with a failure to converge to the “right” solution, or converge at the right rate, followed by a failure to converge to a solution, followed by a loss of stability. The improvement comes from understanding the cause of the failure, and changing the method to expand the range where a better outcome can be achieved. The process of failure is often demonstrably Edisonian, and the challenge is to provide the scientific, structural explanation for the failure to blunt the purely empirical edge. This tension is how progress and knowledge grow, and failure is the engine.
I have not failed. I’ve just found 10,000 ways that won’t work.
Failures are the engine of success. This is widely known and acknowledged, and despite this failure is not encouraged because of superficial fears regarding perception. As a result we have come to accept mediocrity as success, losing almost any conception of what true success looks like. Our success is almost by fiat rather than achievement. We will ultimately pay the price of this orgy of over-evaluation unless something changes in how we view things.
Not knowing that we learn from our mistakes and failures is perhaps the biggest one of them all.
Have no fear of perfection – you’ll never reach it.