Okay, it’s noon on Saturday. I started the morning by looking over the situation for mathematics and for the blog. Then I tried to draft this post. Then my kid interrupted and complained that he wanted to do some mathematics.
(For the blog, as for everything I write, first I write, and then I edit. First I get the ideas out, and then I dress some of them up. In the course of drafting this, I wrote “I want to sit down and do category theory right now, but I really need to put out this post. My kid will just have to wait, and he doesn’t want to…. Maybe he shouldn’t.”
So I’ve just put in some time on category theory.
I’m doing something similar to what what I did for Bloch (see bibliography). The method is the same, but the details differ. For Bloch I chose to read the entire book before returning to work individual chapters. For Lawvere & Schanuel [Lawvere, F. William & Schanuel, Stephen H. Conceptual Mathematics, A first introduction to categories. Cambridge University, 2000 (reprinted with corrections, twice). ISBN 0 521 47817 0], I read as far as I could before returning to work individual chapters. The main difference is that the later chapters of Bloch are familiar material; but by the time I had read 100 pages of the category theory, I had to go back. Too much information had not been assimilated yet. It was possible to go all the way through Bloch, but that doesn’t happen on new material. (It was possible to go all the way through Simmons, too [Simmons, George F. Introduction to Topology and Modern Analysis. McGrawHill, 1963 (reprinted by Krieger, 2003 ISBN 1 575 24238 9)]– I finished him at the end of last weekend — for the same reason: I had enough context and vocabulary in functional analysis for reading all of him.)
One advantage of having some fun before I try to write this post is that it may be a little easier to be casual. Thanks, kid.
There is a particularly relevant quotation I am particularly fond of. (I thought I had already put this out here, but I’ve searched for it and didn’t find it. In any case, it is relevant here and now.)
To take your work seriously is essential.
To take yourself seriously is catastrophic.
Dame Margot Fonteyn.
I want and need to take my mathematics posts seriously; but I do not want to take these diary posts seriously. The diary posts are the place, if not to outright laugh about things, then at least not to cry or complain or pontificate. I do not propose to tell you how to do mathematics; this is how I do what I do.
I almost got out a post during the week, but it kept growing. Even now, I have one or two questions to deal with. Still, one last color post — the last for a while — should go out Sunday evening. And, as I do every Saturday morning, I hope that in addition to finishing that post, I can write two more. It would be so nice to have next Wednesdays’s and Sunday’s posts already written before I return to work this week.
We’ll see. As I said, that hope springs eternal every Saturday morning. And it has almost never happened.
I learned a long time ago that if I feel uncomfortable as I leave the house, I have forgotten something. Now, as soon as I realize I am uncomfortable, the question — what have I forgotten? — and answer — Ah ha! I forgot that! — are almost immediate. A problem arises only if I don’t recognize that I am uncomfortable.
Something similar happens with blog posts. It isn’t so much that I am uncomfortable with a post, it’s more that I have trouble finishing it. I have learned that if I am putting off working on a post, there is something not right. Sometimes it turns out that I have said something wrong; sometimes it turns out that there is a better way to do something; sometimes it turns out that I’ve omitted something. (And sometimes I make a mistake and catch it long after I’ve published it, never having been troubled in the process. I’ve ‘fessed up to every mistake I’ve found.)
The challenge is to realize that I am avoiding working on the post. Once I recognize that, I can look for the problem in it. But it is not always — not often? — that I can analyze a post to figure out what’s wrong, or missing. Sometimes something just needs to percolate through my brain, and the best thing to do is something else, anything else, just not that post, or that mathematics. Sometimes there’s nothing to do for it except to sleep on it. Oh, sometimes it doesn’t take overnight; a walk around the block may be all I need. But put the post down. (I don’t have to put my hands up and step away from the keyboard, but I need to realize there’s a problem with the post and that I can’t force the issue.)
Although I have not written them up, I’ve answered two of February’s questions to my satisfaction: the PCA and the rotational motion equation.
I have some new questions — about wavelets — but I will wait a while longer before posting them. I’ve already computed the “Daubechies D4 scaling function” by three different methods, and gotten the same answers, and I don’t understand even one of the methods. Actually, I’ve computed it a fourth way — a few years ago — and every one of these three is much simpler to execute; I just don’t understand them. I’m delighted that they work, and I’m going to keep looking under the hood until I see why. Well, eventually I might ask for help out on a newsgroup or two. Or possibly at wavelet.org, since that’s where I got one of the algorithms. Or possibly — what a marvelous idea! — ask the authors.
Most other things have not changed. For aircraft control, I have mathematics to do — the actual differential equations — and I have blog posts to do — a magic formula for the angular momentum. For differential geometry, broadly defined, I have mathematics to do — Lie groups and Lie algebras, and an overview of the higher dimensional challenges in TOP-PL-DIFF (topological, piecewise linear, differential) structures — and I have blog posts to do — chapters 3 and 4 of Bloch.
Finally, in an application of mathematics to biology, I’m working on cloning myself.