Methods vs. working on problems

Sometimes problems can be solved by simply following a method. If I'm instructed to multiply two large numbers, I can follow the rules I learned in elementary school to get the answer, and if do each step correctly and in the right order, I will necessarily get the right answer.

Sometimes problems have to be solved with a kind of creative insight. We face problems that have no pre-ordained solutions, no methods for solving them. Professional mathematicians do this-- they try to solve higher-level conceptual questions that have no "plug-and-chug" computational methods for solving them. They just have to figure out the answers.

Now, we can't creatively figure things out on command. If my math teacher orders me to multiply two big numbers on the blackboard, I can do it. If my math teacher orders me to prove a new theorem no one has ever proven before... well, I can't just do it. But, importantly, she can set me to work on the problem. She can show me the problem and say, "Here, work on this and get back to me."

But what is my work? What am I doing when I "work" on a problem that has no method for solving it? I can't simply "be creative." Creativity is not something under my direct voluntary control. But does this mean that there is nothing I can do? No-- I can certainly try things that occur to me, and see that they don't work. But things must occur to me in the first place. Finally I may get the result (and there are objective methods for determining whether I've got the answer right). But it doesn't feel like some process I've worked out. It just... hits me. I might be out for a walk, or cooking, or at a party, or something entirely unrelated to the problem. But I still had to set myself the problem, to "work" on it. It would not have occurred to me if I had just blown off my math teacher's instructions and just gone out and done other things without giving it a second thought.

Now, what does this have to do with Zen? I think it would be helpful to keep this distinction in mind. If I keep pressing my math professor for a method, and my math professor keeps adamantly insisting that there is no method, wouldn't it be a mistake for me to conclude that she is telling me that no effort on my part is needed? When mathematician after mathematician reports (and this is true) that their big insights "hit them" while they were not at the blackboard or otherwise attempting to solve the problem, wouldn't it be a big mistake for me to think that they're telling me I never have to sit down to the blackboard or try to consciously solve the problem? Likewise, if master after master tells us that the "breakthrough" cannot be achieved through methods of practice, that their "insights" come through while they are doing something else like watering the plants or making tea, wouldn't it be a mistake to think they are telling us the effort and practice was not important and that we can just neglect them entirely? Now, we are sometimes told that the insight reveals to them the "wasted time" of practice... but then wouldn't a mathematician say the same thing when the proof finally hits him, and he looks back ruefully on the pages and pages and pages of scribbling in his notebooks and on his blackboard? All those fruitless avenues. Why did he bother messing with any of them? Well... would the answer have occurred to him if he hadn't?

Submitted August 18, 2020 at 08:07PM by Thurstein https://ift.tt/318P4qF