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The Numinous Shore of Partial Understanding
A Scenic Ride with the Base of Natural Logarithms (August 25th, 2009)
(Revised, January 16th, 2010)
"Any sufficiently advanced technology is indistinguishable from magic."
Arthur C. Clarke.
Long ago, I came across a number that I named the Maw of Uncertainty.
The Maw of Uncertainty lies between zero and one. If a certain kind of uncertainty were missing from our world, the Maw would disappear: it would become zero. In the world as it is, you can never quite get to the Maw of Uncertainty. As you'll see, you can only approach it. The Maw is an
irrational
number: it can't be expressed exactly as a fraction or decimal. Without further ado, I'll tell you that the Maw of Uncertainty is approximately 0.36787944123356.
How did I first notice the Maw of Uncertainty?
I was hitchhiking on Route 1 north of Santa Cruz, guitar in hand, San Francisco bound. I wondered how many cars would pass before I got a ride. Suppose the likelihood of any given car picking me up were one in a hundred. Could I then be sure that, among the next 100 cars, one would pick me up? No. Consider simple cases: If the likelihood of any given car stopping were one in two, I'd probably get a ride from one of the next two cars, but there'd be a .25 risk that I would not (1/2 * 1/2). If the likelihood for each car were one in three, I'd probably get a ride from one of the next three cars, but there'd be a roughly .296 risk that I would not (2/3 * 2/3 * 2/3).
Assuming the odds that any given car would pick me up were one in 100, what was my risk of not getting a ride before 100 cars passed by? That calculation (99 to the 100th power, divided by 100 to the 100th power) is a bit hard to do in your head. The simple examples suggest that as the number of cars in this formulation increase, the risk of not getting a ride before that many cars pass increases as well. How large could the risk get? In mathematical terms, what is the limit of (n-1 / n) to the power of n as n approaches infinity? Keep in mind, as 'n' increases, we're decreasing the odds for each car, but we're increasing the number of cars proportionally. You might think these two factors would cancel each other out perfectly, but they don't. As 'n' increases, the risk of not getting a ride increases too, albeit slowly, approaching a limit.
In due course a pickup truck provided an invigorating open-air ride with two brother hitchhikers and a drooling Golden Retriever, none of them vexed by the vicissitudes of variability. We shouted and barked in cheery conversation, yet somewhere on the ride an idea emerged from the vast and weighty deep: the limit I sought might have something to do with
Euler's number
, "e," base of natural logarithms. A strange thought indeed, since I nearly flunked trigonometry during my goof-off senior year of high school.
Turns out the limit I was looking for is 1/e. Euler's number is about 2.71828. What's your risk of not getting a ride before n cars pass by when each car has a 1/n chance of picking you up? The risk rises above .36 (or 36%) at a fairly small number of cars (24), but it never reaches .37 (or 37%), even with ten million cars, each with a one-in-ten-million chance of picking you up.
Click here
to see for yourself.
How does Arthur C. Clarke's observation relate to all this? Well: I was amazed by this sliver of mathematical order, and the universe -- wondrous enough from the back of a pickup on Highway 1 -- seemed all the more divine. My late mother, on the other hand, was a mathematician. Those with little interest in math might conclude from this piece that I am mathematically inclined. Those with mathematical sophistication are wondering what kind of dolt devotes this much thought to a trivial formula. My mother, of course, was of the latter sort. When I called her, she confirmed my intuition and assured me it was all quite simple. I could almost see the wry look on her face as she puzzled over the active yet dim mind of her elder son. It's not that she was bereft of wonder: she and I were the mystics of the family. But for her there was no awe in 1/e as applied to hitchhiking. She understood it too well. For me, this encounter with e was indistinguishable from magic.
Explore something that puzzles you until you find an unexpected connection. You might find yourself borne aloft, a sparkle on the Pacific.
Klatch of other Strange Thoughts:
2009-03-17 -- Background Food: A Modest Business Proposal
2009-02-07 -- A Defense of Massive Conspiracy Theories: ...from one of the few who doubts them all.
2008-07-17 -- Immortality, Part Two: Prisms, Zombies, and Occam's Razor
2008-06-07 -- Immortality, Part One: Identity, Continuity, and the Star Trek Transporter
2008-05-14 -- Tripping out of Trees
2008-05-08 -- Color Octaves: Sight, Sound, and Elegant Recurrence