Level: Introductory Joe Marasco, CEO, Ravenflow
15 Jun 2002 from the Rational Edge: In this article, Joe Marasco debunks popular misinterpretations of scientific laws.
We find many instances these days of scientific language used to "explain" common phenomena. Unfortunately, these usages are often metaphorically or analogically incorrect, devoid of meaning, or just plain silly. In this article we choose some common examples, show the improper application, and then try to illustrate a better way of saying what the author would like to say, stripped of the jargon intended to impress the layperson.
Houston, We Have a Problem
Plato was the first philosopher to point out that achieving a conceptual understanding of the physical world is trickier than we might think at first. In more recent times, Immanuel Kant should be given credit for trying to figure out what knowledge can be known with and without the "filtering of reality." That is, we have a notion that there is an "objective" reality, but in some sense we can never get to it, because it is filtered through our human apparatus for experiencing it -- our senses, minds, and emotions. What we are capable of experiencing, both as individuals and as a species, is a "subjective" reality -- and that limitation introduces lots of doubt as to what "reality" is. This doubt has permeated our thinking, and sometimes its effects can be less than subtle, or even subconscious. Consider how often we say, "Things are not always what they seem."
Our knowledge of physics, in some sense, compounds the problem. As far back as Galileo and then Newton, scientists have been formulating theories that seemed, in their time, very counterintuitive -- for example, that bodies in motion, left to their own devices, would continue in motion forever. As the centuries passed, however, many (but not all) of these counterintuitive ideas became accepted by educated people as reasonable; the ideas became common knowledge and were assimilated as accepted doctrine. Another trendy way of saying this is that certain ideas achieved "mind share"; that is, they became part of our collective consciousness.
But "modern" physics is only about one hundred years old. Although the rate at which the general population accepts new scientific ideas has accelerated, many of these ideas have not actually been absorbed. There are some very good reasons for this. For one, new theories such as relativity and quantum mechanics, because they deal with realms of reality outside our everyday experience -- velocities near the speed of light and subatomic distances, for instance -- turn out to be governed by laws that are very counterintuitive, even to our modern way of thinking. Both physicists and non-physicists alike recognize that it is very hard to explain these fundamental theories, in part because the mathematical apparatus that makes them clear to the practitioner is simply not accessible to most people. So physicists have a problem: Although their theories are correct and powerful, they are not explainable in any detail to the non-physicist.
Nonetheless, people want to understand. So what happens in most cases is that popularizers attempt to explain by analogy. This is perfectly valid. But over the years, the analogies themselves have acquired the status of fundamental truth for the lay public and are bandied about as common clichés. Each time this happens, people like me who have some understanding of the underlying science become baffled by these overused analogies because we don't see the applicability.
Because of my training as a physicist, I want to be sure to state up front that I don't think physics, or science and mathematics in general, is the exclusive province of some exalted priesthood, and that the layperson "shouldn't talk about what they don't understand." I would like everyone to understand science and technology better. But I am also a realist about the magnitude of the problem. What I would settle for is a better understanding of what those scientific clichés actually mean so that people can either use them only when they are appropriate or replace them with more appropriate arguments. In particular, I would like to discourage the practice of smugly quoting scientific jargon to impress an audience with the correctness of one's position. And, for my money, the best way to do this is to point out when the popular analogies actually apply, and when they don't.
So here goes. See if you can spot the faux pas as we proceed. Let's start with some "leftovers" from classical physics, that is, ideas that are still poorly understood, even after a few centuries of simmering in the intellectual pot.
Fig Newtons
Newton's Laws of Motion (or The Three Laws of Motion) are liberally quoted. Here are some of the things one hears from time to time:
From people in general:
"That object is in equilibrium, so by Newton's First Law, there must be no forces acting on it."
From a manager in response to observing a backlash to a recent business initiative:
"We should have known that would happen. Newton's Second Law predicts that for every action, there is an equal and opposite reaction."
From a project manager, remarking on someone else's project:
"That project is definitely in free fall."
Let's look at these one by one.
Misapplication of the First Law
Newton's First Law of Motion says:
A body at rest or in a state of uniform motion (constant velocity) will stay that way unless acted upon by an external force.
Note that this means there are no net external forces acting on the body unless precisely stated. Or, to put it another way, there may be external forces acting on the body, but they (the multiple external forces) cancel exactly. When these external forces balance each other, the object is in equilibrium: static equilibrium if the body is at rest, or else equilibrium in uniform motion -- that is, in a straight line at constant velocity. So remember: Equilibrium does not mean "no forces acting." Equilibrium means, "all external forces balance exactly." Of course, internal forces have no effect, as they cancel in pairs by Newton's Second Law, as we shall soon see.
Let us assume that a lump of coal is moving at constant velocity along the surface of a level table. Ignore for a moment how it came to be in motion, but let's assume it is moving at one inch per hour toward the west. Newton's first law tells us that unless we impose some other horizontal force on the lump, it will continue to move at one inch per hour toward the west forever.
Now, as we pointed out earlier, this defies common sense. In our real world, we would expect the lump of coal to slow down for at least two reasons. One, there is air resistance, and two, there is friction with the table's surface; both of these will tend to retard the uniform westward motion. But of course, there is no violation of Newton's first law here at all; both air resistance and friction are external forces acting on the lump of coal, and the first law states very precisely that the rule does not apply if external (net) forces are acting on the body in question. Now a physicist, used to thinking about and stating conditions precisely, would understand that a force is a force, and you can't neglect any of them. To describe the case above precisely, you would have to state: "The lump of coal will continue to move at one inch per hour to the west in a perfect vacuum on a perfectly level, frictionless, table." The problem is, most of us are not so precise in describing daily phenomena, so it's easy to understand how ordinary folks might misapply Newton's First Law.
A member of the younger generation of physicists recently pointed out to me that these days, students use deep space as a theoretical framework for working out problems, so that they can quickly dispense with the effects of air resistance, friction, "tables," and the gravitational pull of nearby massive bodies. Although this idealized context simplifies the requirements for understanding mechanics, one wonders what will happen when these students are called on to solve real problems "back on Earth."
Misapplication of the Second Law
Newton's Second Law says:
For every applied external force on a body, the body exerts an equal and opposite force.
When something happens in the business world in reaction to an event, someone is sure to bleat out, "For every action there is an equal and opposite reaction." In fact, it is they who are having a knee-jerk "reaction." Rather than applying any thought to the situation, they quote Newton to justify or validate whatever backlash has taken place. The reaction is postulated as something that "had to happen" according to "the laws of physics." In truth, however, what goes on has nothing to do with physics. Not only is the typical reaction unequal to the effect that produced it; often it is not even delivered in the opposite direction, but is rather off at some tangent. Moreover, it may not have been a result of the original action at all.
Once again, Newton's Law is correct, but we must be precise about the force and the body. Often the "equal and opposite" forces people cite in business situations are really an internal force pair that does not exert any external net force on the body. So whenever you hear someone intone, "For every action there is an equal and opposite reaction," my advice is to check to see what the forces are and what bodies these forces are being applied to.
Misapplication of the Third Law
The Third Law says:
A body will be accelerated by an external force in direct proportion to the force and inversely proportionally to its mass.
This one is often quoted as simply "F = ma," which is just a formulaic restatement.
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It is an unbelievably simple and elegant result that applies over an incredible range of phenomena.
But what does it mean to talk about a project "in free fall"? I think managers mean that it is accelerating under the influence of gravity, which means that it is gaining speed and will inevitably collide, inelastically and catastrophically, with Mother Earth. Splat! I "get" the notion that there are no parachute and no brakes, and a sense of rapidly impending doom. Yet, I see here a misuse of the physics analogy. Projects are subject to constraints just as surely as they have mass; the notion that management is so absent that we have effectively yanked the table out from under the lump of coal is certainly disheartening to say the least.
Everything's Relative
OK. Now that we understand the gravity of mistreating Newton, let's try a couple more popular idioms on for size. Two people disagree on something, and one says:
"Well, it all depends on your frame of reference."
Or, someone wants to make the point that "the old accepted laws of nature are no longer true." The usual expression of this is along the lines of:
"Einstein showed that Newton was wrong."
Well, as they say in the Hertz commercial, not exactly. Einstein did a hell of a job with relativity, but his theory has spawned some strange notions.
Frame of Reference
With respect to the first example above, it's true that things can be different depending on your perspective or point of view; however, that is a perfectly classical phenomenon and has nothing to do with Einstein's Theory of Relativity. In fact, things are not different depending on your frame of reference. Relativity emphasizes that when you are within a framework that's moving at constant velocity, you cannot know your velocity as perceived by a stationary observer, because everything else inside your framework behaves according to the laws of physics, and all appears to you just as it would if you were stationary. And, by extension, you also cannot distinguish between, say, acceleration and space curvature.
