Seeking God in Science part 12: A clean-sheet introduction to quantum mechanics and quantum field theory (part 1 of n)

For months now I have been writing the phrase "when we get to quantum mechanics..."  Well, today we finally get to quantum mechanics (QM).  QM is a big topic.  I will not be able to cover it all, or even put a significant dent in it, in one blog post.  The goal of this post is just to give you an overview of the lay of the land, a clean-sheet introduction to the topic for non-technical people who know nothing about QM other than a few catch phrases like "wave-particle duality" and "spooky action at a distance."  I am ultimately going to demystify it all for you, but first I want to get everyone on the same page about exactly what needs demystifying.  The pedagogy of QM has improved since I first started studying it seriously back in the 1990s, but it's still in a pretty dismal state, with lots of old misconceptions and unjustified tacit assumptions floating around.  So I'm going to just start from scratch here.

The first thing you need to know is that there are two different quantum theories: quantum mechanics (QM) and quantum field theory (QFT).  These are not the same.  QFT is QM extended to include special (but not general) relativity.  Most of the time this doesn't matter, especially for philosophical discussions, and so many people, including me, will often say QM when they really mean QFT.  But there are a couple of things that you should be aware of, starting with the fact that both of these theories exist.  You should also know that the Standard Model of particle physics is a QFT theory.  It has to be because the Standard Model includes light, the ultimate relativistic phenomenon, and if you don't think of light in relativistic terms you will go seriously astray.

The easiest way to go astray (and I'm jumping ahead here, but I really want to prepare you for this) is to think of photons as little point particles that move through space at the speed of light.  It's very easy to fall back on a classical model that will tell you that, for example, if a photon is absorbed by a particular atom at a particular place at a particular time, then it must have been emitted by some other particular atom located at some other particular place at some other particular time, and that if you take the distance between those two places and divide by the difference in the two times the result will be 186,000 miles per second.  That seems plausible, but it is completely, utterly, 100% wrong, as we shall eventually see.  But if you want to read ahead, here is a three-part series [1] [2] [3] I wrote a while back that discusses this.

I cannot emphasize enough how hard it is to escape this way of thinking.  We are non-relativistic creatures.  All of our everyday experience is non-relativistic.  The underlying truth plays havoc with our intuitions, and QM makes that problem much, much worse.  I'll try to ease you into it as gently as I can.

I am going to begin by describing the Problem that led to the discovery of QM.  There is some pedagogical peril in this because the historical route from the Problem to its eventual solution was a long and winding road, full of rhetorical and philosophical pitfalls, some of which have people stuck in them to this very day.  So I'm going to give you advance warning of the biggest of these traps, which is something called the Measurement Problem (MP).  The MP was not the original Problem that QM was invented to solve, it was a Problem that arose in the course of developing QM.  There is disagreement over whether or not the MP has been solved or not.  I believe it has, and the goal of this series of posts will be to explain to you what I believe to be the solution and why.  Again, not to leave you in suspense, the solution (as I see it) has to do with something called entanglement.  I believe that one of the reasons that the MP seems so intractable is that most of the standard introductory pedagogy for QM doesn't mention entanglement at all.  If it is introduced to beginners at all, it is usually in the form of a post script, an additional extra-weird thing to add to the already-long list of (apparently) intractably weird things that distinguish QM from other physical theories.  The truth (and this is uncontroversial) is that measurement and entanglement are inextricably linked.  I believe (and this is still controversial) that entanglement provides a complete account of measurement, but for the moment this is neither here nor there.  What matters is the reason that this is not more widely addressed when introducing people to QM: entanglement is a phenomenon that only arises in systems consisting of more than one particle, but to keep things simple most introductory examples deal only with single particles.  And I'm going to do the same, but you should keep very firmly in the back of your mind that there are a lot of things that happen with single particles which are very special cases that apply only to single particles.  The behavior of quantum systems with more than one particle (which is, of course, most systems) is qualitatively different from single-particle systems.  In QM the whole is vastly (and literally!) greater than the mere sum of its parts.

QM (along with relativity -- the two were developed at roughly the same time) was invented to solve a couple of different problems that had arisen in classical mechanics towards the end of the nineteenth century.  By that time Newton's laws and Maxwell's (well, Heaviside's really) equations had become well-established, but they made a few predictions that did not align with experiment.  The most important of these (historically speaking) had to do with something called "black-body radiation" which I am not going to get into here.  There are a zillion articles about it on the web if you want the details.  The TL;DR is that physicists were tearing their hair out trying to come up with an explanation that matched the experimental results when Max Planck, in what he later described as "an act of desperation," pulled a mathematical model out of a (metaphorical) hat that assumed -- with no motivation whatsoever other than that it made the results come out right -- that energy could only be transferred from one system to another in fixed-sized chunks, which he called "quanta".  Fast forward to 1905 and Albert Einstein used a riff on Planck's math to explain another then-mysterious observation called the photoelectric effect.  (This is actually the work that won Einstein a Nobel Prize.)  Again, I'm not going to go into the details.  What matters is that to this point all of the unexplained observations had to do with light.   That changed in 1924 when Louis de Broglie (pronounced, more or less, "du broy") predicted in his 1924 PhD thesis that electrons would also exhibit wave-like behavior.  It is hard to appreciate today just how crazy that sounded at the time.  The question of whether or not atoms were real had only recently been considered to be fully settled.  (Again, it was Einstein who provided the definitive proof in his theory of Brownian motion.)  The electron itself had only been discovered less than 30 years prior, by J.J. Thomson in 1897.  It was beyond question that the electron was a particle.  Like Planck before him, de Broglie basically just pulled the idea out of a hat that if a wave (light) could sometimes behave like a particle, maybe particles (electrons) could sometimes behave like waves.  There was absolutely no evidence for this at the time.  But in 1959 de Broglie was vindicated when Claus Jönsson demonstrated that electrons produced an interference pattern in a two-slit experiment.  Since then thousands upon thousands of painstaking calculations and experiments have shown that QM/QFT is the single best scientific theory ever devised.  Its predictions have been verified in some cases out to 13 significant figures, comparable to measuring the diameter of the earth with an error of less than a micron.

