Have you ever felt a private question light up your mind, before noticing that very same question flashing all across society? Is curiosity a substance that invades from the outside; an unseen wave force of intellectual potential? We are not discussing ESP here, but societal consciousness.

Recently, I was thinking about the invention of calculus by Isaac Newton, and Gottfried Leibniz; trying to remember what it felt like when the concepts of limits, derivatives and integrals first began making sense. During my reverie, a question I’d never asked before came to me. Research revealed several other people discussing the same question, as if it were a new mystery. I found no references to it older than about fifteen or twenty years, an instant in the history of calculus.


The Calculus Wars

Isaac Newton and Gottfried Leibniz seem to have invented calculus independently, and at the same time. History awards Newton the credit, but most of us use Leibniz's notation.

While recalling the interrelation between acceleration, velocity and displacement, I had struggled to define the physical significance of the integral of displacement. A displacement function is a time history of the position of something, like a train, space ship, or planet. Its first derivative is the rate at which the location changes, called velocity, and the second derivative is the rate at which velocity changes, called acceleration. The third derivative, the rate at which acceleration changes, is conventionally called the “jerk.” The fourth derivative is called “jounce” (sometimes "snap") and the fifth derivative is called "crackle." Beyond that are several other whimsical titles.

The other half of calculus, the integral, can be considered the anti-derivative or the inverse of the derivative, in the sense of reversing the mathematical outcome. Take the integral of something and you get the function whose derivative produced it. When you take the integral of the displacement function you get something whose rate of change should be the displacement. But, what does that mean? What is the physical significance of the area under the displacement curve?   I was stumped.



Imagine a car entering a toll road at 45 miles per hour, and gradually accelerating at the rate of 5 mph every hour. This displacement function records how far down the toll road the car has gotten at any point in time. For example, after two hours, it’s at the 110 mile marker.



This is the first derivative of the displacement function, called the velocity, and shows how fast the car is going at each point down the road. For example, after two hours, the car is traveling 50 mph.


This shows the second derivative of the displacement function, the acceleration, which is a constant 5 mph, at each point.



This graph shows the integral, with respect to time, of the displacement function. Dr. Steve Mann has labeled this measurement, the absement, a word that combines absence and displacement. It shows the accumulative time-weighted distance we’ve traveled down the toll road. If driving on the road were a happy experience, perhaps this would be a measure of our accumulative happiness. If the road were particularly bumpy, maybe this would be a measure of wear-and-tear on the car.

Now we get to the stimulus for my original question about curiosity. A search of on-line resources found several people who have recently been asking about the integral of displacement. During the first decade of this century, that question was taken up by Dr. Steve Mann of the University of Toronto. He and a colleague invented a name, absement, for the time integral of displacement. Absement is a combination of two words, absence and displacement, for reasons that can best be grasped by reading some of Dr. Mann’s own work, but the point here is, there was no prior name for it, because no one had thought the question worth asking.

Steve Mann has been recognized as “the father of wearable computing.”


I am a faculty member at University of Toronto, (completed PhD from MIT, degree awarded in 1997).

You can get a feeling for the discussions about absement by perusing various web authors. Dr. Mann himself has invented a musical instrument based on use of the absement property in a fluid flow system. One of his students has won a science contest by demonstrating, or measuring, the property of absement. It is a new term. Your spell checker won’t like it. Even Wiktionary hasn’t caught up with it yet, although Wikipedia is credibly current.

Surprisingly, I turned up no references to anyone who had been asking about the nature of the displacement integral prior to Dr. Mann. I expected to get a clutter of academic, and text-book references dating back centuries, thanks to the Google digitization project. We can read the nineteenth century musings of Riemann on his Zeta function, but nothing about the displacement integral prior to this century. Many of the current references express puzzlement, many are dismissive, saying there is not likely to be any use for the idea, while a few, like Dr. Mann and his colleagues, find exciting innovation buried in the concept.


ECE516 is based on mathematical frameworks that use Integral Kinematics and Integral Kinesiology and the time-integral of distance


"Surveillance" is a French word that means "to watch from above", taken from the word "veiller" meaning "to watch" and "sur" meaning "above".

My lingering question is, what made this exploration of the integral of displacement pop into the collective consciousness over the last decade and a half. We’ve been doing calculus for centuries, and I’ve been doing it for decades. I don’t remember ever thinking about the integral of displacement before.  Were there a few people in prior ages wondering about it, but unable to collaborate until the Internet came along? Or, did something propel this too obvious question forward at this moment?


Imagine being 15 years old and educating Canadian astronaut Chris Hadfield about new advances in fundamental physics.

World's first musical instrument that makes sound from water.


For me, the question just appeared, leaving me wondering why I’d never wondered about it before. I found other people similarly puzzled about why they’d never considered it before. Some of them were fascinated enough to invent language, write poetry, inspire students, create new inventions, develop new theories, and even hint at arcane explorations. One of the characters in a story I’m writing has a theory about all this, but I can’t get into that yet, except to say he is controversial. I’m hoping he will consent to explain it to me.


Arcane Explorations, Indeed!

IPhysmatics and the Polyverse in a nutshell! Mobilis in mobili!


In the third post of this series I will write more fantastic identities related to our friends, the polylogs! (1) and by analytic continuation that equation can be extended to all .

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