My thinking goes (as a non-mathematician) that mathematical insight appears to be more algorithmic, that is to say “systematic without the full abstract realization of the concept under study”.
But speaking as a non-scientist, I’m sure that I’m wrong. I can only envisage that “true” understanding comes to encompass a fundemental realization of the principles, in much the same way that you or I may shout “Eureka!!!” when we comprehend an idea or obtain a solution.
But how correct is this? Is mathematical understanding/perception inherently robotic in this way, and is true perception this “light going off in my head” reaction???
To draw upon on example, here’s Brian Greene mentioning it in Nov 2003 Scientific American issue (read the final paragraph on the page).
It should be noted here that I have read some of Roger Penrose’s work on the inner workings of the human mind and consciousness, and he explains some of the conflict between “mechanical” and “non-mechanical” thought which I am describing here.