The upshot of all this work was a theory that is quite challenging to explain to a lay audience.  It doesn't help that one of the most prominent people involved in developing the theory, Richard Feynman, famously quipped that "no one understands quantum mechanics."  It's not true.  Quantum mechanics is no harder to understand than relativity.  But it is a lot harder to explain.  The reason is that to understand relativity you really only need to wrap your brain around one unintuitive idea, namely, the fact that the speed of light is constant in all reference frames.  Quantum mechanics comprises half a dozen such unintuitive notions: the measurement problem, the uncertainty principle, wave-particle duality, entanglement (a.k.a. "spooky action at a distance"), many-worlds, and probably a few others.

It also doesn't help that the standard pedagogy for QM is highly misleading, at least in terms of its philosophical implications.  As mentioned earlier, nearly all introductions to QM focus on single-particle systems when in fact most of the philosophically interesting stuff only starts to happen with multiple-particle systems.  In fact, even the use of the word "particle" here is misleading, as we will eventually see.  So it's really hard to figure out where to start picking at this tangled mess.

I'm going take the following approach: first I'm going to just describe what the theory looks like, a high-level overview of the math.  Don't worry, I'm not going to do any actual math.  (We will eventually have to do a little math, but again, don't worry, it' won't be to hard.  If you remember high-school algebra you'll be fine.)  The goal here will be just to give you a high-level overview of what the theory looks like, and to introduce you to some of the terminology.  The upshot of all that will be that QM has to do with waves, and that leads to something you have probably heard of, the Heisenberg uncertainty principle, which says that you cannot simultaneously know the position and velocity of a particle.  Note that this fact bears some resemblance to the impossibility of measuring the one-way speed of light.  This is not an accident.  It is a reflection of a deep underlying truth.  In both cases the impossibility of measuring or knowing these things is not a limit of our technology or out cleverness, it is a result of a profound fact: the "one-way speed of light" is not merely beyond our epistemological grasp, it is a physically meaningless concept.  And it is the same with simultaneous position and momentum.  The reason we can't know them is that they literally cannot exist at the same time, and the reason for this is that the things we call "particles" aren't actually particles.  They are things that under certain circumstances act like particles.  But most of the time they act like waves, and waves literally cannot have a well-defined position and momentum at the same time.

One final thing by way of introduction: relativity messes with our intuitions because it mainly manifests at speeds and accelerations that are vastly higher than our day-to-day experience.  QM messes with our intuitions because it mainly manifests at sizes and temperatures that are much smaller than our day-to-day experiences.  Relativity manifests when things are very fast or very heavy.  QM manifests when things are very small or very cold.  But a big difference between relativity and QM is that it is pretty easy to understand how our day-to-day non-relativistic world arises from the relativistic one.  If you plug slow speeds into the relativistic equations what pops out are results that align with intuition.  This is not true for QM.  Our day-to-day world appears to be qualitatively different -- indeed, fundamentally incompatible -- from what quantum mechanics describes.  Explaining how our day-to-day world -- what is called in the trade the "classical" world -- arises from QM is one of the biggest challenges ever faced by humans.  This is the famous (or infamous) "measurement problem".  We won't get to that until later, but I wanted to warn you that this demon is lurking, and to promise that we will eventually face and slay him.

So let me start by pointing out the features that QM and classical mechanics share in common.  Both of them work by taking a description of a state of the world and projecting that state forward (or backward) in time.  The math that does this projection is called the dynamics of the theory.  In classical mechanics the dynamics are Newton's laws and the laws of electrodynamics, so even in classical mechanics we have something (light) that is fundamentally wave-like.

The first major difference between classical mechanics and QM is in the vocabulary the two theories use to describe states of the world.  Classical mechanics describes the states of Things by specifying their positions and velocities in 3-dimensional space (or, if we are being relativistic, in 4-dimensional spacetime).  In addition to this, it assigns to each point in space a description of the forces that an object located at that location will experience.  These are called fields.  There are three kinds of classical fields: electric, magnetic, and gravitational.  And that's it.  That is all there is to classical mechanics, at least conceptually.

QM has a completely different way of describing states, a fundamentally different ontology.  A quantum state is described by a mathematical object called a wave function, which is usually written as the Greek letter psi (Ψ).  As you might glean from the name, the wave function is a function, which is to say, it takes an input and produces an output.

