Memory is a funny thing.
I've spent vast hours this summer (enough time to have produced 174 handwritten pages of work and notes) studying for my real analysis qualifying exam, which I'll take three weeks from tomorrow. I have to take and pass two of these exams to be a PhD candidate. Of all of the exams available (real analysis, complex analysis, algebra, topology, and prob/stats), the word on the street is that the real analysis is the easiest (ha!) and the most memorization-heavy. Many of the other exams rely more on fundamental concepts that you have to cleverly apply to solve the problems (or so I'm led to believe).
I have a stack of the old exams going back to the 1980's. There are a lot of repeat questions, so I've been studying from the list I put together, which has the most often repeated questions of about the past ten years on top, followed by others that have appeared. These are hard questions. Many of the proofs take me 2 to 3 pages to write out, and I'll have to do 8 of them in 8 hours.
Studying for the exam has been difficult but also one of the most fulfilling experiences of my life. I mean that very sincerely. It's been amazing.
One of the amazing aspects, and this always amazes me, is that you can actually learn and remember things. I'm sure if you've taken classes this has happened to you - you had to apparently memorize some amount of material, and it seemed impossible. For instance, if you take calculus, you have to know all of these different derivatives (polynomials, trig functions, natural log, inverse trig functions, etc.) plus things like trig identities, if you didn't already memorize them in a previous class. It seems (to many people, at least) crazy, like a totally unrealistic expectation.
And yet I myself know large chunks of those things, and it doesn't even feel like something I have memorized so much as something that I just know. So I know that it's completely possible to learn that information and internalize it usefully.
The amount of stuff I have to memorize for this qual seems outrageous, but there are a lot of things that, during this past school year when I took the class this qual is based on, I knew I would never be able to remember, but that I now simply know - for instance, Hölder's inequality. When I first saw it, it was totally random garbage. Now, it's something that I know, and that I know some contexts in which it can be used, and so it is just part of my mind.
If you think about all of the things you know, it's actually an amazingly gigantic amount of crap. (Do you know drink recipes? Mathematical formulae? State capitols? Song lyrics? Phone numbers? Email and web addressses? Passwords for various web sites? Avogadro's number? The chemical formula for salt? The plot of "Anna Karenina"? The name of that one British actor who was in that recent "Hitchhiker's Guide to the Galaxy" movie? The details of your sister's divorce? It just goes on and on.) Even though new things sometimes seem random at first, and thus difficult to memorize, it actually seems virtually limitless, our ability to remember things.
The only thing more amazing than how much crap I can actually learn and remember is how easily I forget this, and thus how much despair I feel when faced with new piles of crap to know.