a few things about smallness

The other day I thought maybe it might be time for me to try calculus again.  I didn’t take it in high school the way I might’ve if I’d been born a bit more recently, because there wasn’t a year left for it after 9th grade algebra, 10th grade geometry, etc.  And then I didn’t take it in college because, I think, I was tired.  Or didn’t get along well with the teacher?  I can’t really remember.

But there it is, out there, this un-learned mass with the fancy name. One of the young people I know, with whom I have been doing math for several years because she likes it and doesn’t “need” a tutor but likes having one, is nearing the end of pre-cal.  She and I are really more like math buddies than tutor and tutee.  I help her when she needs help, but a lot of the time we find interesting problems to try to solve together. Now, there she is, about to start calculus, and not only will I not be able to help anymore, she’ll likely be too busy with her calculus to do math with me much anymore.

So I’ve been brushing up on my pre-cal, looking for gaps and rusty spots, and considering  calculus texts.  I’m a Martin Gardner fan, and my library had a copy of Calculus Made Easy, the Silvanus Thompson text with revisions by Gardner, so I tracked it down in the 515s and opened to the table of contents. Thompson’s first chapter is called “To Deliver You from the Preliminary Terrors.” I immediately texted the words to my mom, who is still terrified by most math and went to all sorts of lengths to avoid passing her fear on to her female child.

I didn’t have terrors, per se, but I can’t resist this sort of language.  In this chapter Thompson delivers us from our preliminary terrors by explaining that to learn the two principle symbols d, meaning a little bit of, and ∫, meaning the sum of all the little bits. (He mentions the fancy language as well, but uses the plain to explain.) The second chapter is called On Relative Degrees of Smallness. I will admit that I was so enamored of this title that I had to stop reading for awhile. But then I resumed.  Here’s how the chapter begins:

We shall find that in our processes of calculation we have to deal with small quantities of various degrees of smallness. We shall also have to learn under what circumstances we may consider quantities to be so minute that we may omit them from consideration. Everything depends on relative minuteness.

Here I had to stop reading again and stare out the window.  Who knew this was going to be a philosophy book?

I recently had occasion to look up the word pico.  I have always been partial to words that mean lots of different things and pico is, among other things, the Spanish word for peak (or beak), an island in the Portuguese Azores, an acronym for a method of  locating relevant clinical literature, and a prefix denoting one trillionth. 10−12.  0.000000000001. Very, very little.

The other night I heard about Erika Christakis’ new book The Importance of Being Little.  This book is about very young children, and the way the world view of such people could inform the way we older bigger people interact with them.

This is an awful lot about smallness in one week, I’ve been thinking.  A little hard to ignore.  Also, my father, who is recovering from a massive stroke, cannot remember things I have just told him.  To find the patience and presence to dial back a lifetime of conversational habit in order to sustain an exchange with someone who can cast backward and forward only a few moments is at once an enormous thing and a very very small thing.

I’m not sure how far I’ll get with the calculus, but I can’t help feeling that it’s a terrific thing every time we consider smallness in great depth.