David L. Martin

in praise of science and technology

Archive for the month “August, 2016”

Revolution 3 – Quantum Mechanics

For centuries, physicists have debated about light.  What is it, exactly?  Does it consist of particles?  Or is it a wave?  It turns out, both, sort of.  Light has particle characteristics, and wave characteristics.  In everyday life, we see particles of various sizes.  Dust particles.  Baseballs.  Planets.  What makes a particle a particle is that it’s separate from other particles.  And we see waves.  Water waves particularly are very familiar.  What makes a wave a wave is that it’s continuous.  It’s spread out.  It’s part of a larger whole.


Waves do things that particles don’t.  A wave’s very existence depends on how it interacts with the waves around it.  If two waves collide, and one is as its crest while the other is at its trough, they will cancel each other out.  Think about that for a moment.  The two waves come together and the result is NO WAVE.  They just vanish.  This is called interference.  On the other hand, if they are both at their crests when they collide, the result will be an even bigger wave.

Imagine if particles acted this way – peanuts, baseballs, people – if sometimes, when they collided, they just vanished.  The universe would look very different, obviously.  Particles do not exhibit interference.  Or do they?


Photons, the so-called particles of light, unquestionably behave like particles.  In fact, we have devices that can shoot them, one at a time, like tiny bullets.  When we do this we can see the impacts on a screen, where each photon is absorbed.  Although we can’t see the photons in flight, each photon hits one and only one spot on the screen.  Particles.  No doubt about it.

Now suppose we start shooting photons toward a screen, one at a time.  Between the emitter and the screen we place an opaque barrier with 2 slits close together.  And we fire away.  As always, each photon that passes through hits the screen in only one spot.  But was we accumulate hits, we notice something strange.


What the hell is this?  Instead of 2 distinct clusters of hits, corresponding to the 2 slits, we get this fringe pattern.  Well, it turns out we know of something that exhibits a similar effect – water.  If we create water waves on one side of a barrier with 2 slits, as they pass through the slits they create new sets of waves, centered on each slit.  Waves do not simply strike each other.  They can interfere with each other.  When a crest encounters a trough, they cancel each other out.  When a crest encounters another crest, they reinforce each other.  The result, in a 2-slit situation, is a fringe pattern.


At this point you may well ask, “What are you saying Dave?  That somehow an individual photon can interfere with itself?”  That’s exactly what I’m saying.  Quantum mechanics dictates that the seemingly particulate photon goes through both slits.  Eventually it hits the screen in only one spot.  But where it hits is dictated by wave mechanics.  The photon in flight behaves like a wave.

Now you may notice that the water comparison doesn’t quite hold up.  With water waves, you don’t see bright lines with dark areas in between.   Along each path leading to each bright area, the water waves are still “waving.”  Up and down, up and down.  Along each path leading to each dark area, the water is flat.  How does this translate into photon behavior?


The mathematics of quantum mechanics can be very intimidating.  But it’s really not hard if it’s presented visually.  Suppose we have a series of waves like the blue waves below.  Let’s say that the “waviness” at any given position translates into the probability of finding a particle at that point.  This is represented by the red wavy line (and the intensity of the red dots).  Notice that there are places where we will never find the particle.  This is because of the underlying wave mechanics.


This is really the nuts and bolts of quantum mechanics.  If you understand this figure, you understand QM.  The position, momentum, energy, and other characteristics of particles are described by wave mechanics, and this is why we see a fringe pattern when we shoot photons at a screen.  But the strangeness has only begun.  What if there’s a way to distinguish the 2 paths?  What if we had some way of “marking” the photons passing through, so that we KNEW they went through one slit and not the other?  In fact, we can.  I won’t go into the details, suffice it to say that we can do this, without even actually knowing which slit a particular photon went through, only that it went through one or the other, not both.  What happens now?  When we accumulate lots of hits, we get the pattern at the top of this image:


Instead of the fringe pattern (the bottom pattern), we get photon hits that accumulate in 2 distinct areas, corresponding to the 2 slits.  The photons now behave, in flight, like particles, simply because we have information that tells us each one had to go through one slit or the other, not both.  Notice that many of these photons strike the screen in areas that were pretty dark when the fringe pattern was produced.  It’s important to realize that we only have to place a “marker” in the path through ONE of the slits to eliminate the fringe pattern.  This marker tells us that either the photon went through that slit or it didn’t.  Since the only alternative is the other slit, we don’t have to do anything at all to any photons going through the other slit.


What I’m saying is that the photons that go through the other slit are NOT EXAMINED DIRECTLY in any way, let alone altered.  Yet they fail to accumulate in a fringe pattern on the screen.  They stop “avoiding” some of the areas that are normally dark when the fringe pattern is produced.  It is only because we have information about their path that they change their impact pattern on the screen, not because they are examined or interacted with directly.  It’s as if they “know” what’s going on along the other path, which they never follow.

Think this is strange?  I’ve only just begun.  What if we could somehow “erase” the information about which path each photon went through?  It turns out we can do this too.  Instead of using a double slit, let’s really separate the two paths.  In the upper part of the figure below, we are shooting individual photons from the lower left through a beam splitter (the green object).  This is simply a half-silvered mirror.  An individual photon has a 50% chance of going straight through, and a 50% chance of bouncing off.  We have ordinary mirrors in each of these alternative paths, which cause the paths to cross at the upper right.  But if we put a screen in the red path, it can only be struck by photons following that path.  So we should see a particle-type pattern.  And we do.


But what if we place a second beam splitter right at the point that the two paths cross, like in the lower part of the figure?  It shouldn’t make any difference at that point right?  The individual photons have supposedly already “committed” to one and only one path by then.  They can’t back up and retrace their paths, changing their behavior.  We still get the “particle” pattern on the screen, right?  Wrong.  If we place a screen in the path, we get the fringe pattern.  This is called a quantum eraser, and it has been observed many times.


Obviously, water waves do not behave like this.  Saying that tiny particles have wave-like behavior doesn’t fully explain quantum phenomena.  The strangeness goes much deeper than that.  Another experiment illustrates just how deeply it goes.  What if we could somehow delay the acquisition of information about which path the photons took, until AFTER they had been detected at their destination?  Surprising as it may seem, we can do that too.  We can do it by splitting a photon into 2 new photons.  The way we do this is by sending it through a special crystal.  The photon strikes an electron within the crystal.  The electron absorbs the photon, causing the electron to jump to a higher energy level.  Very quickly it will drop back down to its previous energy level, but this time it does it in two steps.  Each step produces a new photon.


Each of these new “daughter” photons is said to be entangled with the other.  That is, the characteristics of one must always complement the characteristics of the other, no matter how far they travel.  So here’s the experiment.  We send the “parent” photon through a barrier with a double slit.  Then we send it through our special crystal, splitting it into 2 entangled photons.  We then send these 2 photons through two separate experiments.  One photon (we’ll called it photon 1) simply goes to a screen.  The other (photon 2) is directed through a carefully positioned set of mirrors and beam splitters.  Depending on what path it takes, it will hit one of four detectors.  If it hits detector A, the “parent” photon must have gone through slit 1.  If it hits B, the “parent” photon must have gone through slit 2.  If it hits C or D, we don’t know which slit the “parent” went through.

But here’s an important point:  Photon 2 will only reach one of these detectors AFTER photon 1 hits the screen.  Thus the “marking” only occurs after photon 1 has completed its journey.  So what happens?  Do the original photons behave like particles or waves?  Answer:  It depends on whether one of their entangled “children” will eventually be marked.  “Wait a minute!” you might say.  “What do you see on the screen? Some of the photons behave like waves and some don’t?  You can’t have a fringe pattern AND at the same time, a particle pattern!”  Actually, you can.  Some of the photons go through slit 1.  Their “daughter” photons produce a particle pattern on the screen.  Others go through slit 2.  These “daughters” produce another particle pattern.  Still others go through both.  These produce a wave pattern, a fringe pattern.  But all of these patterns are superimposed on the screen.  Only by identifying which photons hitting the screen correspond to which photons hitting each of the other detectors can we tease apart these patterns


In other words, it’s as if each photon “knows” whether one of its entangled “children” will or will not eventually be marked, and insists on behaving like a particle or wave, accordingly.  Even if we delay the acquisition of this “knowledge” until after one of its “daughters” has hit its screen, been absorbed, and is long gone, this still affects the behavior of the “parent.”  This is called a delayed choice quantum eraser.  In principle, we could delay the acquisition of the “which slit” information for a HUNDRED YEARS.  The “parent” photon would have to adjust its behavior according to actions taken a century later.


It hardly needs to be said that explaining this requires something pretty strange.  And the standard quantum mechanical explanation is just that.  Here it is:  The photon doesn’t follow a particular path, UNTIL the experiment is complete and the fate of all of its entangled “daughters” is established. A physicist would say it’s in a superposition of states.  It is spread out, and it will stay spread out until the fate of its “offspring” is decided.  It is in more than one place at once.  The implication of this is astounding – the screen itself is in multiple states, until the fate of the entangled photon is known!

Don’t misunderstand me.  I’m not staying that if you look at the screen, you will see the same photon hit multiple spots.  I’m saying that the screen AND you, if you’re looking at it, are in a superposition of states.  There are other versions of “you,” looking at other screens which are in different states.  Any given version of you only sees one of them.  This seems to be the only way to account for what we observe in these experiments.


