## Sunday, May 29, 2005

I feel like I'm in a minor rut. I was able to do some actual work today, though I gained no ground or progress. To sum it up, there seem to be three stages to reading a mathematics text:
1. A lot of smiling and nodding and page-turning, because everything makes sense. Usually no paper or pen is involved at this stage.

2. A lot of frowning and confusion, because what made sense before no longer does. Usually paper and pen have manifested themselves, and in particular, there are many pages covered with hasty diagrams and scrawls .. all of which contribute nothing to one's understanding (but seemed like good ideas at the time).

3. A lot of groaning and self-incrimination, because everything does, in fact, make sense. Usually by now extraneous ideas are set aside and one peers into the heart of the matter. Furthermore, two sub-cases occur: (a) the result is clear and you wonder why you didn't see it sooner, or (b) you were right all along but your paranoia caused you doubts and in particular, your counter-example is not a counter-example, after all.
I'm currently on Chapter 4 and Stage 2: what fun and joy, I tell you. q:

However, thinking about these various stages of learning and understanding, it makes me wonder how any of us got into the mathematics business. Are we mental masochists? Fanatics? Or is it that we have nothing better to do, and mathematics is the least of all evils?

Don't mind me; those previous questions were meant to be rhetorical. \:

As alluded in a comment to my previous post, I believe that I entered mathematics because I couldn't leave "well enough alone" and certain problems in analysis and geometry were so tantalizing that I couldn't let them go. Now I'm trying to make a career out of it.

I don't know about other students, but this inclination of mine causes untold frustration when I'm learning something, which at this stage of the game is always abstract theory. I begin to wonder how this or that definition came to be, and why anyone would ever think of such a thing. Textbooks often emanate an illusion that as presented, this theory is fact and for lack of a better term, "God-given." It has been written and taken this form. Understand it so.

It's easy to forget that books are written by people who have specific perspectives and ideas on how to view a certain phenomenon in mathematics, whether it is the structure of a particular space subject to its metric, or the nature of functions in accordance to their regularity. It's equally easy to forget that these ideas and intuitions take different shapes, from how they appear in print and under the full regalia of logical rigor. If there is one meta-lesson I've learned from reading mathematical texts, it is this:

Always draw a picture or work out an example, but be careful not to trust the picture or example too much.

I'm lucky to be in a department where my professors are willing to explain to me the motivations of how these theories were designed: as frameworks to overcome difficulties and issues in solving problems. They patiently draw pictures for me and show me how these matters reduce in simple cases, and my nerves settle.

All of this lead me to wonder: Will I ever be able to do this for myself? Moreover, if I want to be in the business of solving problems, then such skills are key, and I must be able to do this for myself.

I suppose there is no underestimating the power of negative thinking. \: