The Numbers Game
Life’s a crapshoot, one hears. That may be the way folks see it today, but for a wink of time, between the publication of Isaac Newton’s Principia in 1687 and the formulation of the Uncertainty Principle by Werner Heisenberg in 1926, it was believed by many in the natural sciences community that given sufficient information about the individual parts of creation—locations and motions, specifically—the future of the parts and the future of the whole could be known. Life was but a game of billiards writ large. It was not a big jump to the two-sided coin: Nothing happens without reason; without reason, nothing happens.
.jpg)
.jpg)
The Twentieth Century was barely on its feet before confidence in those determinative beliefs was delivered a crippling blow. When viewed through microscope and telescope, the depth and breadth of creation was revealed to be vastly smaller and vastly larger than previously thought. The numbers of objects seen (and surmised) staggered the imagination. The very idea of counting the molecules of gas trapped inside a balloon or cataloging a heaven’s weight of stars—some of those “points of light” said to be universes unto themselves—was preposterous, a word whose time had finally come.
Naked-eye counts of stars placed their number within manageable thousands, not the billions of thousands the telescope revealed. The number of atoms in a billiard ball? Ten trailed by twenty-five zeros might fall short by a zero or two, and ten trailed by fifty zeros might prove insufficient for the number of atoms making up Earth herself. These simple tens with their swarms of accompanying zeros are numbers so large they have yet to be named. (Just this side of infinity lives one number with so many zeros it has yet to see practical use: a googol—google is a misspelling—is one followed by one hundred zeros or 10100.) The sensible observer could not be blamed, then, for despairing of knowing nature in her parts, for even if measuring could be reduced to mere counting, the job appeared quite beyond the grasp of mortal man.
Enter the mathematics of statistics and probabilities. Nature would be subdued, if not in her parts, then in her aggregates and averages. Rules stood in for Laws and eventually came hard up against the scientific community’s most cherished belief, Cause and Effect.
Perhaps it is time to examine the notion of “derivative” properties and hopefully learn a thing or two about viewpoint and perspective. We travel to the gambling world of Las Vegas but bring with us a game new to the Nevada scene: Heads or Tails.
(Theoretical physicist and Nobel Laureate Albert Einstein loved to create thought experiments designed to probe and perhaps destroy a targeted thesis. For what follows, Einstein’s spirit provides us with the perfect penny and the perfect penny flipper—after a million flips we can expect half a million heads and half a million tails, plus-or-minus a handful either way.)
Inside one of the Strip's oldest casinos, Mister Lucky, we find space between gaming tables and plug in our penny flipper. We hit the switch and watch Einstein’s penny flip and fall, now heads, now tails . . . and tails again, now heads. We see something like this going on over at the roulette wheel—reds and blacks, odds and evens. At both locations the action is discrete and final and unpredictable. Oh, there is a kind of predictability. We know the casino owner will make money tonight. What we don’t know, what we can’t know, is whether we will make money tonight.
Each flip of the penny is radical and exclusionary, either this or that, but never both and never somewhere in between. Each flip is a statistical event, sometimes heads, sometimes tails. But what are we to make of the growing number of events, duly tallied and summarized? If we begin to see a pattern in those events, do we presume a rule, a rule that perhaps suggests a hidden force or law of nature? If we speed things up a bit so that we are watching the millionth flip of our penny, the tally is indeed so close to half heads, half tails we risk little in establishing once-and-for-all the Rule of 50:50. The question now becomes, can we say that this rule deduced from careful observation is something more, a force or law governing in some mysterious way the outcome of future flips?
Downstairs in Mister Lucky's counting room the owner, Mr. Lucky himself, smiles as the penny flips continue to support his own thesis, one he is betting on, that given a “fair” penny, there is indeed a Rule of 50:50 that will not backfire and betray. He is banking on this “derivative” property of penny flips. A careful man, Mr. Lucky keeps a close eye on a bank of digital displays beside his desk. Topmost, a pair of counters tabulate accumulating heads and tails. Under the counters is a single display labeled La Difference. It flickers with each flip, expanding to +17 at one moment, then passing through -6 a short time later. This number, Mr. Lucky smugly informs us, measures the actual fairness of Einstein’s penny, not its presumed fairness. Fair enough.
The propensity of La Difference to hover around zero mirrors a derivative property of Einstein's penny, its propensity to fall heads and tails equally, and thus the Rule of 50:50. This property is derivative because it is somehow inherent in the penny itself or the penny flipper or the penny-flipping process or all three (or, to be fair, any two of the above). This derivative property is first discovered and then defined by penny flipping events; it persists along with the flipping, and is seemingly unaffected by the hour, the weather or the outcome of the potato race in Sweet Sam Hill.