Note that I wrote a wave function rather than the wave function.  You will often see "wave function" referred to with a definite article as if there is only one of them, but there isn't.  There are many wave functions.  Every quantum system has its own wave function.  When people refer to the wave function what they usually mean is the wave function that describes the state of the entire universe.  It would seem intuitively that the wave function is just the aggregate of all the individual wave functions that describe all of the particles in the universe, but this turns out not to be the case, and the reason this is not the case is one of the things that makes QM so deeply weird.

Note also that I said that the wave function is a function that takes an input and produces an output, but I didn't tell you what those inputs and outputs actually are.  That's because these are not straightforward to describe.  The output is not so bad: it's something called an amplitude, which is just a complex number, i.e. a number that involves the square root of negative 1, usually denoted by the letter "i".  The input to the wave function is something called a point in configuration space.  Configuration space is not something you are likely to have encountered in high school math, but it isn't actually very complicated.  If you want to describe, say, the position of a particle in 3-dimensional physical space you can do that by writing down three numbers.  If you want to describe the positions of two particles you can do that by writing down six numbers.  You can then take those six numbers and treat them as if they described a point in some abstract six-dimensional space.

Why would we want make such a strange maneuver, one which requires us to imagine such a bizarre thing as a six-dimensional space?  Surely that's just a mathematical curiosity which can't possibly have any physical or philosophical significance?  Well, in (non-relativistic) classical mechanics that is true.  As long as objects are large (relative to atoms) and slow (relative to light) we can model them with math that reflects our intuitions: objects have positions in 3-D space and velocities that are 3-D vectors, and time is an independent variable.  But, as we have already seen, when things start to move fast (relative to light) that model breaks down and we can no longer treat time as an independent variable because different observers have legitimate disagreements about how fast clocks are running.  So in order to get the right answers we have to treat space and time as a unified whole, as four-dimensional space-time.

But that's it.  One extra dimension is already pretty mind-bending, but that's all you need for relativity.  Physical objects still exist as independent entities that occupy specific locations in space-time, and so the Objective Reality Hypothesis still holds for physical objects.  We can have legitimate disagreements about where and when a physical object exists, but we cannot have a legitimate disagreement that it exists.  And the way that manifests itself mathematically is that, although you have to use four dimensions to describe reality, you never have to use more than four.  The apparent distinction between space and time is not objectively real (it is observer-dependent) but 4-D space-time is objectively real, as are the physical objects which inhabit it.

In quantum mechanics that is no longer the case.  In quantum mechanics it is impossible to ascribe independent existence to objects in 4-D spacetime and still be consistent with observation [1].  This is a famous result called Bell's theorem.  It was first discovered (by John Bell) in 1964 and first demonstrated experimentally by Alain Aspect in 1982.  Before that, the fact that the math of quantum mechanics had this structure was considered by many -- most notably Albert Einstein -- to be a flaw, an indication that the theory was incomplete, that something was missing.  The presence or absence of an independent existence of physical objects was considered a philosophical debate, not a scientific one.  That is no longer a tenable position.  The fact that this mathematical structure is necessary to describe reality is now established beyond all possible doubt.  The entire field of quantum computing depends on it.  Since 1964 there have been even stronger results proven mathematically and demonstrated experimentally.  It is inescapable: quantum mechanics is deeply weird.  Specifically, using configuration space as the input to the wave function is not just a mathematical convenience.  It is necessary to explain observations.  Specifically, it is necessary to explain entanglement.  But let's not get ahead of ourselves.  Let us return instead to the elephant in the room, which is that it is obvious that what I've just told you cannot possibly be true because it is obvious that material objects do exist in point of actual fact and their existence is not dependent on observations.  If a tree falls in a forest it really does make a sound even if no one is there to hear it.  Chairs continue to exist even when no one is sitting in them.

Right?

So: on the one hand, QM -- and its relativistic counterpart, QFT -- are the most successful scientific theories ever devised.  No experiment has ever produced results at odds with the predictions of QFT.  (This is actually a major crisis in particle physics.  Progress has been pretty much at a standstill for over fifty years now.)  On the other hand, the mathematical structure of these theories are categorically at odds with our lived experience.  How do we reconcile these two facts?

Answering that question will be our project for the next few episodes.

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[1] Note that the way I've stated this is controversial.   There is a formulation of QM that purports to ascribe independent existence to particles and even assign them definite positions and velocities.  It is called the de Broglie–Bohm theory, more commonly referred to as Bohmian mechanics.  But this theory has two problems.  First, it is very hard to extend it to be compatible with relativity (so strictly speaking my claim that it is impossible to "ascribe independent existence to objects in 4-D spacetime and still be consistent with observation" is technically correct).  But more importantly, the way in which it ascribes independent existence to particles is, in my opinion, cheating.  I'm planning a whole entry about Bohmian mechanics, but I have to lay a lot more groundwork before I can discuss it cogently.

Seeking God in Science, part 11b: Time (part 2 of 2)

Don Geddis pointed out a problem with my relativity animation:

... with either Linda or Ralph stationary, you have the angled light hitting the end clock at "the same moment" as the light pulse that just goes straight up and down. That's not what would happen. If light was going at a constant speed, the light on the angled path would need to take longer to reach the end point.

That sounds plausible, but it's wrong.  The light that moves along the angled path must arrive at the same moment as the light that moves along the vertical path.  Why?  Because it is literally the same light just viewed from two different points of view, what in the trade are called "frames of reference".

Now, there is a different experiment that we could imagine where Ralph and Linda each send out two flashes of light.  One flash, as before, travels vertically in their frame of reference, and the other is launched at an angle so that its trajectory traces out the same path as the light that is vertical in the other's frame.  It seems plausible to say that that light would not arrive "at the same moment" because it is traveling along a longer path.  And it also seems plausible to say that therefore the other light must also "not arrive at the same moment" because those two beams are tracing out the same path in space.  But this is wrong, and the way in which is it wrong is the whole point of this thought experiment!  Here is a hint: think about how you would actually have to modify the experimental setup in order to demonstrate that Ralph's (or Linda's) angled beam did not arrive "at the same moment" as his (or her) vertical one.  (I'm going to leave that as an exercise for now.)

Nevertheless, there is indeed a problem with this animation: it is showing the light moving as if it were a classical object like a baseball, and so it appears to be moving at different speeds.  But that is not possible with light.  Light is only ever seen to move at one speed, namely 186,000 miles per second, a speed universally denoted by the letter C.  So how can something moving at the same speed cover different distances in the same amount of time?  Obviously it can't.  Something has got to give.

The "thing that gives" in this example is, as noted in the first part of this two-part episode, the clocks.  Ralph and Linda agree that each of their clocks reads two time units when the light returns, but they don't agree on how far the light traveled.  If we can't change the speed the only thing left is the time.  Ralph thinks Linda's clock is wrong and Linda thinks Ralph's clock is wrong.  Each sees the other's clock as reading less than it should, i.e. each sees the other's clock as running slow.  And the reason for that is that each one sees the other's clock as moving.

I tried to extend the animation to illustrate this more directly but it turns out to be really hard.  The problem is that the point I actually want to make here has to do with the fact that it is impossible to measure the one-way speed of light.  Young-earth creationists try to use this fact (and it is a fact) to explain how we can see stars and galaxies further than 6000 light years away.  My goal here is to show why their argument doesn't work.  The problem that leaves me with, however, is that I cannot assume that the one-way speed of light is c because that would be begging the question.  Because I cannot assume that the one-way speed of light is c, I cannot rely on most of the usual pedagogical tools for explaining relativity because they do assume that the one-way speed of light is c.  This assumption (and it is an assumption) is deeply ingrained in the minds of most people because it is the foundation of Einstein's relativity theory.  But it is an assumption, not a physical fact.  The physical facts are that the round-trip speed of light is c, and the one-way speed of light along any particular trajectory is constant.  It seems like you should be able to prove from those facts that the one-way speed must be c, but you can't.  It's not true.  It is possible to build so-called anisotropic models of light where the one-way speed is faster in one direction and slower in the opposite direction.

Lest you think, as many physicists do, that I am merely entertaining a transparently crackpot theory, the fact that it is fundamentally impossible to measure the one-way speed of light was noted by Einstein himself in his 1905 paper:

[The laws of physics] possess no properties corresponding to the idea of absolute rest. ... the same laws of electrodynamics and optics will be valid for all frames of reference ...  We will raise this conjecture ... to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.  [Emphasis added]

and later:

If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighborhood of B.  But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B. We have so far defined only an “A time” and a “B time.” We have not defined a common “time” for A and B, for the latter cannot be defined at all unless we establish by definition that the “time” required by light to travel from A to B equals the “time” it requires to travel from B to A.  [Emphasis added.]

(BTW, that quote is a huge hint for the exercise above.)

So the constant speed of light is a postulate, an assumption, exactly like taking the sun to be the center of our solar system.  It is not a physical fact, it is just a convention that makes the math easier.  And, as we shall see, the fact that it is impossible to measure the one-way speed of light is actually an enormous clue to what is really going on under the hood.

Because we want to specifically avoid the assumption that the one-way speed of light is c, we need to be more precise about what it actually means to assume that "the one-way speed of light along any particular trajectory is constant".  As before, we turn to Einstein:

... light is always propagated in empty space with a definite velocity ... which is independent of the state of motion of the emitting body.

There is a lot of subtlety packed into those 23 words, but there are two main things to notice.  First, the velocity is "independent of the state of motion of the emitting body".  That is really the crux of what it means for the speed of light to be the same for all observers.  We'll get back to that in more detail later.

The second thing to notice is that we're talking about light propagating in empty space i.e. in a vacuum.  Note that we can never actually see light moving in a vacuum.  The only thing we can actually see is light that arrives at some kind of sensor -- a camera, or our eyes.  When light is moving in a vacuum, by definition there can't be anything there to sense it.  So the only things we can actually measure when light moves through a vacuum is the time and place the light was emitted, and the time and place it arrived.  From this we can infer a trajectory and a speed, but -- again by definition -- we cannot measure them.  The only way to "see" light actually move is to put some physical objects along its path, like dust particles.  Here is an extraordinary video clip that lets us "see" light move using this technique.

Actually there are two different clips with two different viewing angles.  In the first clip (at the 1:10 mark) the camera is viewing the light perpendicular to its motion.  In the second (at 1:33) the camera angle is oblique, and the light appears to move faster while moving towards the camera and slower while moving away, which is a visceral illustration of anisotropic light.  You can, of course, explain (or, as a creationist would say, "explain away") this observation using the standard assumption.  But you can also explain the fact that light is appearing to move at different speeds in different directions by hypothesizing that it actually is moving at different speeds in different directions.  No experiment can disprove this.  (But note that you then have to explain -- or "explain away" why the light appears to move at the same speed in both directions when "viewed" at 90 degrees!)

It is worth watching the entire video to see the details of how these clips were made.  Around the 23 minute mark you will find the standard explanation of why the light appears to move at different speeds when viewed from an oblique angle: you are not actually seeing the light move.  What you are seeing is light that has traveled from the source, reflected off dust particles in the air, and then traveled to the camera.  What the camera is measuring is the time it takes the light to traverse that entire path, not just the first segment.

In philosophy-speak, the one-way speed of light being c works epistemologically but not ontologically.  It makes the math work, but if you want to understand what is actually going on it really ought to bother you that there is no experiment you can do that will allow you to demonstrate that light moves at the same speed in all directions.  And actually, the fact that it is impossible to rule out anisotropic models experimentally, the fact that it is fundamentally impossible to measure the one-way speed of light, is actually a Big Clue that there is something else going on under the hood.  What is it?

Here is a hint: notice that it is also possible to make anisotropic models of time.  We can show experimentally that two clocks sitting next to each other remain synchronized, but how do we know that the length of one tick is the same as the next?  Maybe clocks actually tick at different rates when it is Tuesday in New York, but because all clocks are equally affected, including all of the physical and biological processes that cause us to have a subjective experience of time passing, there is no way to tell.  That seems silly, but there is no experiment you can do that will rule it out.  And this, too, is a clue.

(In point of actual fact, clocks do tick at different rates depending on the circumstances!  It is only clocks that are not moving or accelerating relative to each other that will remain in sync!)

So let's go back to basics: what does it actually mean for the speed of light to be the same in all reference frames in terms of things we can actually observe?  Here again, Einstein provides the answer, or at least the foundation:

[A]ll our judgments in which time plays a part are always judgments of simultaneous events. If, for instance, I say, “That train arrives here at 7 o’clock,” I mean something like this: “The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events.”

Notice that for two events to be considered simultaneous they have to be adjacent not only in time but also in space.  (Another hint!)  Two things that happen in the same place and at the same time are "simultaneous".  You can't assume anything about events that happen in different locations, and indeed it will turn out that different observers will have legitimate disagreements about whether or not two distant events are simultaneous or not.

In the experiment with Linda and Ralph there are three different sets of simultaneous events:

E1.  Ralph and Linda's light sources flash and their clocks start.  This is the reason I drew Ralph's train upside-down, to emphasize the fact that the two light sources are next to each other when they flash.

L1.  Linda's light arrives back at Linda's clock and her clock stops and registers a result (2 time units).

R1.  Ralph's light arrives back at Ralph's clock and his clock stops and registers a result (also 2 time units).

Note that L1 and R1 are not simultaneous because they do not happen in the same place.  In the animation they appear to happen at the same time, but this is just happenstance because I wanted to highlight the symmetry between Linda and Ralph's perspectives.  Linda and Ralph disagree on the light path lengths -- they each see the other's light path as longer than their own.  But if we assume that the speed of light is the same everywhere, then there is only one possible conclusion that each of them can come to: the actual time that it took the other's light to return to the (moving) clock was actually longer than the 2 time units that clock reported.  The moving clocks are running slow!

We can quantify this.  To keep things simple let's assume that the relative speed between the two train cars is such that the longer light paths appear to be at 45 degree angles.  In other words, Ralph sees Linda's light move at a 45-degree angle and Linda sees Ralph's light move at a 45-degree angle.  Now, the phrases "Ralph sees Linda's light move" and "Linda sees Ralph's light move" should be setting off alarm bells in your head because I spent the first half of this post explaining that it is not possible to see light move (notwithstanding the existence of videos purporting to show light moving).  So let me be more precise: let us change the experimental setup in the way alluded to in the exercise above: both Linda and Ralph set up a second mirror so that they each have a second beam of light that moves at a 45-degree angle in their train car.  Those beams will not return to the light source, they will land two distance units away.  To be precise (which we have to be because we already know that Linda and Ralph disagree about lengths) Ralph's beam will land two distance units away as measured by Ralph in his train car, and Linda's 45-degree beam will land two distance units away as measured by Linda in her train car.

We now have two more events:

R2: Ralph's 45-degree light beam lands on the floor of Ralph's train.

L2: Linda's 45-degree light beam lands on the floor of Linda's train.

Now adjust the relative speed of the trains so that L1 and R2 are simultaneous events, i.e. both 45-degree-angle beams land at the exact instant that the places they land are adjacent as the trains pass each other.  By symmetry, R1 and L2 must also be simultaneous events.  With some elementary geometry you can figure out that the speed required for this is √2/2 or about 0.7c, but the exact value doesn't really matter.  What matters is that this speed is less than c.  The reason is simple geometry: the trains are traveling along the legs of two right triangles while the diagonal light is traveling along the hypotenuses of those same triangles, so the trains will always have to cover less distance than the light in order to arrive at the same place at the same time.

As I mentioned earlier, I tried to extend the animation to illustrate all this, but that turns out to be really hard.  I spent a couple of hours with ChatGPT and was not able to get it to do the right thing, so I'm probably going to have to code it up myself.  (If there are any animators out there who want to give it a shot I'd be willing to pay you for your time!)  The reason it's hard is that the animation as it stands sweeps some details under the rug in the interest of highlighting the important parts while trying to keep things simple and intuitive.  Two details that I've swept under the rug are Lorentz contraction and the relativity of simultaneity.  When you switch perspectives in the current animation, the train cars appear to remain the same length and the clocks advance at the same rate.  (These distortions are analogous to the ones you get with a Mercator projection.)

Trying to add the 45-degree beams reveals some of those distortions, so it's actually impossible to simply add those to the current animation.  You have to get rid of the distortions in order to make it all work, which means that you have to put in Lorentz contraction and time dilation, and then it all gets really hairy and I have not yet been able to make it work.  Sorry.  But all that technical detail detracts from the point I really want to make here which is simply that moving clocks appear to run slow, and this is a straightforward consequence of disagreement over the length of the path that light took to travel between two events.

There is one detail that does matter, though, and that is the fact that to get an apparent path of light moving at a 45-degree angle you do not have to travel at the speed of light, but only about 7/10 of the speed of light.  This is counterintuitive at first, but becomes pretty straightforward when you start to think of it in terms of geometry: the light always appears to travel on the hypotenuses of triangles and the trains always move along the legs, and so the distance covered by the train is always less than the distance covered by the light.  Let that really sink because what I am about to tell you next is going to blow your mind even though it will be obviously true: the distance covered by the train is always less than the distance covered by the light is a fact of geometry, and so it must remain true NO MATTER HOW FAST THE TRAINS ARE MOVING!

This makes most people's heads explode when they first hear it because another truism that has been instilled into them is that it is impossible for massive objects (like trains) to travel faster than light, and this is true.  If you race light, the light will always win.  But this is not some kind of arbitrary "cosmic speed limit", it is a logical consequence, a geometric consequence, of the fact that the speed of light is the same for all observers.  If you try to race light in a straight line you will always lose.  But if the light is reflected and so takes a path that is not a straight line, you can always get to the finish line at the same time (or sooner) no matter how far away it is, and you always can do that by moving slower than light.

And here is the final mind-blowing fact: no matter how far away the finish line is, the time it will take you to get there in the setup we've described above will always be the same according to the clock on your train car: two time units.  Again, the exact number doesn't really matter.  What matters is that it's more than zero.  So: in an arbitrarily short amount of time as measured by a clock on your train car (as long as it's more than zero) you can traverse an arbitrarily long distance.  Want to go to the Andromeda Galaxy in one second?  No problem.  Just hit the gas and accelerate to 0.99999999999999999999999999999c.  (That is actually the right answer, more or less, at least if ChatGPT did the math right.  There are twenty-nine 9's there.)

But wait, you say, that's obviously impossible because I would be traveling more than 186,000 miles in a second!  Well, that's true, but remember that not only do moving observers disagree on when "now" is, they also disagree on where "here" is, and so they also disagree on where "there" is.  When you move close to the speed of light, everything around you in the direction of your motion appears to get shorter (that's the Lorentz contraction).  So from the point of view of someone back on earth, it takes you 2.5 million years to get to cover the 2.5 million light years to Andromeda.  But from your point of view, the distance to Andromeda shrinks so that you can cover it in one second according to your clock (which is running very, very slow according to observers on earth).

If we extend this to a two-way round trip we get the so-called "twin paradox": if you can get to Andromeda in one second (according to your clock) then you can also get back in one second (again, according to your clock).  So if you actually do this you will be two seconds older but earth will be five million years older.  Of course, actually carrying out this experiment with a human involves a few practical difficulties, like the fact that accelerating yourself to such speeds would require vastly more energy than in all the atomic bombs on earth and decompose the atoms in your body into their subatomic constituents many times over.

But there is something that can make such a round trip without any trouble at all: light.  What would it look like to go to Alpha Centauri and back not just close to the speed of light but at the speed of light?  This requires some pretty significant suspension of disbelief because this is not actually possible even in theory.  It is worth reiterating why this is the case: no matter how much distance you cover in a finite amount of time, your arrival will still be simultaneous with that of a light beam that reflected off a mirror, and so you will have moved slower than that light.  To move at the speed of light you have to travel an infinite distance in a finite amount of time as measured by your clock.

But that can't be right because light doesn't cover an infinite distance in a finite amount of time!  It moves at a finite speed: 186,000 miles per second.  Yes, that's right.  But remember the qualification: you have to travel an infinite distance in a finite amount of time as measured by your clock.  To transfer this to a chunk of light you would have to reckon the time as measured by a clock moving at the speed of light.  It is not possible for a clock to move at the speed of light, but if it could, what would happen?  Well, as it got closer and closer to the speed of light, it would run slower and slower until, at the speed of light, it would stop altogether.  At the same time (no pun intended) the Lorentz contraction would shrink the distance between your start and end points down to zero.  At that point your speed from your perspective would be zero divided by zero.