The “daughter” photons discussed above, in the delayed choice quantum eraser experiment, are linked.  They are part of a combined system that is spread out in space.  The photons are said to be entangled.  When they interact with a larger system that has information about both of them, they will always be found to have complementary states.  Even if the system measures the 2 photons when they are miles apart, and measures them AT THE SAME MOMENT, they will always be found to have complementary states.  It’s almost as if they are in instantaneous contact with each other.  In quantum mechanics we call this non-locality.  Einstein called it “spooky action at a distance.”  The only way to explain it, seemingly, is to say that the 2 particles are not really separate – that they are parts of a combined system that is in multiple states until it is measured.  It means that we can’t speak of a particle as having an existence in isolation from the rest of the universe.


Now at this point, dear reader, you may interject, “Well, okay, this is pretty weird, but light is special.  It’s not matter, it’s energy.  Matter is made of particles, not “wave/particles.”

Matter is made of particles all right.  Subatomic particles.  Protons, neutrons, and electrons.  It turns out we can shoot individual electrons, one at a time, just like we can shoot individual photons.  What happens when we shoot some through a double slit apparatus?  This:


Electrons exhibit the same kind of behavior that photons do.  We see fringes when we don’t know which slit each electron goes through.  We don’t see them when we do.  All of the strange effects I have described apply to matter as well as light.  An electron isn’t actually a particle, until it interacts with a large system.  Until then, it’s a wave/particle.  And what it does is influenced not only by what you do to it directly, but what you do to any particles it’s entangled with, even in the future.


Some physicists have argued that actually, particles like electrons and photons always behave like particles.  It’s just that these particles are GUIDED by a “wave function” that gives them wave-like behavior, just like a ball in flight is guided by mathematical rules.  This is called the pilot wave theory.  The problem with this is that this pilot wave has to somehow take into account what the entangled “children” of the particle will encounter in the future, even after some of them are long gone.


The only way out of this conundrum, seemingly, is to assume that the wave/particle is actually doing all of these different things at once – in other words, that it is in superposition.  When it interacts with a larger system, that system becomes entangled with the wave/particle.  The system only “sees” one of these possibilities.  That’s what we call measurement.  But our measurements can be affected by what happens to other parts of the wave/particle that we don’t measure directly, even if those events are far away, or in the future.  In this sense, quantum mechanics does involve a kind of time travel to the past.


Because matter obeys the laws of quantum mechanics, we can get some very strange effects.  Let’s take radioactive material for example.  The reason some substances are radioactive is that their atoms are falling apart.  They are losing some of their subatomic particles, like electrons.  But whether a particular atom will lose an electron is only a probability, not a certainty.  Quantum mechanics dictates that as long as the atom is observed, it is “trapped” in a state of non-decay.  When it is not observed, it enters a state of superposition between being intact and having decayed.  When it is observed again, it will be seen to be in one state or the other.  But the atom will never show ANY SIGN that it’s about to fall apart.  It will always be either intact or decayed.  Nothing is “breaking.”  Nothing is “wearing out.”  It is all or nothing.


Let’s say the atom has a 50% chance of decaying within 1 hour.  Within 24 hours, it’s almost sure to fall apart.  But until this happens, the atom doesn’t degrade, decompose, wear down, or anything of the sort.  Quantum mechanics dictates that, as long as it is observed, it is “trapped” in a completely intact state.  What I’m saying is that as long as the atom is observed, IT WILL NEVER DECAY.


Yet at any time, we can stop our observation.  Within an hour, if we observe again, there’s a 50% chance that the atom will have decayed.  If we wait 2 hours, 75%, and if we wait 3 hours, almost 90%.  Simply because we didn’t look at it!  There’s an old expression:  A watched pot never boils.  It refers to the fact that it seems to take water longer to boil when you’re watching.  In everyday life this is simply due to the fact that our sense of time is subjective.  It doesn’t actually take longer.  But when you look at very small chunks of matter, like atoms, IT REALLY HAPPENS.  We can inhibit the decay of an atom simply by observing it frequently.  We can actually keep it from decaying by observing it continuously.  This is called the quantum Zeno effect.  It is not some wild theory.  It is a well-documented principle.

 I’m not done.  Because small chunks of matter obey the principles of quantum mechanics, rather than classical mechanics, they can do some unexpected things.  Let’s say an electron approaches a barrier.  If the electron were merely a particle, if would either bounce off, embed itself in the barrier, or rip through.  But an electron isn’t a particle, not most of the time.  It’s a wave/particle.  As it approaches the barrier, some of the wave/particle can actually appear on the opposite side.  At some point, the electron will “commit” to being on one side of the barrier or the other.  There’s a chance it will appear on the opposite side of the barrier.  And sometimes, it does.  This is called quantum tunneling.


It’s important to realize how different this “tunneling” is from what we normally think of as penetration.  The electron doesn’t necessarily pass through the barrier.  It can actually APPEAR on the opposite side.  It can do this because until it “collapses” into a particle state, it is actually in more than one place at once.  Furthermore, the parts of the wave/particle that are on opposite sides of the barrier can be completely disconnected from one another.  And this is one of the strangest things about quantum mechanics.


Let’s say you have two water waves, with a completely undisturbed, tranquil water surface in between.  The undisturbed area is literally wave-free.  No wave exists there.  But this doesn’t seem strange to us, because there is still a physical connection between the two waves.  In a sense, there IS a wave there – It’s just that its amplitude is zero.  The entire surface of the water is continuous, and this explains why waves behave the way they do – how a wave can split into two waves, with no wave in between, and how two waves, one at a trough and one at a crest, can destroy one another.


But imagine that the condition of the water waves will actually translate into the existence of a particle.  If there is no wave at a given location and time, a particle will not appear there.  That is what happens with quantum mechanical waves.  They determine the position and other characteristics of particles when they are observed.  The waves themselves are never directly observed.  We never actually see particles “spread out.”  What we see is that particles never appear where the mathematics tell us there is no quantum mechanical “waviness.”  And the “wavy” areas may be completely disconnected, with only “flatness” in between.  This is what quantum non-locality means.  In its superposed state, one part of a wave/particle can be on one side of a barrier, while another part is on the opposite side, WITH NO PART OF IT inside the barrier.  In many situations, the particle quite literally CAN’T EXIST within the barrier.  So it doesn’t rip through the barrier.  It is NEVER IN THERE.  It merely appears on the opposite side.  Classical mechanics allows nothing of this sort.  Only quantum mechanics explains such behavior, and it is routinely observed.


We can illustrate how truly strange this is by looking at a simple atom.  A hydrogen atom.  It consists of one proton and one electron.  Remember that the proton has a positive charge and the electron a negative charge.  So why doesn’t the electron simply fall toward the proton?  Answer:  Because it has angular momentum.  The reason the earth doesn’t fall into the sun is that it has angular momentum.  In the case of the earth, this is easy to understand.  The earth is in constant motion around the sun, traveling in an elliptical orbit.  The earth’s “sideways” momentum balances the sun’s pull.

But in the case of an electron, we have a problem.  Remember that in the absence of an external force, any object tends to continue moving in a straight line.  Any object that is changing direction is being accelerated.  Since orbital motion is a constant change in direction, an object in an orbit is constantly accelerating.  We know that if an electron is accelerated, it gives off a photon.  But an electron in a hydrogen atom doesn’t.  How can this be?

 The resolution of this problem is that an electron is not a particle in an orbit.  It is a wave/particle.  Its angular momentum is not like that of a planet in orbit.  It really can’t be understood in that way.  In a sense, the electron isn’t moving at all, because it is ALREADY IN MANY PLACES AT ONCE.  When observed, it will be found to have a specific location.  But when not observed, it is “spread out.”  This is why it doesn’t fall toward the proton.


Electrons can move to higher energy levels within an atom, called orbitals.  But this term is misleading.  In fact, the whole sentence is somewhat misleading, because an electron is NEVER OBSERVED to be in between energy levels.  It is either in one or the other.  It doesn’t “move” between levels in the way we normally think of movement.  It shifts from one level to the other.  And within these levels, these orbitals, it isn’t “orbiting.”  It is spread out within the orbital.  Notice that many orbitals have gaps – areas where the electron WILL NEVER BE FOUND.


Over short distances, an electron can appear to move faster than light.  It can do this because it ISN’T REALLY MOVING that distance.  Instead, it is spread out over the space in question when it is not observed.  When it is actually observed, it will be somewhere within that space.

As you can see, the quantum mechanical world is very different from our everyday world.  Some people are fond of arguing that quantum mechanics only applies at very small scales.  We don’t really have to worry about the strangeness of quantum mechanics, they say, because we never see it at large scales.  The problem with this argument is simple – it’s false.  This is illustrated by the quantum eraser experiment we looked at before.  Suppose the 2 mirrors in this experiment are 20 feet apart.


This means that when the photon reaches these mirrors it is spread out over a distance of 20 feet.  And notice – the photon has no existence in the space between the 2 alternate paths.  In principle, we could spread this experiment out so that these 2 mirrors could be miles apart.  It shouldn’t change the outcome of the experiment.  This is the nature of quantum non-locality.  Quantum mechanics dictates that a photon isn’t a particle until it interacts with a larger system.  Until then it is spread out as a wave/particle, spread out over miles if conditions are right, with no “existence” in between.  Only when it interacts with a large system does it actually become a particle, with a particular location.  And I’ll say it again.  All matter is composed of particles that obey these same principles.  When these particles interact with larger systems, when they are measured, they behave like particles.  But the rest of the time they behave like wave/particles.


Now suppose we take a particle, like an atom, and connect it to a large system, so that the fate of the large system is intimately tied to the fate of the atom.  One of the founders of quantum mechanics, Erwin Schrodinger, imagined doing just that, and his thought experiment is one of the most famous aspects of quantum mechanics.  It’s actually pretty simple.  We put a live cat in a box, which completely isolates it from the surrounding environment.  We also put a vial containing poison gas in the box.  Adjacent to the vial is a small hammer, connected to a switch.  When the switch is activated, the hammer will break the vial, releasing the gas and killing the cat.  The switch is activated by a detector, which sends a signal when it detects the radioactive decay of a single atom.


An important point here is that the detector is not monitoring the atom continuously.  It is merely “watching” for a decay product.  So the quantum Zeno effect doesn’t apply.  At some point, the atom will decay, and the cat will die.  But since the cat is isolated from any outside observer, quantum mechanics dictates that the ENTIRE SYSTEM – the atom, the detector, the switch, the hammer, and the cat – is in a state of superposition.  The cat is both alive and dead, until we open the box.  The experiment is called Schrodinger’s cat.  This actual experiment has not even been attempted, for all kinds of reasons.  But the principle is far from ridiculous.  In fact, superpositions of “large” objects have been achieved, and appropriately, they are called cat states.  One example is a resonator, consisting of about 10 trillion atoms, which vibrates and does not vibrate at the same time.


Notice that from the point of view of the cat, there is no superposition of the hammer and the vial, only of the atom itself, which is not being continuously monitored.  If the atom decays, the cat dies, period.  Only from a point of view outside the box is there a superposition of the cat.  As soon as a larger system interacts with the atom, it is entangled with it.  But superposition remains, from the point of view of other systems that remain isolated.


This brings up a very crucial question.  Suppose I have a device that is constantly monitoring a radioactive atom.  The quantum Zeno effect dictates that the atom will never decay, from the point of view of this device.  Now suppose I have a second device, isolated from the first, that also monitors the radioactive atom.  But this device only checks the atom after 24 hours.  If the substance in question has a half-life of 1 hour, after 24 hours it is almost certain that the atom will decay.  Yet the quantum Zeno effect dictates that it CANNOT have decayed, because the other device has been monitoring it continuously.  So what happens?

Answer:  BOTH RESULTS HAPPEN, as long as the two devices are isolated from one another.  The device that monitors the atom constantly will observe that it never decays.  The device that checks the atom after 24 hours will almost certainly see it decayed – as long as they don’t exchange information in the future.  But if information about the state of the atom is EVER exchanged between the two, even thousands or millions of years later, the atom will be found to have not decayed by BOTH devices.


This illustrates two critical points about quantum mechanics.  The first is that it is all about INFORMATION.  The flow of information between systems is what determines the behavior of particles.  And second, it does allow a kind of backwards in time effect, because what we see today is a function of what information will be exchanged in the future.


Now you might think this leads to an inherent contradiction.  Suppose I store the 24-hour reading from the first device in a computer file (let’s say as a zero, meaning no decay) and the 24-hour reading from the second device in a different file (let’s say a zero for no decay or a one for decay).  Then I have someone examine the second file, someone who has no knowledge of the experiment.  It’s quite likely that the file contains a 1.  He tells me so.  Then I decide to examine the first file.  It MUST contain a zero, because of the quantum Zeno effect.  So the two results contradict each other!

But this doesn’t happen, and understanding why it doesn’t happen helps us understand the real strangeness of QM.  Remember that I have knowledge of the nature of the experiment.  So when my friend gives me the result from the second device, we are creating an informational connection between the two devices.  The two devices are now entangled, so the quantum Zeno effect applies to both.  My friend will invariably see a zero.  The behavior of the two devices MUST comport with their informational connections, even if those connections happen in the future.


Notice something very crucial though.  The behavior of an atom could be said, in this scenario, to depend on nothing more than what knowledge a person has about the experiment.  Information, and information alone, determines the outcome.  This is the real revolution in physics that quantum mechanics represents.  The flow of information controls the behavior of physical systems.


I repeat – These are not wild speculations.  They are very much in line with mainstream physics.  Many theoreticians have tried to “save” quantum mechanics from its strangeness.  They have failed.  Quantum strangeness seems unavoidable.  The exchange of information, even in the future, affects the behavior of physical objects in the present.  Physical objects can pass “through” other physical objects without affecting them.  And there is no such thing as an observation that has no effect on what is being observed.   Quantum mechanics is so revolutionary that, decades after it was formulated, the most brilliant physicists are still trying to get their minds around it.  Our common sense is a very poor guide to the universe.  Understanding it requires us to stretch ourselves.  The universe is strange.  Very strange.










Revolution 2 – Chaos Theory

Edward Lorenz was a mathematician and a meteorologist.  Back around 1960, he was running a program to simulate the weather, and he noticed something strange.  If he took the output of the program at a given time, and plugged this number into a new simulation, the 2 simulations would soon produce vastly different results.  At first he thought there was something wrong with his computer.  But soon he realized what was happening.  In taking the output from one simulation and plugging it into another, he was using rounded-off numbers.  Although the numbers he was plugging in differed by only the tiniest amounts from the ones the original simulation was using, this was enough to quickly drive the two simulations apart.


Lorenz eventually called this the butterfly effect.  The flapping wings of a single butterfly would affect the movement of the air surrounding it, which would in turn affect the characteristics of the air surrounding that, and so on.  Because the mathematics of the atmosphere are mostly non-linear, any differences would grow exponentially over time.  The tiniest local effects would spread outward and build dramatically.  To precisely predict the weather in Australia next week, you would have to know what every butterfly on earth was doing today.


Lorenz was one of the pioneers of what we now call chaos theory.  One of the most common misconceptions about this theory is that it involves randomness.  It doesn’t.  In fact, Lorenz and others showed us that systems like the weather tend to stay within fairly well-defined boundaries.  We don’t suddenly see supersonic winds one day, or see the temperature shoot up to 200 degrees Fahrenheit.  This is because of what are called attractors.


The basic idea of an attractor is pretty simple.  Think of a thermostat that controls both heating and cooling systems.  It “attracts” the temperature to a specific value.  Such attractors are all around us, and within us.  Our bodies regulate their own temperature, glucose, sodium levels, and so on.


But seemingly simple systems can produce some unexpected behavior.  A pendulum, for example, is a simple system that behaves very predictably.  But what happens if we connect 2 pendulums together?  You can see the result here:  https://en.wikipedia.org/wiki/Chaos_theory#/media/File:Double-compound-pendulum.gif.  As you can see, the tip of the combined pendulum traces out a pretty wild path.  But notice that over time, this path is concentrated in particular areas.  In other words, the system is attracted to specific trajectories, but not to a single point, like a thermostat.  This is called a strange attractor.


Strange attractors allow us to predict the general behavior of a system.  But they do not allow us to precisely predict where it will be at some future time.  Changing the initial conditions by the tiniest amount leads to huge differences later on.  Since we can’t measure the initial conditions with infinite precision, we can’t possibly make precise predictions very far into the future.  This is true even though the system is following VERY SIMPLE mathematical rules.  It is following them faithfully.  No randomness is involved.  Yet the system’s future state is unpredictable beyond a certain point.


The behavior of many systems depends on what the specific conditions are.  A system that behaves in a very predictable way can become very unpredictable if we change the conditions.  Take the growth of a population for example.  Let’s say we have a population of birds.  The environment can only support a given number.  That’s called the carrying capacity.  As with any population, the birds produce offspring at a certain rate.

If the rate of reproduction is low, the population will grow slowly until it approaches the carrying capacity.  Then it will start to level off.  After that it will fluctuate a bit around the carrying capacity, but the opposing forces of reproduction and mortality with always drive it toward this value.  The system is quite predictable.


However, if we increase the rate of reproduction sufficiently, the population will grow so quickly that it will overshoot the carrying capacity.  In other words, the mortality will not have enough time to “catch up” with reproduction.  But after the overshoot, mortality will be high.  The population will drop back down sharply.  Then it will rise back up sharply, again overshooting.  This process will continue, producing an oscillation.  The population will never steady down to a specific value, but it will oscillate regularly back and forth.  Again the system is predictable.


If we increase the rate of reproduction to a certain point, though, the mortality will never be able to properly compensate for the reproductive rate.  The population will overshoot badly, then crash to a low level.  Then it will quickly recover and overshoot again.  But how much it overshoots is very much dependent on how far it crashes.  Instead of oscillating regularly, the population will exhibit wild, irregular swings.  The future state of the population will be unpredictable, even though it is still obeying the very same, very simple, mathematical rules.


In this situation, the reproductive rate is what is called a stress parameter.  Many systems only exhibit chaotic behavior if you increase the stress parameter enough.  Water for example.  When water flows slowly, it usually exhibits what is called laminar flow.  In other words, the water molecules are staying within specific parts of the flow.  But as the flow rate increases, the water will eventually reach the point where laminar flow can no longer occur.  Instead, the water breaks into turbulence.  Turbulence is one example of chaos.  In this system the flow rate is the stress parameter.


Naively, one might think that the transition of a system from non-chaotic, predictable behavior to chaos would occur quite randomly.  But one of the astonishing revelations of chaos theory is that this isn’t true at all.  A given system makes this transition at very predictable points as you increase the stress parameter.  Some systems, like the bird population described above, first transition to a cyclic mode before reaching the level of chaos.  The point of transition is very consistent, very predictable.  And the reason is simple.

Remember that chaos often comes from very simple mathematics.  What chaos mathematicians discovered is that these equations produce very consistent results when you change the stress parameter.  Take this equation for example:

Xn+1 = rxn(1-xn)

Notice that this equation gives a value of x (xn+1) as a function of a previous value of x (xn).  Notice also that this equation contains non-linearity, because it can also be written like this:

Xn+1 = rxn – rxn2

You can start with any value of x (let’s say 1), plug it into the equation, and get the next value.  Then you can take that value in turn and plug it in again.  In this way you can predict future population levels of animals, or the state of any dynamical system.  Each step is called an iteration.


