| I think having a tool but not being able to know how it works can be both amazing and frustrating at the same time (depending on the perspective of someone using the tool vs someone wanting to make the tool better). Both perspectives are equally valid. As a concrete example, suppose an alien cube suddenly appeared on Earth that, with a push of a button, would emit a pill. It was discovered that the pill could cure half (but only half) of all known forms of cancer. However, no matter how hard anyone tried, it was impossible to understand how the cube or the pill it created worked. For those with cancer that took the pill and were cured, the cube was an amazing magical device. For the people that took the pill and it didn't help them, the cube was worthless, since doctors couldn't tell them anything about why it didn't work, or what they could do to make it work. For researchers it was frustrating because the cube showed some cancers are curable, but gave no reason why. For the cancers it did cure, they didn't know why, and for those it didn't cure they also didn't know why. Further it gave no insight into why only half were curable. Was it because the cube was imperfect, or was that really the best that could be achieved? For all intents and purposes, cancer research and treatment didn't really change. If you had cancer, you tried the magic pill. If it worked great. If it didn't, then doctor's were back to traditional techniques. Thus from a knowledge perspective of one day curing all cancer, the cube didn't help, because the knowledge of how it worked couldn't be expanded to cure all cancer. For someone cured by the cube, it was an amazing device, and nothing could change their mind. For a cancer researcher, the cube was useless, and the only thing that could change their mind was if its mechanisms could be understood. It is similar for math. As a tool, it is good enough to just know that the theorem is valid. However, if you want to expand on that tool, develop more tools, or create different ways of looking at the problem, you need to understand why the theorem is true. |