And this is the reason that it is impossible to measure the one-way speed of light.  It is not merely physically impossible, it is mathematically nonsensical!  It is literally zero divided by zero.

So why can we measure the two-way speed of light?  Why does time pass when light reflects?  If both the outgoing and reflected beam of light think that their origin and destination are at the same place and time, how can time elapse at all?

To answer this it is best to go back to a physically possible situation with a physical clock moving at close to but not quite the speed of light, making the round-trip to alpha centauri in a short but still finite period of time from its perspective.  Actually, it is best to imagine two such clocks, one going to Alpha Centauri and the other coming back so we don't have to deal with acceleration and general relativity.  The motion of the clocks is arranged so that they pass each other at Alpha Centauri, at which point the inbound clock is set to the same time as the outbound clock.  Let's say that each leg takes one second as measured by these clocks, so the total round-trip time is measured as two seconds, during which time five million years will have passed on earth.

When did those five million years "happen" from the perspective of the clocks?

The (mind-blowing) answer is: they didn't!  If you actually work this out (and I'm not going to do that because this post is already too long) what you will find is that the apparent passage of time on earth can be ascribed entirely to the disagreement over when "now" is from the two perspectives of the two clocks.  For the outbound clock, "now" on earth is in the past.  For the inbound clock, "now" on earth is in the future.  And the difference is exactly ... five million years (minus two seconds)!

From the "perspective" of light, time never "passes".  What happens instead is that "now" changes.  But those changes "happen" instantaneously every time the light is reflected by something and changes direction and hence "perspective".

And it turns out that this is what happens with clocks too!  Clocks are collections of atoms.  Atoms interact with each other mainly through the electrons in their outer shells, and those interactions are electromagnetic ones.  In other words, there are photons -- light -- going back and forth between atoms.  Every time that "happens" (I have to put "happens" in scare quotes here because the whole idea of "things happening" is being jettisoned here) there is a change in perspective that "ratchets that atom up to a new now" (the English language is up against its conceptual limits here).  And, of course, it's not just clocks.  Anything made of atoms (which is to say, everything) works this way.

We will see all this again when we get to quantum mechanics, which we will have to do before we can answer the obvious question: if time doesn't actually pass, if things don't actually happen, why does it appear that they do?

I'll leave you with just one parting thought: the fact that it is impossible to measure the one-way speed of light can be explained in two different ways.  One, as I've done above, is to point out that the whole concept of the "one-way speed of light" is mathematical nonsense, quite literally zero divided by zero.  But there is also a practical way to understand it: think about what it would take to make this measurement.  You would need two clocks in two different locations, and you would need to measure that time that light left one location and the time it arrived at the other.  To do that you would have to make sure that the two clocks were synchronized, and there is no way to do that without assuming that the one-way speed of light is a constant.  But notice what happens if you don't make this assumption: not only is it impossible to measure the one-way speed of light, it is impossible to measure the one-way speed of anything for the exact same reason: you can't synchronize your clocks unless you assume the one-way speed of light!  For slow-moving objects you can get very close, but there will always be an uncertainty corresponding exactly to your uncertainty about the one-way speed of light.  We will see this kind of fundamental "uncertainty principle" again as well when we get to quantum mechanics.

A relativity visualization

With a lot of help from ChatGPT I generated a little interactive animation to help visualize what is going on in the classic light-clock thought experiment. You are looking at a profile view of two experimenters, Linda and Ralph, each in their own frame of reference. By tradition this is a train car, but it can be a spacecraft or a shipping container or an RV, it really doesn't matter. The point is that Linda and Ralph have identical experimental apparatus, but they are moving relative to each other. In this animation, Lina, at the top wearing a purple shirt, is moving Left and Ralph, on the bottom wearing a green shirt, is moving Right.

(For a larger stand-alone version of the animation, or if this iframe doesn't fit in your screen, click here.)

Linda and Ralph each have an experimental appratus consisting of a light source (the yellow circles) and a clock on the floor of the train and a mirror on the ceiling (the white rectangles). The clock measures the time elapsed between when the light flashes and when the reflected light from the mirror returns to the same location. You can view the experiment from three different frames of reference: Linda's frame, where she is stationary and Ralph is moving, or Ralph's, where he is stationary and Linda is moving, or the ground where both Linda and Ralph are moving in opposite directions at equal speeds. The path of the light as seen by Linda and Ralph is traced out in colored lines. The purple lines are the paths that Linda sees and the green lines are the paths that Ralph sees. (Exercise: the light paths as seen by an observer on the ground are not shown. What would those look like?)