This simple equation describes the population dynamics of many animals.  You only need to know 2 things.  The letter x represents the population level in relation to the carrying capacity of the environment.  The letter r represents the reproductive rate.  That’s all we need to know.  What is rather astonishing is that this simple equation produces some amazingly complex behavior as we change the stress parameter r.  If r is below 3, the population always approaches the carrying capacity, eventually fluctuating around the carrying capacity modestly.  But if r is between 3 and about 3.5, the population will oscillate back and forth, above and below the carrying capacity.  If r is between about 3.5 and 3.57, the population will oscillate between multiple levels, the number of levels increasing rapidly as we increase r.  But its behavior is still quite predictable.  Then, at about r = 3.56995, something quite dramatic happens.  The population abruptly shifts to chaotic behavior.  The population levels become highly sensitive to the initial conditions.

Note that this is simply a result of the MATH.  Seemingly simple mathematical equations produce these very complex, unexpected results.  No randomness is involved.  We can produce a graphical representation of this, called a logistic map:


Again, as r increases to 3, there is only one stable state.  Then, at an r of about 3, the system oscillates between 2 states, then 4, then 8, and so on.  At an r of 3.56995, it suddenly transitions to chaos.  All of this complexity comes from a simple equation.  We plug in an initial value into the equation, get a new value, plug that one in, and so on.  When we do lots of iterations, we see these kinds of patterns.


Different mathematical equations produce different transitions.  In the 1970’s, mathematician Michael  Feigenbaum discovered something quite amazing.  Notice again that the logistic map has these points where it splits.  One stable state splits into 2, then 4, and so on.  If you look at the points at which these sudden transitions happen, you notice something.  The transitions happen closer and closer together.  It turns out that this ALSO follows a very precise mathematical pattern.  The differences in the stress parameter at these transitions are declining at a very precise rate.  That rate we now call one of Feigenbaum’s constants.

Far from giving us mere unpredictability, chaos theory shows us that there are “hidden” mathematical patterns in simple equations.  Take real numbers.  Real numbers can be positive or negative.  We can represent the real numbers by a horizontal line, centered on the number 0.


But some numbers do not fit here.  For example, 12 = 1.  -12 = 1.  So the square of any real number is a positive number.  Then what is the square root of -1?  This is called an imaginary number.  It doesn’t fit into the real numbers.  So how can we represent imaginary numbers?  Easy.  By drawing another line, perpendicular to the real number line.  But this line, instead of being centered on 0, is centered on 0 TIMES the square root of -1 (which is represented by the letter i).  Notice that 0 times i is 0, so the 2 lines cross at the real number 0, which is the same as the imaginary number 0i.


Now we have a plane.  The numbers along the horizontal axis are real numbers.  The numbers along the vertical axis are imaginary numbers.  What about all of the points that don’t fall on these axes?  It turns out that any point on the graph can be represented as the sum of a real number and an imaginary number.  If a is a real number, and bi (b times i) is an imaginary number, then any point on the graph is the sum of the two – a + bi.  This sum is represented by the letter c, and is called a complex number.  A point on this graph represents what’s called a complex number, a combination of a real number and an imaginary one.


It might seem like we’re just playing a silly little game here, but this is not the case.  Complex numbers have everything to do with chaotic dynamics.  Suppose we have this equation:  xn+1 = c + xn2.  The letter c is a complex number.  Remember that the logistic function looks like this:  xn+1 = rxn(1 – xn).  This can also be written as xn+1 = rxn – rxn2.  As you can see, this looks quite similar to the first equation.  In fact, we can iterate the first equation, starting from an xn of zero, using different values of c.  Let’s start with c = 1.

X1 = 0

X2 = 1

X3 = 2

X4 = 5

X5 = 26

Notice that the numbers just keep getting larger.  In fact they keep getting larger at a faster rate.  The values do not steady down.  Now let’s try c = -1.

X1 = 0

X2 = -1

X3 = 0

X4 = -1

X5 = 0

Here we get a very different result.  The values bounce back and forth between -1 and 0.  What if we try c = 0.2?

X1 = 0

X2 = 0.2

X3 = 0.24

X4 = 0.2576

X5 = 0.2664

Although the values are increasing, the rate of increase is slowing.  No matter how many iterations we do, the values will stay within certain limits.

In other words, there is one set of complex numbers, that when you plug them into this equation and iterate starting at 0, produce a dynamic that trends away from zero, indefinitely.  There is another set of complex numbers, that when you plug them into this equation and iterate starting from 0, produce a dynamic that stays within a well-defined area.  If we explore this with many, many different complex numbers, we get this.


The black area of this graph contains all of the complex numbers that, when plugged into the equation, iterate to well-defined points on the graph.  The white area of the graph contains all of the complex numbers that iterate away from 0 indefinitely.  The black area is called the Mandelbrot set.  This mathematical object has remarkable significance and has a lot to teach us, not to mention amaze us with its hidden complexities.

First, instead of just black and white, let’s use some color to enhance the edges of the set.  The black area is still the area of the Mandelbrot set.  The other colors are outside the set.


For one thing, the boundary of the black area is INFINITELY LONG.  It is like a shoreline, that no matter how much you magnify it, you just see more little inlets and outlets.  For example, here is a zoomed-in look at one area of the graph.


Note that the edge of the set has a similar pattern that repeats at different scales.  The most obvious pattern is a shape that looks kind of like a bulb.  In fact, mathematicians call it a bulb.  The Mandelbrot set has innumerable bulbs occurring at lots of different scales.  They are similar but not identical.  The boundary of the set just keeps showing this incredible complexity as we zoom in closer.  Here is an even closer look at one area.


And here is another look, zoomed in even closer.


Notice that this bulb shape, along with others like spirals, just keeps repeating as we look closer and closer at the boundary of the set.  No matter how much we zoom in, we will keep seeing a highly complex boundary, infinitely long.  In fact, here is a still closer look at one tiny part of the boundary of the Mandelbrot set.


Notice that this tiny portion of the Mandelbrot boundary closely resembles the boundary we saw above, before we started zooming in.  This kind of pattern, in which similar shapes are repeated at different scales, is called a fractal pattern.  Many things in nature exhibit patterns like this – river systems, mountain ranges, lightning bolts, seashells, the list goes on and on.  In fact, computer-generated landscapes are often produced with the help of fractal geometry.  The stunningly complex and hauntingly beautiful geometry of the Mandelbrot set boundary is a marvel to behold.  Many programmers have produced animations, showing the “camera” zooming in ever closer to some portion of the boundary.  You can watch an example here:  https://vimeo.com/12185093.  That a simple mathematical equation could produce so much complexity is astonishing in itself.  But we’re not done.  Notice something about these bulbs.  Each bulb has a smaller bulb attached to it, and a still smaller one attached to that one, and so on.  If we look at these progressions, we find something quite remarkable.  Let’s go back to the original, zoomed-out graph:


Remember the logistic map, which describes the behavior of many animal populations?


Notice something?  The points at which the logistic map branches, first into 2, then 4, and so on, come faster and faster as we increase r, the stress parameter.  The bulbs of the Mandelbrot set get smaller and smaller, and therefore repeat more frequently, as we move to the left away from 0.  I wonder if there is a connection?


No question about it.  If we flip the logistic map over, the branching points (called bifurcation points) correspond exactly to the starting points of each bulb of the Mandelbrot set along the real number line.  This is only one way in which the Mandelbrot set gives us insights into patterns we see in nature.


If you’ve been paying attention, you may realize at this point that “chaos theory” is a bit of a misnomer.  Chaos theory is actually about finding highly complex, but orderly behavior hidden within seemingly simple mathematical equations.  Chaos theory gives us a handle on dynamical systems.  But for many systems, under many conditions, it doesn’t give us PRECISE PREDICTION well into the future.  It only shows us why we can’t have it.  Newton’s clocklike, predictable universe was a pleasant fantasy.  But the very mathematical equations he gave us rip the clock to pieces.






Revolution 1 – Relativity

Relativity can be a tricky concept to get your mind around, so much so that even science educators sometimes get things wrong.  But understanding the strangeness of relativity isn’t too hard.  Let’s start with the speed of light.


The speed of light is just that.  It’s the speed at which photons, the “particles” of light, travel.  Photons are not like little bullets shooting out of a gun.  A bullet accelerates as it is pushed forward by the expansion of gases in a gun barrel.  Photons DON’T ACCELERATE.  They ALWAYS travel at the same speed.  Always.  People are fond of saying, “everything is relative.”  The irony is that Einstein’s relativity tells us that one very important thing is NOT relative.  The speed of light.  The speed of light is a constant.


Now you may have heard that light travels more slowly through water or glass than through a vacuum.  But this is misleading.  The photons ALWAYS travel at the same speed.  They don’t speed up or slow down.  They spring into existence traveling at light speed, and they wink out of existence traveling at light speed.  What looks like a speed change in glass is actually caused by photons getting absorbed by atoms in the glass, and new ones being created.  The time lag between the two is responsible for what looks like a change in speed.


The speed of light doesn’t change, regardless of how you measure it.  You can be at rest as a beam of light shoots past you, or you can be traveling alongside it at close to its speed.  Either way, you will measure its speed as being the same.


By contrast, all other velocities are relative, and this is where people often get confused.  What is your current velocity?  Answer:  You don’t have ONE velocity.  You have an infinite number.  It all depends on what you’re measuring in relation to.  Right now, my velocity relative to my chair is zero.  Relative to the center of the earth, hundreds of miles per hour.  Relative to the sun, thousands.  And so on.