Einstein Proved Newton Wrong?
As for the second claim above, Newton's Laws are perfectly valid at velocities that we encounter in our daily lives. Change comes only when things are moving at or near the speed of light. Then you need to apply different rules. And that's where Einstein comes in. Newton works at low speed (that is, most of the time), and Einstein's Relativity Theory kicks in when you start to go very fast. If you use "just Newton" for too long, then you will get progressively more incorrect results as you approach the speed of light, and your answers will be completely wrong when you actually reach the speed of light. Just remember that, relative to our daily experience, the speed of light is a very, very, big number.
The effects of relativity are completely negligible in our common experience. You can compute them if you'd like, using Einstein's Theory of Special Relativity, but you will find that your results don't change at all.
That is because the speed of light is so great. On the other hand, the speed of sound is something we relate to daily. You can observe that the speed of light and the speed of sound are very different by doing this experiment the next time you are playing golf. When you are about 250 yards or more down the fairway (you have just hit your second shot and are walking toward the green), look back and watch (and listen) for the next group's tee shots. You will see the club hit the ball, and then a split second later you will hear the impact. You can compute this discernable interval by using the speed of sound in dry air at sea level,
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and by assuming that the speed of light is infinite; that is, it takes zero time for the light to travel from the golf club to your eyes. This will give you the right answer.
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If you do the calculation using the actual speed of light, then you will get basically the same answer.
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So although you can legitimately apply Einstein's Theory of Relativity here by using a finite speed for light, it won't buy you much.
There is a real life situation in which you can experience the speed of light as finite. When making an international phone call, you are sometimes unlucky enough to go up to a geostationary satellite and back down to Earth, and that takes about half a second. That's long enough to give you the impression that your interlocutor is pausing; you might misinterpret that pause as dissent, hesitation, apprehension, or the like, depending on the conversation.
And Another Thing
When it comes to Einstein, we have just scratched the surface. All the phenomena we've just discussed are manifestations of his Special Theory of Relativity, which holds only for bodies moving at constant velocity. When bodies actually accelerate relativistically, then you have to use his General Theory, with consequent additional heavyweight mathematical baggage. Yet popularizers invoke the General Theory with equal impunity. In fact, there are only a small number of experimental tests that we know of to test the General Theory, all of them involving very, very, small effects.
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None of this diminishes the magnitude of Einstein's accomplishments. However, applying his brilliant discoveries to situations in which they do not really apply in a sense cheapens them.
Quantum Nonsense
Let's move on from relativity to quantum mechanics. Recently I had someone who was unwilling to make a forecast say to me:
"It's just like quantum mechanics. All I can give you is a probability."
Although the second part of his claim was most assuredly true, I am certain that it had absolutely nothing to do with quantum mechanics.
About twenty years after the relativity revolution, circa 1927, quantum mechanics burst upon the human race with equally momentous and unsettling effect.
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All you have to remember about quantum mechanics is all you have to remember about relativity. Neither theory replaces Newton's Laws. Whereas Einstein's Relativity Theory extends Newton's Laws into the domain of the very fast (velocities near the speed of light), quantum mechanics extends classical physics into the domain of the very small. That is, when we get down to subatomic dimensions, new rules come into play. That's when we need to use quantum mechanics. For everything else, the rules of quantum mechanics still apply, but the effects are so small that they are irrelevant.
The reason it took so long to discover both bodies of knowledge is that we could not measure either stuff that went really fast or things that were really small much before the second half of the nineteenth century. Actually, it was the invention and perfection of the vacuum pump -- an engineering feat -- that facilitated measurement in both arenas. This also explains why the effects that required the application of either Einstein's theory or quantum mechanics were not observed; except for the conundrum about the wave-particle duality of light, nothing in our plodding macroscopic world hinted that anything was "wrong."
The "It's just like quantum mechanics" reveals an interesting misconception. Because quantum theory involves calculations involving probabilities, many people think that predictions based on quantum mechanics are somehow imprecise. The reality is just the opposite.
For example, we can determine  , the fine structure constant
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, experimentally. Now this number is quintessentially a "modern" physics number: It is made up of, among other things, the charge on the electron, Planck's constant (see more on this later), and the speed of light. When you are measuring it by any method, you are doing quantum mechanics, and the theoretical predictions of the number involve some of the deepest applications of quantum theory we know. Yet we can do experiments that measure its value to about one part in 108. Now that is pretty good in anybody's book.
