It’s odd, I know. But this morning I was sure that my right foot was home to six toes. Like many of you, I imagine, I fall out of bed—which is actually a couch despite my having access to a bed à la Fox Mulder—onto the floor. I then slump across the room to shut off my radio alarm clock before waddling over to the bathroom for a morning shower. However, this morning, after I had found the appropriate water temperature, I slipped my right foot into the tub. Because my footing is not especially sure in wee hours of the early dawn, I have to muster my full concentration not to fall. And then I saw it: a sixth toe. It was tucked nicely between my middle toe and I guess what would be called my ring toe. It wasn’t especially offensive, just, well, weird. I looked up and back down to refresh my visual sense. Yep, still there: six.
There was a problem with the way I was counting, though. I do this a lot. People think that because I have a degree in math, I’m naturally good at arithmetic. I’d like to dispel that rumor here. In fact, I’m going to let you in on a secret. You’ll be privy to some of the inner-workings of my mind. Most of the time, I don’t actually add or count: I see and intuit. Let me explain. When I was much younger, I used to stare down at the tiles on my bathroom floor for tens of minutes. Sure, it doesn’t sound like a lot, but you try it. It’s a lot harder to do than it sounds. Each tile measured about an inch and a half across and varied in color from a dark brown with a smooth finish to a rough, speckled beige and white. My current bathroom floor reflects a similar choice in design, except the palette runs across the blues. There in the bathroom, I’d sit on the toilet and stare down at the floor, trying first to focus on one tile, then a group of two, then three, and so on. Then I’d close my eyes and try to visual groups of dots. I’d work hard to make sure that I had the right number by arranging the pattern, rather than counting. I played other tricks, too. I still do. I try to focus only on one color or to spell things with the contours of the tiles. I wanted my visual arithmetic to become automatic. Unfortunately, I was unaware of certain natural cognitive impediments, and so, I must report that like most others, I was stuck within the realm of 7+/-2. (I could distinguish the numbers zero through nine; now I’m up to twelve. After that, the dots in my head become unwieldy.)
I do the same thing with written arithmetic—up to a point. I try to intuit the response, though it is much harder to explain just how I go about doing it. Especially this morning. The point is, I didn’t count my toes, I just recognized a pattern that represented six. (What is the number six? It’s six fire trucks without the fire trucks.) At last I resorted to convential counting. To my relief I counted only five. Moving among the toes individually had done the trick. So much for that Gestalt business. It’s too complicated.
But this is interesting. I had long been aware that my audio senses can become numb to stimuli. As a kid I discovered that if I repeated the word “enter” quickly and without pause it would temporarily lose its meaning even though I knew what I was saying. So often I forget that these sorts of rules ought to apply more generally. In fact, my sixth toe convinces me more of a fact (which is really a conjecture) that I stumbled upon the other night after tutoring introductory physics: training a computer to see and understand a scene in motion is probably a lot easier than training it understand a static scene.
Last Friday I met my tutee Katy, for the first time. She’s a pre-med post-back with an MFA in the visual arts. Somehow vectors and derivatives have been taken out of the art history curriculum. So we drew lots of position-time, velocity-time, and acceleration-time diagrams. And we talked about jerk (the slope of acceleration) and higher order derivatives, though we didn’t quite use that language. But what I stessed most is this: Absolute position doesn’t make a whole lot of sense. In fact, there’s a lot of ambiguity in the way we measure. Because of the ghosts of our coordinate systems, we should be careful only to pay attention to changes in quantities we measure, and not to the quantities themselves. If I’m here now, and five feet over there later, I’ll have moved five feet no matter if you started measuring my position from the fire hydrant across the street or from that sketchy all-night barber shop in the back corner of Beijing. It’s possible that your eye knows this, too.
I’ve since learned that some video compression techniques take advantage of change to keep the space it takes to store video down. If a pixel doesn’t change its color, why record it twice? Only report the things that aren’t the same from frame to frame. It makes sense. And when I attempted to learn the piano, my teacher said to watch the jumps an interval, not the notes. I bet computer eyes (if not your human eyes) can track trajectories. Most visual landscapes are complicated. Last night’s asparagus dinner occluded most of my dish at the beginning, yet I was not surprised to find it at the end of the meal once the asparagus was gone. My eye (and my mind) were able to reconstruct the plate even though I couldn’t see most of it. It’s like it was there the entire time. And so it was! Computers have a harder time with that sort of thing. How are they to know what to fill in behind or under or around? (I love using prepositions without objects. Where is it? Oh, it’s between.)
Anyway, I’d just like to report that my right foot only has five toes. That is all.