Solar System orbits, artwork

Furthermore, if you are in a plane flying overhead at 100 mph, and I’m standing “still” on the ground, it doesn’t actually make sense to say that your speed is 100 and mine is zero.  It’s convenient for us, obviously, but it actually DOESN’T MAKE SENSE in relativity.  Because it implies that I’m not moving.  But I AM MOVING.  The earth, the sun, the galaxy are moving through space.  There is no “unmoving spot” for us to measure our absolute velocity against.  This is a critical element of relativity.  Velocity is relative.  If you are moving in relation to me, I’m moving in relation to you.


Since the speed of light is constant, I can always compare my speeds with that of light.  The speed of light is represented by the letter c.  Remember, though – I don’t have ONE velocity.  It’s all relative.  If I measure my velocity in relation to my chair, it’s zero – in other words, 0c.  Let’s say I measure in relation to the sun.  I get about 67,000 mph.  The speed of light is about 670,616,628 mph.  So my speed in relation to the sun (and its speed in relation to me) is only a tiny fraction of the speed of light – less than 0.0001c.  Large, everyday, nearby objects in our lives travel at very slow speeds in relation to light.  This is why we don’t typically see relativistic effects.


Some very distant objects do travel at high speeds in relation to us.  Distant galaxies for example.  In some cases they are flying away from us (and us from them – remember, it’s relative) at large fractions of the speed of light.  At such speeds, the effects of relativity become noticeable.  Because of the mathematics of relativity, it turns out that relativistic effects are very minor, until you get close to the speed of light.  But as you do, they increase dramatically.  The passage of time, for example.  At everyday speeds, the flow of time does not change noticeably.  Your friend’s watch ticks along at the same rate, whether he’s standing next you or flying overhead.  But if your friend were traveling at 99% of the speed of light in relation to you, his watch would tick away at a rate about 7 TIMES more slowly than yours.  Similarly, he would observe your watch ticking away at a rate 7 times more slowly than his.

Just to be clear – When you looked at your own watch, you would see nothing unusual.  If another friend was standing next to you, he or she would look perfectly normal.  The effects of relativity are just that – RELATIVE.  That doesn’t make them any less real.  Relativistic effects are inescapable, and some of our technology has to take them into account.


Another dramatic effect of high speed is that mass, something we think of as very constant, actually changes.  Again, this effect is relative.  Your friend flying overhead doesn’t get more massive from his own point of view.  But from yours, he does.  He is in fact 7 times more massive than he would have been standing next to you.  And vice versa.  And if he flies into something at 99% of the speed of light, his momentum will reflect his greatly increased mass.


In particle accelerators, subatomic particles are often accelerated to near light speed.  When these accelerators are built, they have to take this into account.  According to relativity, the particles will have far greater momentum at near light speed than we would calculate, based on their rest mass.  And they do.  This is only one of the ways that Einstein’s theory is routinely confirmed in today’s technologically advanced world.


Up to this point, I have been speaking about what is called SPECIAL relativity, which deals with the effects of velocity.  But Einstein went beyond that, to what is called general relativity.  In general relativity, gravity becomes very important.  It is not merely a force.  It is crucial to the very shape of space and time.  In general relativity, the presence of matter alters the shape of space.  Objects follow the paths they do in space simply because they are following the local shape of space itself.  Time is also affected.  Time actually flows more slowly the closer you are to a massive object.


Again, in everyday life we don’t see these effects.  It takes some STRONG gravity to change the shape of spacetime itself so strongly that the effects on time would be obvious.  Even being close to our sun wouldn’t do it.  But there are objects in the universe that produce incredibly strong gravitational fields.  Black holes for example.


If you were some distance from a black hole, and your friend was close to it, you would notice that his watch seemed to be running more slowly than yours.  That’s because it is!  And to him, your watch looks like it’s running fast.  Notice that unlike the relativistic effects of velocity, these effects are not symmetric.  Your friend is actually aging more slowly than you, as long as he’s closer to the black hole.  If he stays there a long time, when you reunite you might be years older, while he might have aged only hours.


It turns out that we can achieve a similar effect without a black hole, in a scenario called the twins paradox.  We take a set of identical twins.  One of them gets in a space ship and takes off, the other stays on the earth.  The space-faring twin accelerates to near the speed of light.  Eventually he turns around and heads back to earth.  When he arrives, he notices something.  His twin is now much older than himself!  This happens because unlike velocity, acceleration is NOT relative.  Only one of the twins accelerates.  (A physicist would say that he changes his inertial reference frame.)  This creates the asymmetry.  If they both accelerated in the same ways and reunited, they would still be the same age.  But since they didn’t, you get this kind of effect.  Depending on how closely one of the twins approached the speed of light, and for how long, he might return to earth months later to find that millions, even billions of years had passed.


All of the effects I’ve described are based on solid, widely accepted physical principles.  Some of our technologies, such as particle accelerators and GPS systems, are designed with relativity in mind.  They have to be, because Einstein’s theories are not just theories.  They are well-established, well-documented principles governing our physical universe.  They are also strange, because they violate our common sense intuitions about how things should work.  We don’t see twins aging at noticeably different rates, because one of them flew to Australia this week and the other didn’t.  We don’t see space ships running up against a universal speed limit.  We don’t see people’s watches running slowly because they are in a valley and we are on a mountain.


We don’t see these things because we experience life at very slow velocities and very low gravities.  A commercial jet flies at about 600 mph, which is about 0.0000009c.  The entire earth is pulling on my body with a force of only about 200 pounds.  A pit bull can generate more force than that with its jaws.  Our experience is therefore highly skewed.  Much higher velocities and much stronger gravities exist, but we don’t experience them.  We don’t realize how truly strange the universe is.





The Strangeness of Nature

Isaac Newton was a genius.  He didn’t learn calculus.  He INVENTED calculus.  He showed us that simple equations predicted the behavior of everything from apples falling to the orbital path of the moon.  Classical physics and Isaac Newton are virtually synonymous.


Suppose I throw a ball.  Classical physics gives us simple equations that precisely predict the behavior of that ball.  Gravitational forces will cause it to follow a parabolic path through the air.  How far it travels depends on how much force I use and the angle at which I throw it.  The equations that govern these things are pretty simple, and the very same equations govern the motions of moons, planets, and stars.  Because these objects obey these simple mathematical rules, we can accurately predict exactly where you can see a solar eclipse years from now.


Newton envisioned the universe behaving like a clockwork mechanism.  Knowing the state of the “clock” at any given moment, you could, in principle, predict its exact state at any time in the future.  All you had to do was plug the numbers into the equations and get an answer.  The only uncertainty was due to the fact that you couldn’t measure the initial state of the clock with INFINITE precision.  But if you could measure it with a lot of precision, you could make an equally precise prediction.


In Newton’s view of the universe, space and time are inflexible.  That is, space has the same “shape” everywhere, and for everyone.  The same with time.  The flow of time doesn’t change for different observers.  In the Newtonian view, space and time are kind of like “background.”  Objects move around, energy is converted from one form to another, but space and time are always there, inflexible, unchanging.


Newton showed us that if you push an object twice as hard, it accelerates twice as rapidly.  If you push it four times as hard, it accelerates four times as rapidly.  And so on.  The acceleration of a given mass is a simple function of the force applied to it.  Presumably, if you push an object with enough force, you can make it go a billion miles an hour.  Right?


It turns out that Newton was wrong.  Newtonian physics is fundamentally incorrect.  In the 20th century, 3 revolutions came about in physics, which destroyed Newton’s framework.  That’s not to say that Newton’s equations are worthless.  In the everyday world, they are often a VERY good approximation.  We can indeed predict the motion of a ball very precisely.  We can indeed predict solar eclipses years into the future.  BUT – The universe is not like a clock.  Space and time are changeable.  And it turns out that we can’t go a billion miles an hour.


If we fire up a spaceship and head out into the universe, we CAN make it go a million miles an hour.  This is actually not as hard as you might think, because space doesn’t have the problem of air resistance.  Accelerating constantly at the very modest rate of 1 g, in less than 14 hours we will be traveling at 1 million miles an hour.  (Incidentally, even at this speed, it would still take us about 3 weeks to get to Jupiter – That’s how big space is.)  If we kept it up, at this rate, in less than a year we could achieve 500 million miles an hour.  But if we keep pushing and pushing, we start to notice something strange.  It gets harder and harder to make the ship go faster.  As we pass 600 million miles an hour, it’s taking a lot more force to add any more speed.  When we get to 660 million miles an hour, we’re having to use enormous amounts of force to increase the speed.  It’s as if, instead of going into more velocity, the energy we’re putting in is making the ship more massive, increasing its inertia.  At about 670 million miles an hour, the resistance of the ship to any further speed increase is becoming almost impossible to overcome.  It turns out we will never get the ship past about 670,616,629 miles per hour.


If we can see outside, we notice something strange.  The stars ahead of us all look bluish.  Some of them in fact seem to have vanished.  The stars behind us, the ones we can see, all look reddish.  If we have a telescope to look back at our friends on the earth, it seems like we’re looking at everything through a red filter.  The oceans, which should look blue, now look yellowish.  The people look reddish.  And we notice something that’s even stranger.  Everything seems to be moving in slow motion.  The earth itself seems to be spinning more slowly.  If someone on earth has a telescope to look at us, they see something similar – our ship and everything in it looks more reddish than normal, and WE seem to be moving in slow motion.


Albert Einstein predicted all of this in the early 20th century.  And every experiment, every observation we have made since, indicates that he was correct.  Einstein’s special and general theories of relativity destroyed the Newtonian view of things.  Einstein showed us that mass, and even space and time are not constants.  But the strangeness gets worse, much worse.