By contrast, G, the universal gravitational constant, a perfectly "classical" quantity known since the time of Newton, has been experimentally measured to only about one part in 104. That's not bad either; it corresponds to 0.01 percent precision. Yet we know with several thousand times more precision. Somewhat ironic, isn't it?
So much for the probabilistic nature of quantum mechanics and its relation to making predictions.
More Quantum Nonsense
Actually, my pet peeve is the frequent misuse of "Heisenberg's Uncertainty
Principle." If you are interested in a particularly droll example of this,
see Freddy Riedenschneider's monolog in the Coen brothers' movie "The
Man Who Wasn't There."
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It is easy
to see why Billy Bob Thornton got the chair after his lawyer tried to
use the principle to convince (or confuse) a jury.
A common lament when someone is asked to make a difficult measurement:
"We're screwed. Heisenberg tells us we can't measure something without disturbing it."
Another example: Software people now talk about "Heisenbugs."
"Man, it took us weeks to track down that defect. Turned out to be a Heisenbug."
These are bugs that are very hard to eliminate, because in the process of trying to do so we change the working of the program, and our original bug is further hidden by the actions of the debugging apparatus.
What is really going on here?
Measuring Stuff
The fundamental issue is this: can you measure something without at the same time disturbing the thing you are trying to measure? That is, when you go to take the measurement, do you influence in some way the very thing that you are trying to determine? If so, then you have a problem, because your measurement will be contaminated by your perturbation of the system you are trying to measure.
Now, this is not an extremely "deep" problem. Medical diagnosticians have to deal with it all the time. They spend a lot of time and energy making assessment procedures as minimally invasive as possible. Yet we know that some people's blood pressure goes up the minute a cuff is put on their arm. Ergo, their measured blood pressure is higher than their normal resting blood pressure.
In software, we work very hard to make debuggers "non-intrusive." Nonetheless, sometimes the act of debugging changes something that causes the program to behave differently than when it is running without the debugger. Whether this is the fault of the program or the debugger is somewhat moot; in either case, the programmer has a big problem.
And in the 1920s, Elton Mayo discovered the Hawthorne Effect; he demonstrated that in studies involving human behavior, it is difficult to disentangle the behavior under investigation from the changes that invariably occur when the group under study knows it is being studied.
Note that these phenomena are perfectly "classical"; you don't need quantum mechanics or the Heisenberg Uncertainty Principle to explain them.
Before we delve more into Heisenberg, we might ask the following question: Is it possible to do any measurement, even a macroscopic one, that is totally "non-intrusive"? If we can find just one example, then we can debunk the idea that it is impossible.
So here's my example. I wake up in a hospital bed in a room I have never been in before. I want to figure out how large the room is. So I count the ceiling tiles. There are sixteen running along the length, and twelve along the width. I know that ceiling tiles are standardized to be one foot by one foot. Hence I know that the room measures sixteen feet by twelve feet for an area of 192 square feet. Bingo! I have performed a measurement without even getting off my back, and I claim that I have not disturbed the room at all.
Applying Heisenberg
Where the Heisenberg U.P. applies is in the atomic and subatomic realm. Basically, it posits that it is impossible, quantum mechanically, to specify both the position and momentum of a particle to arbitrary precision. If you want to make your knowledge of the particle's position more exact, then you will have less precision on its momentum, and vice versa.
To observe said particle, you have to "shine a light on it." But in so doing, the light itself affects the particle's momentum, and therefore makes it impossible to know the particle's position exactly. So "non-intrusiveness" is impossible at quantum dimensions, and Heisenberg supplies you a formula to compute just how much the intrusion will affect your measurement.
One caution: Heisenberg's U.P. uses Planck's constant, which is very, very, small. So small, in fact, that the Heisenberg U.P. yields nonsensical results once you get into anything greater than atomic and subatomic distances. That is, if you shine a light on an electron, you will affect its position, measurably. On the other hand, if you shine a light on me, you are not going to affect my position much. Shining a light on the ceiling tiles of the hospital room affects them not at all. So using the Heisenberg Uncertainty Principle for macroscopic objects is just nonsense.
Heat Death
On to the last misuse. Some say that there have only been four or five fundamental watersheds in physics. Sandwiched in between Newton and the twentieth century behemoths of relativity and quantum mechanics is the science of thermodynamics.
