Understanding something.

I am revising my 2003 book (Doing Mathematics) for a second edition. There was lots of stuff I did not understand then, and I still don't understand, not to speak of the new stuff I don't understand. But...

An article I must have read 5 times, finally made lots of sense, at least 4/5 of it.

I discovered a nice way of putting much of it together. Right now it is in a three or five column chart, and I know that some of the stuff belongs in different columns. But I can fix it.

I finally figured out a way of describing something in everyday terms. I looked at the first edition, and I more or less said the same thing--but now I can be clearer, I hope.

There were various places where I had elided over difficulties. I did not realize I was eliding, for I did not appreciate the deeper issues. Now I can insert better explanations and details.

[Note that I am willing to believe that all my changes will not be much noticed by most if not all readers. But scholarly authors are usually an authority on what they are saying. Still, they might well be found out by a reviewer or a competitor. But often not!]

There are some bloopers in the first edition, but fewer than I was anticipating. I suspect that the prospect of going over something you wrote a dozen years ago makes one both vigilant and apprehensive. It's always better than I remembered, and surely there are places where the elisions are now realized, the mistakes now standing out.

My experience is that I understand something because I eventually find a perspective on that something that mirrors something I understand already. Not exactly, but close enough. What I need to do is to keep thinking about what I do not understand (better, a few things I do not understand), do lots of reading, and perhaps I will encounter an exposition that opens my eyes. I'd like to ask a real expert, but I am not sure how to do so: namely, what I want are the keys to the kingdom, and those are supposed to be earned by advanced degrees and having a member of the club show you the key closet. Club members may be pleased to be of help, but they don't know that they might know what I need. One day, I might be able to ask the right questions, and they respond readily and without pretense, and let me know what's up. But asking the right question only is clear after you know the secret answer.

Put differently, when I reread stuff I have written, I am often surprised by the quality of my understanding, it's that good. Not all the time, but often enough. I forget all the work I did then so that I could figure out what was going on.

As far as I can tell, this act of understanding something cuts across the disciplines, the sciences, the humanities, social sciences, the professional schools. Every field is esoteric to outsiders. Some are more forbidding, but it all depends on where you feel most vulnerable.