If you plug the current state of the sun, moon, and earth into Newton’s equations, you can indeed predict their future positions quite accurately.  But even Newton realized that this was a special case.  The moon and earth have very stable orbits.  But lots of orbital scenarios lead to very messy, unstable orbits that look like tangles of spaghetti.  Prediction becomes an impossibility.  These results come about because some of Newton’s equations, although simple, contain non-linearity.


A linear relationship is one like this:  F = ma.  The acceleration of an object is directly related to the force applied to it.  Twice the force produces twice the acceleration on a given mass (ignoring relativity for the moment).  Gravity, however, is a force that exhibits non-linearity.  The force of gravity is a function, not of the distance between 2 objects, but the SQUARE of the distance between them.  The force of gravity between 2 objects that are twice as close is 4 TIMES as great.  The force of gravity between 2 objects that are 4 times as close is 16 TIMES as great.  This is an example of non-linearity.


Because of this non-linearity, a tiny change in the initial state of the system can lead to big differences later on.  The differences multiply exponentially over time, and when you add more objects, these complexities multiply to the point that the system quickly becomes unpredictable, even though the objects are still obediently following simple mathematical rules.  This is called chaotic behavior.  Chaotic behavior is something we see in everyday life.  The weather is chaotic, which is why it becomes unpredictable after only a short time.  Chaos is a reality we have to live with.  Even with simple mathematical rules, we often get unpredictable behavior.  So much for Newton’s clocklike universe.


Now you might think that if only we could measure the initial state of a system very, very, very precisely, we could get around the chaos problem.  These systems are still obeying simple mathematical rules.  We just have to pin down the initial conditions, and then we can make predictions.  In principle, Newton’s clocklike universe still holds, right?  Unfortunately, there is something that throws a huge monkey wrench into this, and it is the most significant of the three 20th century revolutions in physics.


Because of chaos, even the current position of an individual electron might have a significant effect on the weather on the other side of the earth a year from now.  The problem is, an electron doesn’t always HAVE a precise position.  Welcome to quantum mechanics.  Everything material is made of subatomic particles.  The problem is that when we break matter down into its smallest parts, things get very fuzzy.  There is a fundamental, inescapable randomness in nature at the level of the very tiny.  Since these random variations will translate into big differences later on, there is a fundamental randomness at the macroscopic level too.  We’re stuck with it.


This picture of nature, which has been confirmed in experiment after experiment, blows Newton’s “clock” idea right out of the water.  The universe has fundamental unpredictability built into it.  We cannot, even in principle, know the precise state of every subatomic particle in a system at any point in time.  Therefore we can’t precisely predict the state of that system indefinitely into the future.


We are accustomed to thinking of nature as ordered.  Our intuitions lead us to think that we can predict tomorrow by looking at today.  That’s because the everyday world IS often predictable.  Newton’s ideas work quite well in many situations.  But these are always approximations of what is actually going on.  Nature doesn’t actually conform to our intuitions.  Nature violates “common sense.”  Badly.  In other words, nature is strange.  It’s important to understand this, because physical reality is one of those things that we tend to take for granted, and uneducated people tend to think that “common sense” is a good guide to it.  But relativity, chaos theory, and quantum mechanics, the three physics revolutions of the 20th century, tell us otherwise.

Understanding the Book of Genesis in Context

This is a rather long post, written for those of you who want to understand the Book of Genesis, as Biblical scholars understand it.  If you have no interest in this, you might as well stop here.  At the outset, let me say that any piece of literature can be interpreted to mean whatever the reader thinks it means.  For example, suppose I write, “My car is black.”  Seems pretty straightforward.  But you can interpret this literally or figuratively.  You might say for instance that I actually mean black in the figurative sense of invisible, or concealed, in the same sense that people use “black op.”  You might argue that I meant that my car is perfectly concealed.  So depending on what background my car is against, it might look cherry red, sky blue, or even snow white.  Or you could argue that I meant the word car figuratively – that the whole sentence is an allegory, alluding to the fact that my life is a journey in which I float through society almost like a ghost, being unnoticed.  But armed with a knowledge of me, my other writings, and the culture and time I live in, you might well argue that some interpretations are pretty far-fetched, while others are quite reasonable.


Of course, if your language is very specific, it gets harder to broadly interpret what you write.  If I write, “My car reflects the wavelengths of visible light in amounts as follows…,” it seems you are pretty limited in your interpretations.  But of course the Bible is not written in such language.  Instead, we are left with our understanding of the ancient cultures of the region and time, and our knowledge of the process by which “the Bible” was created, including many ancient writings in the original languages.  And we are left with a fundamental choice – whether to be reasonable and let the evidence speak for itself, or start with a lot of baggage of dogma and doctrine and try to force the evidence to fit that.  My grandfather was a Methodist minister, and like many Americans, I grew up surrounded by Bible-believers who skimmed over many parts of the Bible without much explanation.  Generally, they themselves had never been educated in the historical context of it, or the process by which “the Bible” was created.


What does this have to do with science, technology, and critical thinking?  Nothing perhaps, if you haven’t been indoctrinated at a young age by Bible-believers who approached the scriptures without critical, open-minded skepticism.  If I didn’t think a large number of Americans were having their critical thinking skills dulled or even squashed by such indoctrination, I wouldn’t bother.  Examining the Bible with a critical, exploratory mindset, as many Christian and Jewish scholars have, is often an important step in applying critical thinking in general, especially if you have been raised by Christians or Jews.

Why Genesis?  Because I think it’s a great place to get clarification and context.  It is about origins and world views and the nature of the creator.  And of course it has had an enormous influence on our society.  I will be quoting other related parts of the Bible as well.  Needless to say, I will not go through the whole book of Genesis.  For my purpose it is sufficient to focus on the first few chapters.


Let’s start with the basics.  What we call the “Book” of Genesis was not originally composed as a single body of work.  No serious Biblical scholar, religious or secular, believes it was written by Moses.  A broad consensus of scholarship holds that it, like Exodus, Leviticus, and Numbers, was built from a number of sources, modified over the centuries.  All of these were originally written in the Hebrew language.  This is actually a bit trickier than it seems, because there were different forms of Hebrew.  It’s quite possible that some or all of it was originally written in Paleo-Hebrew, a writing derived from Phoenician.  This language was used by the Israelites from about 1000 BCE to about 500 BCE.  It looks quite different from what most people think of as Hebrew writing (see above).  Parts of what we now call Genesis (as well as parts of other Old Testament books), in Paleo-Hebrew, were among the Dead Sea Scrolls.


Around the same time that Paleo-Hebrew developed, what we call Biblical Hebrew also came into being.  This is the ancient Hebrew script that most people would recognize today (see above).  It used a modified Aramaic script.  It was written from right to left, and originally had no vowels, although vowel points were eventually added.  There are no chapters, verses, or “books” in this “original” material.  Most of the Hebrew scripture that corresponds to the Book of Genesis survives in this form – for example, the Dead Sea Scrolls (https://en.wikipedia.org/wiki/Dead_Sea_Scrolls).  Scribes preserved these writings, but also changed them, over the centuries, from about 500 BCE to the time of Christ.  The Dead Sea Scrolls date from about 300 BCE to about 100 CE.


A few hundred years before the time of Christ, the scriptures that would become the Old Testament in some Bibles (and some related texts) were translated into Greek.  This translation is called the Septuagint.  It is this translation that the writers of the New Testament, such as the apostle Paul, had access to and quoted, and it was the basis for early Latin translations that would evolve into most Catholic Bibles.  But it would NOT become the basis for Protestant translations of the Old Testament.


Between 700 and 1000 CE, the texts that would eventually form much of the basis for the Old Testament in Protestant Bibles were created.  These are collectively called the Masoretic Text.  They were written in both Hebrew and Aramaic, translated from Biblical Hebrew.  It was here that the scriptures were divided into books, verses, and so on.  By this time, the Catholic Church was already using its own collection of scriptures, in Latin, called the Vulgate, translated from the Greek Septuagint.  Keep in mind that most people could not read Hebrew, Latin, or Greek at this time.  Bibles were not intended for the masses of people, but for priests and other clergy – in fact ordinary people were forbidden to even attempt to read the scriptures.


In 1611, the King James Bible was published.  The scriptures were translated from Hebrew, Aramaic, and Greek into English.  This was only 5 years before Shakespeare’s death – more than 100 years after Columbus discovered the New World.  Protestantism had existed for less than a century.  The King James Bible introduced the beautiful, flowery style many people associate with the Bible.  For example, “Vanity of vanities, saith the Preacher, vanity of vanities; all is vanity.  What profit hath a man of all his labour, which he taketh under the sun?  One generation passeth away, and another generation cometh; but the earth abideth forever,” as it says in Ecclesiastes.  The high literary quality of this version undoubtedly contributed to its success, and many more recent versions of the Bible are built on it.


Through this whole process, there were numerous edits, syntheses, canonizations, and of course translations.  Different sects of modern-day Christianity (not to mention Judaism) have included and excluded different scriptures.  For example, the King James Bible, Protestant of course, includes 39 books in the Old Testament.  The most commonly used Catholic Bible contains 46 Old Testament books, and the Orthodox Bible contains 49.  So where should we focus our attention if we want to understand “the Bible”?  Well, since most Americans have grown up with English Protestant Bibles, built on the King James Bible, that seems a logical place.  But we also have ancient scrolls in Hebrew and Aramaic that can help us dig a little deeper.  And critically, we have a lot of other archaeological information, including written information, from the region during the Iron Age, which can help us immensely with the context.

So let’s begin.  But instead of just skimming over the words, let’s REALLY LOOK AT THEM.