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Thermodynamics really shook things up. Here is where our modern ideas of energy conservation come from. Here is where the relationship between work and heat becomes clear. Here is where we show that perpetual motion machines are impossible. But most intriguing of all, here is where we get an entirely new concept: entropy.
So today we hear statements such as this:
"Large companies are doomed to failure, because entropy inevitably takes over."
Entropy is a measure of disorderliness. And one of the key tenets of thermodynamics is that entropy is always increasing. The most common example given to students is intuitively very appealing. Take a box that has a partition down the middle, and fill one side with oxygen molecules and the other side with nitrogen molecules. Remove the partition, and the molecules will continue to move about spontaneously. After some time, we observe that there is a uniform mixture of oxygen and nitrogen in the box. We can wait forever, and the molecules will never, of their own accord, find themselves back in the state with all the oxygen on one side and all the nitrogen on the other.
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The mixed state is considered to be more "random" or disordered; the segregated system more ordered. The entropy of the final system is greater than that of the initial system.
In any closed system, entropy spontaneously and naturally increases. Eventually the system reaches maximum entropy, or total randomness. In applying this phenomenon to the universe, physicists refer to it as "heat death"; hence the title of this section.
Seems logical, and by and large it is. Most systems, left alone, will tend to a more disorderly state. Just look at my desk.
But somewhere along the line, lay people started extending the concept to economic and social systems. And this is where I think it took a wrong turn. Note that the quotation above is not entirely without merit. It is certainly true that the larger an organization becomes, the more communications links it must support; as Kenneth Arrow pointed out many years ago,
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this may eventually limit its growth. Certainly it becomes harder for large organizations to coordinate activities, and even more difficult for them to respond quickly to changing circumstances. On the other hand, it is a mistake to assume that entropy must inevitably win.
Although it's certainly true that a closed system, left alone, will tend to a state of maximum entropy, economic systems -- such as the company you work for -- are not "closed." They are open to the flow of matter and energy. And we don't tend to leave our companies alone. We add raw materials to them all the time; we do work on the system; we expend energy to combat entropy. Just as I work to clean off my desktop and make it more orderly (less entropic), I can invest work in the communications channels and mechanisms in my company to reduce the disorder.
Now this work is roughly equivalent to the energy a machine might expend
to overcome friction; it is not, in some sense, "useful" work. On the
other hand, it does provide us with a rationale for continuing human enterprise.
With the correct balance, we can at least hold off entropy while we make
progress on the "real" objectives. One philosopher, long since forgotten,
pointed out that we spend our whole lives combating entropy. It's how
human beings and societies survive. In effect, the social organizations
-- countries and enterprises -- that do a better job of beating back entropy
ultimately win over those that are less successful in this fundamental
enterprise.
The key thing to ask when someone refers to the inevitability of entropic disorder is: "Is it a closed system?" If not, then the spontaneous and inevitable increase in entropy is not a given.
Other Examples
There are, unfortunately, lots of other examples I could delve into.
Each one would require a few paragraphs, and this article grows long already.
I have heard numerous misstatements concerning the dual nature (wave-particle)
of light. The discoveries over the last forty years in Chaos Theory have
been incorrectly quoted to (once again) invalidate Newton's Laws. Gýdel's
Incompleteness Theorem, around seventy years old, is sometimes used to
justify our inability to prove something. And in the computer science
arena, Turing's Machine is frequently used to demonstrate undecideability
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in areas where it has no applicability at all. The recently deceased scientist
Stephen Jay Gould wrote extensively on how Darwin's Theory of Evolution
has been widely quoted and generally misunderstood. All of these fundamental
theorems are profound, and all of them are betrayed when used in situations
in which they absolutely don't apply.
Good Science
Scientists and mathematicians have given us some incredibly powerful tools to help us understand our physical world. These tools are wonderful triumphs of human intellect, allowing us to start with first principles and explain a wide variety of phenomena, right down to the existence and behavior of elementary particles. As the phenomena get farther and farther away from our common experience, however, the theories become more abstract and require more esoteric mathematics for their exposition. It is at this point that we sacrifice to inaccessibility much of what we stand to gain in fundamental understanding. Nevertheless, although the average Joe cannot really appreciate all of the subtleties, he can certainly benefit from the trickle-down effects of these discoveries, embodied in practical products that come into his life. In this sense, it is all "good science."
