Annotations, Ch 70
Additive inverse-given some number x, an additive inverse is a number -x such that x + (-x) = 0. For example, -1 is the additive inverse of 1, since 1 + (-1) = 0.
Open and closed sets-again, not going to go into set theory, though this is pretty basic. You can sort of conceptualize them as yin and yang.
Spock contrasts between two different kinds of opposites. There are opposites that cancel each other out and produce nothing, and there are opposites that complement each other to produce something more than themselves.
“Friendship is a serious affection; the most sublime of all affections, because it is founded on principle, and cemented by time.” From A Vindication of the Rights of Women by Mary Wollstonecraft.
A note on calculus
“If one may speak of this as a calculus, the Riemann sum can only give an approximation of the integral. It will never yield the true answer until the size of one’s partitions vanish to zero-an impossibility to count an infinite number of infinitesimal moments.”
There’s lots of ways to think about calculus. One of the most basic is to think of taking the area under a curve. That is to say, you have some curvy line on a graph and you want to calculate how much area it covers from certain points, say a to b. Well, how do you go about doing that? We all know how to take the area of a square or a triangle, or even a circle, but how the hell do you find the area of some arbitrary blob?
Well, you can approximate. You can essentially divide up the blob into a bunch of rectangles of different sizes so that they kind of cover the whole thing. But any approximation is going to be off, since rectangles have corners while curves are completely smooth.
Say you have a curve, the parabola y = x^2, and you want to know the area under that curve starting from x = 0 to x = 6. You can use something called a Riemann sum, which takes a bunch of rectangles of equal width to approximate the area. So I might take rectangles of width 0.5 to end up with 12 terms (rectangles, or areas) in my Riemann sum and I get an approximation. It’s not very good-0.5 is a large interval. In order to get a better approximation, I can make my rectangles smaller, say 0.3, and get 18 terms in my Riemann sum. I can keep doing this until I get really really thin rectangles (I could go up to 0.0000001, if I really wanted to) and a whole bunch of terms in my sum, getting closer and closer to the actual value. But the sum of the parts still does not equal the whole. (wikipedia has good illustrations of this, as does any calculus text, I’m sure: wikipedia (dot) org/wiki/Riemann_sums)
Then how do I get the true value of the area under that curve? The answer kind of falls out of the way we’ve been doing our approximation. You keep taking smaller widths of rectangles until the width approaches zero. Remember limits? That’s what this is about. You can’t take the area of something that has width zero-it’s impossible. But mathematicians (well, Newton, who was mostly a physicist/alchemist, and Leibniz, who was a mathematician) got around that with the idea of the infinitesimal-something that’s so small that there’s no way to measure it. Of course, this means that you’ll have an infinite number of terms in your Riemann sum, but at this point you have what’s called an integral. And with some hand waving (ie magic) you get the real answer for this area under the curve you’ve been seeking.
I only slightly joke when I say that calculus involves magic. In a regular maths class, all of this is left as a vague idea of things that are really really close to zero (or whatever number it’s approaching), magically integrating functions, etc. When you get to analysis, you rigorously prove the ideas that calculus is based on by examining the properties of real numbers, doing δ-ε proofs, looking at the Fundamental Theorem of Calculus and all that good stuff. Read Walter Rudin’s Principles of Mathematical Analysis.
All right, fine, calculus is good, you’re messing around again with zero and infinity, sums of parts and wholes. What does this have to do with Spock.
Think of it this way. Spock’s relationship with Jim, with Leonard and Nyota, with the crew of the Enterprise-it’s like a curve. He’s trying to figure out when all of this happened, how did this curve form? What’s the area under it? Examining it in discrete chunks of time (look at it in little rectangles), he can see how it’s been forming, but it doesn’t show the complete picture. The sum of the parts does not add up to the whole. If he makes the chunks smaller and smaller, eventually he’ll get down to examining every second, but that doesn’t provide anything satisfactory either. Tiny moments isolated don’t have any meaning. They only have significance in the context of the whole picture.
So Spock is stuck. He can only approximate how, when, why this friendship progressed and grew, knowing that any answer he provides can never cover the whole experience. And what can he say of moments like this: “As we walk, his fingers brush the small of my back.”
Time passes-an infinitesimal moment, an everlasting interval.