In the beginning God created the heaven and the earth.

And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters.

The Hebrew word for God here is Elohim.  It is simply a generic word for god.  Later in Genesis, you see the phrase “Lord God.”  This corresponds to the Hebrew words “Yahweh Elohim.”  The name Yahweh is often translated in the King James Bible as “Lord,” while Elohim is usually translated as “God.”  Yahweh was only one of a number of gods worshipped by the early Israelites.  Much of the Old Testament is a strong admonition to abandon other gods and worship only Yahweh.  There is a distinct break between Genesis 2:3 and Genesis 2:4.  Before this point, only Elohim is used.  After this, Yahweh Elohim is used (see here:  http://biblehub.com/interlinear/), and there is a distinct style change.  Why?  Modern scholarship has the answer.  There are at least 2 different authors, with 2 different origin stories, that have been spliced together.  This kind of editing is commonplace in the Old Testament.  These 2 origin stories are distinctly different:  In the first, plants and animals are created before man.  In the second, Yahweh Elohim clearly makes man before the animals.  He brings each animal to the man as it is created, so that he can name it.


Then there is the reference to “the deep.”  The Hebrew word for this is tehom or tehowm.  (I will not try to reproduce the Hebrew writing here; see the reference above.)  This has a very specific meaning in the Hebrew culture of the time, and a connection to nearby cultures.  The “Great Deep” or “Abyss” was the primordial ocean.  Indeed, later in Genesis, the very same word tehom is translated as “great deep.”  The Hebrew tehom is almost certainly derived from the Sumerian word Tiamat.  In Sumerian religion (as well as Babylonian, Assyrian, and Akkadian), Tiamat was the goddess of the primordial abyss, a symbol of the chaos of primordial creation.  Tiamat was sometimes represented as a sea monster.  She was defeated by the god Marduk in Babylonian myth.  In Canaanite religion, the god Baal does battle with 4 enemies, 3 of which are related to the sea.

Jumping to Psalm 74 for a moment:

Thou didst divide the sea [Hebrew: yam] by thy strength: thou brakest the heads of the dragons [Hebrew: tannin] in the waters.

Thou brakest the heads of leviathan [Hebrew: liuyratan] in pieces, and gavest him to be meat to the people inhabiting the wilderness.

Passages like this must seem rather bewildering, unless you understand the context and the likely origin of such references.  This is not the only place the Bible speaks of sea monsters.  Why would God be breaking the heads of sea dragons?  Because these dragons represent the chaos of the Great Deep  The very names of 4 enemies fought by the god Baal in Canaanite (Ugaritic) culture have corresponding names of 4 enemies fought by Yahweh in later Hebrew culture:  sea (Ugaritic ym, Hebrew yam), leviathan (Ugaritic ltn, Hebrew liuyratan), tunnanu (Ugaritic tnn, Hebrew tannin), and “death” (Ugaritic mot, Hebrew mawet).  The sea was undoubtedly seen as chaotic and disorderly by the peoples of the region.  The common theme is the chaotic primordial ocean, with great monsters, which is tamed by a god.

The origin myths of Sumer and Babylon both begin with the primordial ocean, which is chaotic.  The waters of this ocean are then divided into waters above and water below, the waters above held by a dome over the earth.  This brings order out of the chaos.  Let’s see what happens in Genesis:

And God said, Let there be light:  and there was light.

And God saw the light, that it was good:  and God divided the light from the darkness.

And God called the light Day, and the darkness he called Night

And the evening and the morning were the first day.

And God said, Let there be a firmament in the midst of the waters, and let it divide the waters from the waters.

And God made the firmament, and divided the waters which were under the firmament from the waters which were above the firmament:  and it was so.

And God called the firmament Heaven.  And the evening and the morning were the second day.

Note that the sun has not yet been created – that won’t happen until the 4th day.  Yet day and night now exist.  There is not the slightest hint that the earth moves, nor that the day/night cycle has anything to do with the earth’s relationship to the sun.


The word firmament is translated from the Hebrew word raqia.  This word refers to a solid material, like metal, that is stamped out into a broad, thin structure.  This concept of the sky as a solid dome was common amongst Middle Eastern cultures of the time.  In Babylonian myth, the god Marduk, after slaying Tiamat, splits her body in two like a clam shell, forming the solid dome of the sky out of one half.  Thus the primordial ocean is divided into the water above and the water below.  If you’re wondering what the “water above” refers to, remember that the sky is blue, and yes, water does fall from above.  The word heaven is translated from the Hebrew shamayim.  This concept of heaven is, as stated above, that of a solid dome being held up by pillars and/or mountains.  Again, this is similar to the cosmology of nearby civilizations – the ancient Egyptians had the same idea, as did the Sumerians.  Other scriptures flesh it out a little more.  Job 26:11 reads, “The pillars of heaven tremble and are astonished at his reproof.”  2 Samuel 22:8 reads, “Then the earth shook and trembled; the foundations of heaven moved and shook, because he was wroth.”  Such language is bound to be confusing to those who don’t know the context.  Naturally it’s tempting to rationalize and say it’s all symbolic.   Or we can look at the cultures of the region at the time, and not impose our modern understanding of the universe on the text.  Then it falls readily into place.  Many Middle Eastern civilizations of the time share this cosmology – There is a large body of water above our heads, held there by a solid dome.  This dome is supported by pillars and/or mountains.  The sun, moon, and stars are affixed to the dome.  There is another large body of water below us, continuous with the seas that surround the earth.  The earth is a flat or slightly domed disk which sits on the ocean below.  There are “fountains of the great deep” (referred to later in Genesis 7) connected to this ocean – presumably they were referring to springs.  The earth does not move – unless of course it is shaken by God.

Let’s move on:

And God said, Let the waters under the heaven be gathered together unto one place, and let the dry land appear: and it was so.

And God called the dry land Earth; and the gathering together of the waters called he Seas: and God saw that it was good.

And God said, Let the earth bring forth grass, the herb yielding seed, and the fruit tree yielding fruit after his kind, whose seed is in itself, upon the earth: and it was so.

And the earth brought forth grass, and herb yielding seed after his kind, and the tree yielding fruit, whose seed was in itself, after his kind: and God saw that it was good.

And the evening and the morning were the third day.

Notice that there is still no sun, no moon, no stars.  Picture it in your mind – you have dry land and sea, plants but no animals, and there have already been 3 days and 2 nights, but there is no sun during the day, and no moon or stars at night.  There is not the slightest hint of lands or oceans beyond the Middle East.  Nowhere does it say, “Let the places to the far north and south be ice covered, and let most of the earth be covered with water.”


And God said, Let there be lights in the firmament of the heaven to divide the day from the night; and let them be for signs, and for seasons, and for days, and years:

And let them be for lights in the firmament of the heaven to give light upon the earth: and it was so.

And God made two great lights; the greater light to rule the day, and the lesser light to rule the night: he made the stars also.

And God set them in the firmament of the heaven to give light upon the earth,

And to rule over the day and over the night, and to divide the light from the darkness: and God saw that it was good.

And the evening and the morning were the fourth day.

Finally the sun and moon are created on the 4th day, and here we see some sloppiness in the creation story.  The sun and moon are merely lights.  The Hebrew word is maur or maowr.  It simply means a light, in the sense of something that gives off light, nothing more or less.  The Hebrew word for light is aur or owr, showing that the sun and moon are seen as nothing but sources of light.  These “lights” are set in the firmament to give light upon the earth, and to divide the light from the darkness – but the light was already divided from the darkness on the first day.  There is absolutely no hint that the sun, moon, or stars are anything but lights, set in the dome above the earth.  In fact 3 days had already passed without them – the earth, dry land, and plants were created before the sun, moon, and stars.  This may seem strange to us today, but it made perfect sense to people who thought the earth was the immovable center of all things – the sun and moon were merely lights placed in the firmament.  Everything in the sky is directed toward earth, and specifically toward humans.


And God said, Let the waters bring forth abundantly the moving creature that hath life, and fowl that may fly above the earth in the open firmament of heaven.

And God created great whales, and every living creature that moveth, which the waters brought forth abundantly, after their kind, and every winged fowl after his kind: and God saw that it was good.

And God blessed them, saying, Be fruitful, and multiply, and fill the waters in the seas, and let fowl multiply in the earth.

And the evening and the morning were the fifth day.

An interesting feature here is that fowl may fly “in the open firmament of heaven.”  This might lead us to think that the firmament is not the solid dome we have been supposing – but in fact, the Hebrew reads, “al pene raqia hashamayim.”  The Hebrew word pene means face, so this translates as “on the face of the firmament of heaven.”  There is not the slightest hint that this “heaven” is anything more than a dome above the earth.  The sun, moon, and stars are not far off.  Even the birds can approach this “heaven.”  When you get away from city lights and see how bright the night sky is, it’s not hard to imagine that the moon and stars are indeed very close.

Milky Way Over Crater Lake Ultra HD

And God said, Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creepeth upon the earth.

So God created man in his own image, in the image of God created he him; male and female created he them.

And God blessed them, and God said unto them, Be fruitful, and multiply, and replenish the earth, and subdue it: and have dominion over the fish of the sea, and over the fowl of the air, and over every living thing that moveth upon the earth.

And God said, Behold, I have given you every herb bearing seed, which is upon the face of all the earth, and every tree, in the which is the fruit of a tree yielding seed; to you it shall be for meat.

And to every beast of the earth, and to every fowl of the air, and to every thing that creepeth upon the earth, wherein there is life, I have given every green herb for meat: and it was so.

And God saw every thing that he had made, and, behold, it was very good. And the evening and the morning were the sixth day.