What is not good science is using shibboleths from science to explain things that are clearly unrelated to the physical principles that underlie those shibboleths. Human beings are not just like fundamental particles; there is no reason to believe that they obey quantum mechanical laws as macroscopic beings. Such analogies really are misleading, and we should be wary of those who would use them to convince us that their positions are valid.
Likewise, we should all be careful about using pseudoscientific jargon in our daily communications with others. The least harmful result is that they will believe us without thinking, because we have "snowed" them with technical lingo. A more harmful result, which you should carefully consider, is that they will nod in agreement, and secretly conclude that you are a carnival huckster. And you will never know that you have lost credibility when you thought you were gaining it.
Dedicated to Mark Sadler, 1945 - 2002
Notes
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While "F = ma" is the commonly quoted
formula, the more general equation is "F = dp/dt," which says that the
force is proportional to the rate of change of momentum. This only matters
if the mass of the system does not remain constant, as in the problem
of a rocket becoming lighter as it burns its fuel and thus loses mass
during its flight. "F = dp/dt" is more general than "F = ma," but the
latter formulation is the one you hear the most. Just remember when you
hear it that it contains the assumption that the mass doesn't change.
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Notice how precision just starts to creep into our language when we want to be careful about using physics. The speed of sound depends on a lot of things, including the temperature, which I did not specify.
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I hope you were curious enough to do the calculation yourself. If not, here it is. At 68 degrees Fahrenheit, the speed of sound is 1127.3 ft/sec. Two hundred and fifty yards is 750 feet, so the sound will reach you in 0.665 seconds, or roughly two-thirds of a second. This is a noticeable interval. And by the way, I used English units here, not metric, because golfers, by and large, are "calibrated" in yards, not meters.
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Another common example of this calculation is determining how far away a lightning bolt is by timing how long it takes to hear the thunderclap after you see the lightning. Same idea.
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When I was doing physics around thirty years ago, there were only three. They were (and are still today) 1) the precession of the perihelion of the orbit of Mercury, 2) the gravitational bending of light as it passes by a massive object, and 3) the gravitational red shift of light as it climbs out of the gravitational field of a mass. My sources tell me that since then, several more have been added; one involves measurements on binary pulsars. All these effects are extremely small and hard to measure, and have very little connection to our everyday lives.
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Some date the origin of quantum mechanics back to Planck's work in the early 1900's, which was contemporaneous with that of Einstein. I use 1927, because the papers that Schrýdinger's published in 1926 were publicized in early 1927, giving us Schrýdinger's Equation. That formalized things and really launched the revolution.
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The fine structure constant comes up when considering the separation of lines observed when doing spectroscopy on the atoms of an element. Quantum mechanics evolved as physicists tried to explain the various separations for different elements; later, quantum theory was used to predict higher-order effects on the spectra when, for example, the atom in question was subjected to an electrical or magnetic field.
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http://us.imdb.com/Title?0243133
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We might mention in passing that around the time of the American Civil War, our friend James Clerk Maxwell, building on the empirical work of Michael Faraday before him, recast electromagnetism in a beautiful mathematical formulation. It made electricity and magnetism understandable as aspects of one theory, and it is stunning. It enabled, in some sense, modern telecommunications to be born; for example, Marconi and his radio came after. So it is not to be downplayed. Yet to me Maxwell's equations are a mathematical tour de force; much of the physics and phenomenology was well understood at the time of Maxwell's work. For example, we know that the laying of the transatlantic cable was interrupted by the Civil War, so that telegraphy was in place well before it.
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Theoretically, you can compute the probability that this will happen. It is very close to zero, believe me.
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See http://www.amazon.com/exec/obidos/ASIN/0393093239/qid=1022704048/sr=1-6/ref=sr_1_6/002-2515674-9292830
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The notion of "undecideability" refers to the impossibility of writing a computer program to determine the result of a problem or class of problems.
About the author  | 
| | Recently appointed CEO of Ravenflow, an Emeryville, California-based company delivering precision requirements validation for software developers, Joe Marasco served as senior vice president and business unit manager for Rational Software prior to the company's acquisition by IBM. He held numerous positions of responsibility in marketing, development, and the field sales organization, overseeing initiatives for Apex and Visual Modeler for Microsoft Visual Studio. After retiring from Rational in 2003, he published The Software Development Edge, a collection of his essays on software project management originally published in The Rational Edge. He holds a Ph.D. in physics from the University of Geneva, Switzerland, and an M.B.A. from the University of California, Irvine. |
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