mathematics and the subconscious
I have observed something very curious about studying mathematics. Ideas that seem very complicated and hard-to-grasp at first seem to make much more sense a day or two later, even if I haven’t done any additional studying. For example, when I first read about simplicial homology, it was very confusing and frustrating. But the very next day, everything just clicked, and it all made sense. It is somewhat difficult to describe how this change happened, but nevertheless I felt very comfortable with the material after sleeping on it.
There are two possible explanations for this, and I have seen different mathematicians arguing for both. The first possibility is that you just get used to the material. As the famous saying goes, “You don’t ever learn mathematics, you just get used to it.” While that quote seemed somewhat crazy to me 2 years ago, now I know why such an aphorism exists. There really is a process that most math students go through, where even very complicated ideas suddenly become very comfortable after a day or two. Proponents of this theory would hold that, where as upon first learning one is concerned with the reasons and justifications for the new ideas and theorems, one begins to shift focus on the results themselves rather than the reasons for them as comfort settles in. And by shifting focus to the “what” rather than the “why”, the topic seems “easier”.
The second explanation is that your subconscious works out details while you are sleeping, hence you have really understood the material better the next day, even if you haven’t deliberately studied. Apparently Poincare was a big proponent of this idea, and hence he never worked on problems for more than short spurts at a time. I think I have also read somewhere that Einstein was sleeping almost 14 hours a day when he was working on Relativity.
I think both explanations are plausible, and a little of both are probably true. One thing is for sure: cramming for math doesn’t work. And studying for extended periods of time also doesn’t work. Frequent breaks are needed to solidify the material, which is exactly what I am doing right now. 