If you’re wondering about the use of the plural here, this is often rationalized by saying that it is a “royal plural,” analogous to a king saying, “We are pleased with you,” meaning himself.  Those who subscribe to the trinity doctrine suggest that it this is the explanation.  And if you’re wondering what happened to the story of woman being created out of the rib of man, that’s an entirely separate creation story that starts a few verses down.  It is only there that the Garden of Eden is mentioned (even located), and only there that the phrase Yahweh Elohim (translated as Lord God) makes its first appearance.  You may also notice that God gives “every green herb for meat,” in fact this is repeated.  Yet of course we know that some herbs are poisonous.  Like many creation myths in many cultures, there is sloppiness.


Many modern Christians and Jews argue that none of this should be taken literally, that it is all symbolic.  The problem is that we have no evidence that the ancient Israelites, or the people of nearby cultures, thought so, and plenty of evidence that they thought otherwise.  To this day there are many who believe Genesis is a factual history of the origin of the earth, sun, moon, stars, and man.  And the majority of independent scholars, and even some religious scholars, would agree that this was what the authors intended.  Just as countless societies all over the earth created origin stories, the ancient Israelites created origin stories.  It is readily apparent that many elements were borrowed from nearby societies.


Lest you think that that the interpretation I have given is that of godless atheists bent on destroying people’s faith, I point out that Biblical criticism goes back more than 200 years, and many if not most of the people who built this picture of scripture were religious people.  Julius Wellhausen, for example, the son of a pastor, developed the documentary hypothesis that teased apart the multiple authors of the Genesis, Exodus, and the other books of the Pentateuch.  Much of the research into ancient scriptures, as well as the ancient cultures of the Middle East, has been performed by devout people.  Mainline Christianity has long accepted that the stories in Genesis are not literal histories.  Reasonable, educated people, religious or not, do not dispute that many of these stories have been adapted from those of surrounding cultures.


The cosmology I have described is fleshed out more in other scriptures.  Above the firmament of heaven, there are chambers in which God holds such things as snow, hail, and pestilence, in readiness for when they are to be released.  You can read more about this here:


The second creation story focuses on the creation and fall of man.  It doesn’t deal with the sun, the moon, or the stars at all.  The Genesis story of the great flood also has roots in surrounding cultures.  And like the origin story above, it contains some sloppiness.  These statements are not controversial among thoughtful, reasonable people.  Much of what I have written here can be found on Wikipedia or in the writings of people who consider themselves Christians or Jews.  People who are committed to reason and intellectual integrity, religious or not, are unwilling to say something is white if it’s clearly black.  Of course, you, dear reader, are free to make your own interpretations and form your own conclusions.


One of the interesting things about Bible interpretation is that even the most fundamentalist among Christians and Jews usually reject the cosmology above, interpreting the scripture so as to conform to modern evidence of the nature of the solar system, galaxy, and universe.  At the same time they often insist that the 7 days of creation are 7 literal days, and that the earth is about 6000 years old.  In this and many other ways, so-called fundamentalists reject straightforward, reasonable, literal interpretations of scripture, because they conflict with so much that is obvious and irrefutable in the modern world.  But it’s hard to gain a lot of adherents, even among ignorant Americans, to the idea that there is a solid dome above the earth, covered with water, that the sun and moon are affixed to it, and that the earth is a flat disk.  It is much easier to convince the ignorant that the earth is only 6000 years old, that the day/night cycle existed before there was a sun, and that everyone living is descended from only 2 original people.


It is also worth expanding on what I have already alluded to, that much of the Old Testament is an attempt to crush the polytheism of the early Israelites.  In fact, the very name Israel is almost certainly related to the god El, who was the chief of the Canaanite gods.  With his consort, Asherah, his military subordinate Baal, and a host of other deities, he wielded power over the earth and its people.  Early on, Yahweh was brought into this collection of gods as a son of El.  There is a curious passage in the 32nd chapter of Deuteronomy:

Remember the days of old, consider the years of many generations: ask thy father, and he will shew thee; thy elders, and they will tell thee.

When the Most High divided to the nations their inheritance, when he separated the sons of Adam, he set the bounds of the people according to the number of the children of Israel.

For the LORD’s portion is his people; Jacob is the lot of his inheritance.

Someone referred to as the “Most High” divided to the nations their inheritance, and “the Lord’s” portion, his inheritance, are the descendants of Jacob.  Who is this “Most High”?  The Hebrew name is Elyon.  Who is “the Lord”?  As usual, the Hebrew here is Yahweh.  In most parts of the Old Testament, El and Yahweh are presented as one and the same, but here (and a few other places) it is fairly clear that they are 2 different beings.  Here it’s pretty hard to get around the conclusion that Elyon is dividing up the inheritance, and Yahweh is receiving his share.  This passage among others in the Old Testament reflects the older belief that Yahweh was a son of El, the son who becomes the national god of Israel.

There is another interesting passage in Psalm 82, in which “gods” are described as being in an assembly, judged by a yet higher “god:”

God [Elohim] standeth in the congregation of the mighty [El]; he judgeth among the gods [Elohim].

This is almost certainly a remnant of polytheistic El-related worship.  A bit later, it says:

I have said, Ye are gods [Elohim]; and all of you are children of the most High [Elyon].

But ye shall die like men, and fall like one of the princes.

The Hebrew god acknowledges that these other gods are indeed gods, “children of the most High,” yet he will destroy them all.  The so-called “false” gods in the Old Testament are often referred to as if they truly were gods – they simply can’t stand up to Yahweh, just as other peoples can’t stand up to his chosen.  Biblical and archeological scholarship has demonstrated that there is a progression from Canaanite/Ugaritic to early Israelite to late Israelite worship, from polytheistic worship of El and his pantheon, eventually to worship of Yahweh alone.  It is in this religious/political context that much of the Old Testament was written.  It describes battles with peoples such as the Canaanites, the Moabites, the Midianites, and so on, peoples that worshipped gods other than Yahweh, often gods that were part of El’s pantheon, such as Asherah, Baal, and Moloch.


Ironically, there is no archaeological evidence of military conflict between the Canannites and Israelites – instead, Canaanite culture appears to have simply evolved into Israelite culture.  The Hebrew language is one of the Canaanite family of languages – why would the Hebrews adopt the language of an enemy they had supposedly destroyed?  Over time, the names El and Yahweh became essentially synonymous, and the name El simply became a generic word for god.  Yahweh, who had been one of a number of gods, was promoted as the only god deserving of worship and sacrifice.  Time and again we see stories of the Israelites themselves turning to other gods, invoking Yahweh’s wrath, and turning back to him.  There are long passages extolling the power of Yahweh, his superiority to other gods, and his great might as a warrior god, leading his people through victory upon victory.  And it is worth noting that whether it was polytheistic El-related worship, or monotheistic Yahweh worship, there was nevertheless a “divine council” or “heavenly host” of lesser beings at the side of the Almighty, mirroring the royal councils and bureaucracies of the earthly kingdoms that worshipped these beings.


Yahweh is often portrayed as neither all-powerful nor all-knowing – for example, Judges 1:19 states:

And the LORD [Yahweh] was with Judah; and he drave out the inhabitants of the mountain; but could not drive out the inhabitants of the valley, because they had chariots of iron.

Yahweh is hot-tempered and can be talked out of his wrathful intentions.  In the 32nd chapter of Exodus, Yahweh becomes angry at the Israelites and tells Moses, “Now therefore let me alone, that my wrath may wax hot against them, and that I may consume them.”  Moses reminds him of his covenant with Abraham, and points out that the Egyptians will think he is a mischievous god, delivering his people out of Egypt only to destroy them.  It works.  Yahweh “repented of the evil which he thought to do unto his people.”  Such passages reflect a conception of God that is quite human, with human emotions, particularly jealousy, anger, and a desire for self-aggrandizement.  The Old Testament God, Yahweh, is portrayed much like a very powerful king, who has issued edicts and expects nothing less than absolute obedience and loyalty from his people.  He demands tribute, and advocates, even insists upon, genocide against peoples who worship other gods.  In a song, Moses even refers to Yahweh as “a man of war.”


By the time we get to the New Testament, there is no mention of Yahweh at all.  Remember that the writers of the New Testament did not read the Hebrew scriptures, but the Greek translation of these, the Septuagint.  In the Septuagint, the name Yahweh, the national god of Israel, was translated as theos, which is just a generic word for any god.  New Testament scriptures that make reference to God often use the word Lord, which is translated from the Greek word kurios, meaning master.  For example, when an “angel of the Lord” appears to Mary, this is translated from the Greek aggelos kuriou, which actually means “messenger of the master.”  This new approach to God reflects the belief of these writers in a single, universal God, for Jews and Gentiles alike, whose name has become irrelevant because there are no other gods.  The New Testament theos, in contrast to the Old Testament Yahweh, expresses no favor toward the Hebrew people – he is no longer their national god.  Nowhere does the New Testament refer to the Hebrews as God’s chosen.  Instead, everyone is now condemned except those who follow his beloved son, Jesus – in fact the word kurios, which means master, is used to refer to Jesus as much as it is to God.


As I said at the beginning, any piece of literature can be interpreted any way you like.  The more than 30,000 denominations of Christianity are proof of this.  But if you have any tendencies toward rational inquiry, you will probably ask yourself at some point how much intellectual twisting and contorting you have to do in order to make the scriptures conform to your interpretation.  All other things being equal, the simplest explanation tends to be the correct one.  One can almost feel, reading and listening to the interpretations of many Bible-believers, the discomfort they are enduring as they desperately try to contort the words of scripture to make them fit into their pre-packaged theologies.  Of course I encourage you to do your own research.

My car is black.  Know what I mean?

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