Dare we call the Rule of 50:50 a Law?
A bell goes off. Our casino owner is suddenly on his feet peering closely at his displays. La Difference flutters near -113, an oddly high value, then drifts back to safer ground at -99, -98. All is silent in the counting room. Mr. Lucky studies his displays for a few more minutes. So maybe the Rule of 50:50 is not a law after all.
But there is one more display, at the bottom of the bank of displays. It is not clear at first just what the number displayed there represents. It too flickers with change, but not at every flip. The action hovers around the last of a series of digits—0.000017 at the moment. A few minutes observation indicates a persistent tendency for this number to diminish, though at times it does reverse course. When asked the meaning of the number Mr. Lucky explains that his own play in the penny game (in support of his grandson’s college tuition fund) resides with this steadily shrinking number, a second derivative property of perfect penny flips. That number, a value which inexorably approaches zero, is the result of dividing the running difference between heads and tails by the total number of flips. “As close to a sure bet," Mr. Lucky assures us, “as you will find this side of the law.”
If Einstein’s penny proved a phony and plopped down Heads every time, our first derivative property would equal the number of throws—growing larger with each flip rather than hovering near zero—and the second derivative property would equal 1.00. Long before total flips hit one thousand our casino owner would have thrown us out on the street for playing him the fool.
The lesson to take home from all this is simply, Zilch is not Zero. In plain language, the casino owner was comfortable with a second derivative property steadily approaching zero (the Rule of 50:50 was solidly in play) but fully understood and accepted small trend-line reversals. When asked by a coffeehouse friend just what were the odds of his poking his finger through our tabletop given that the atoms in his finger and the table are composed mostly of space, this author answered—you guessed it—“Zilch but not zero.” Rules are not laws. Where the two notions get confused is in the numbers game that modern science must play if it is to advance our understanding of cosmos and microcosm.
Which brings us to—hold your breath—Quantum Entanglement and Fuzzy Action at a Distance.
[For your consideration and bemusement I place before you an hourglass, rather typical of the species but special in this way—sealed inside both ends are paddlewheel counters that mark each inversion. The number 101 is slowly disappearing under fine white grains as we continue this discourse.]
O.K., Quantum Entanglement. That expression is popping up all over the place these days, and not just in scientific journals. As I bang away at my keyboard I take breaks to page through Richard Bartlett’s Matrix Energetics: the Science and Art of Transformation. On the cover, Mr. Bartlett’s name is followed by a D.C. and an N.D. (Doctor of Chiropractic and Naturopathic Doctor) so I guess he knows what he’s talking about. It seems our bodies are made up of photons in one form or another all of which manifest “electromagnetic gauge-symmetry states” governed by “Heisenberg’s Uncertainty Principle.” I’ll take the good doctor at his word but turn to the physics community for an understanding of the big Q.E.
In the mysterious world of the infinitesimally small where quanta shake their thing, there are occasions when a pair (or even a triad) of mass objects—electrons and their kin—or energy objects—photons—form an association of like property states. These little guys are then said to be entangled, a snug relationship that may survive but fractions of a second or months and months.
With Werner Heisenberg’s guidance early on and the mathematical heavy lifting of Erwin Schrödinger, the nascent community of quantum physicists settled on a statistical description of the quantum world and turned to “wave functions” to help communicate what they were observing and learning. This area of science is dense and frequently runs counter to everyday experience, but suffice it to say, when two quanta become entangled they share certain property states such that discovering a particular state in one is to know at once that state in the other (whether an identity, a mirror image or a complementary state is not important here). Anyway, to know one is to know the other. It seems that peering into this universe of small things so perturbs the observed object that learning something meaningful results in a statistical reordering, in effect a destruction of the previous wave function. An entangled photon tested for its polarity, say, is rendered again a mystery, but happily, the polarity of its twin is known, and without looking. The twin’s wave function is said to have collapsed to a known state, yet it hasn’t been touched.
.jpg)
[Just a moment. Time to invert my hourglass. Did I mention I’m timing each event to see if one end empties faster than the other? Perhaps the Rule of 50:50 applies with hourglasses as aptly as it does with perfect pennies.]
If we address a particular property of photons—spin, so called—which exists in one of two possible states, spin up or spin down, we must assume that the state is either up or down until we learn otherwise. Quantum physics goes one step farther in declaring that both states mysteriously co-exist until that state is tested for, and then and only then the probability function collapses and becomes the certainty that the state is up or down but no longer up and down. This business of states coexisting until tested is called Quantum Superposition and, in some quarters, has taken on the status of Law.
No comments:
Post a Comment