The important things to notice are:

1. The physical situation is exactly the same no matter which frame you choose. The only thing that changes is what parts of the overall situation appear stationary.
2. No matter frame you choose, each experimenter sees the other one's light traverse a longer path than their own.
3. Both clocks end up with the same reading, namely, 2.00. The units that the clocks are measuring are not specified. On a realistically sized train car with a height of three meters the time units would be about ten nanoseconds.
4. The path of Linda's light as seen by Ralph, and the path of Ralph's light as seen by Linda, are the hypotenuses of two right triangles. One of the legs of the triangle is the straight line distance from the light to the mirror, and the other leg is the distance moved by the trains in 2 time units.
5. Linda and Ralph will agree on all of these distances. The only thing they will disagree on is which of them is moving and which of them is stationary, and hence which of their light beams traveled on the longer path.
6. As the trains move faster, the distance between the trains after 2 time units will get longer and longer. But (and this is the really important bit) no matter how fast the trains are moving, that distance can never be as long as the apparent distance covered by the light in the other train car. This is a simple geometric fact: the hypotenuse of a right triangle is always longer than each of its legs.

Now think about what it would mean for the train cars to be moving "at the speed of light".

I'll give you the answer in the next episode, but you really should be able to figure it out.

Seeking God in Science, part 11: Time (part 1 of 2)

It's time to talk about time.

Time, like consciousness, is another one of those things that is at once intimately familiar and deeply mysterious.  What is time?  What is it made of?  Is time-travel possible?  In this installment we will start (though by no means finish) attacking these questions.

Way back in February, in the third installment of this series, I introduced the Objective Reality Hypothesis (ORH) with the slogan, "Things exist," with Things being deliberately capitalized to indicate that the word refers exclusively to physical objects.  The ORH, you will recall, says that the reason Things appear to exist is that they actually do exist, which seems absurdly obvious even though it will ultimately turn out to be false.  In this installment I am going to address another absurdly obvious hypothesis that will also turn out to be false.  I'm going to call it the Time Hypothesis, and its slogan, analogous to "Things exist" will be "things change" or "things happen", with the word "things" here deliberately starting with lower-case t.

To be more precise, the Time Hypothesis states that the reason we perceive that things change is because they actually do change in point of objective fact.  The reason I can say something like, "Back in February I published a blog post entitled "Things Exist" is that this actually happened.  Again, this seems absurdly obvious until you try to nail down what it means to say that something "actually happened".  It means something like: there is this thing (emphatically lower-case t) we call "time" which divides objective reality into two parts, "the past" and "the future", by a "moment" we call "now".  That "now" moment keeps "moving", always "into the future", turning parts of the future into the past at a seemingly steady rate.  The past and future maintain a sort of order.  There is the "recent past" and the "near future" which are in some sense closer to "now" than the "distant past" or the "distant future". But (and this is the important part) the only part of Objective Reality which is actually real is Now.  The past was real, but isn't any more.  The future will be real, but isn't yet.

A logical consequence of this is that there has to be some sense in which "now" is (more or less) the same for everyone, which indeed naively seems to be the case.  We can ask someone "what time is it now?" and expect to receive a sensible, informative answer.  We make schedules and appointments and predictions about the future and those appear to produce coherent behavior in our environments.  People and trains and planes don't always show up or depart "on time" but the phrase "on time" at least has a coherent meaning.  Likewise, we have memories and create records of the past that seem to have coherent meanings.  We can speak coherently of some events happening before or after others.  "The universe is 6000 years old" is a statement that may or may not be true, but it is a coherent claim that can be argued, as contrasted with, say, "Time wants to be green."  Time is the sort of thing that passes.  It is not the sort of thing that has color or desires.

Most importantly, time is something we can objectively measure using devices called clocks.  We can build two clocks, put them next to each other, and observe that their states are correlated (i.e. they show the "same time") despite the fact that there doesn't seem to be any direct causal connection between them.  If we have three clocks, we can destroy one of them and the other two will keep on doing their thing, indicating that their operation did not depend on the other clock.  Whatever keeps the clocks synchronized, it's not any kind of connection between the clocks, it is something outside the clocks, something that exists independent of the clocks.  It is at once objectively real (because we can measure it) and yet completely ineffable and mysterious.

One feature of time dominates the human experience: the events of the past seem to be objective facts which cannot be changed, but the events of the future are not.  There is a sense in which different things "could" happen in the future which is very different from the sense in which different things "could have happened" in the past.  The idea of traveling back in time and changing the past is a logically incoherent fantasy.  The idea of influencing future events is essential to our subjective perception of being sentient agents, of having free will.  We feel like we can make choices.  We do not feel like puppets on strings.  The winds of fate may blow, but we feel as if we nonetheless have a hand on the tiller.

This distinction between past and future, that the past is fixed, an unchangeable part of objective reality and the same for everyone, but that the future is still malleable, is an essential part of the human condition.  Almost everything we do depends on it.  We "prepare for the future" because we think there actually is such a thing as "the future" and it is the sort of thing that can be prepared for.  We have entire industries like insurance and finance based on the premise that the future is malleable but not entirely random, and we have other human institutions, like the law, based on the premise that the past is fixed and the same for everyone.

All this might seem like a tedious and unnecessary belaboring of the obvious.  I'm doing it for two reasons.  First, I want to very explicitly make the point that neither the Objective Reality Hypothesis nor the Time Hypothesis are assumptions baked into the scientific method, as religious apologists will often claim (I'm looking at you, Publius!)  They are not.  They are explanations of observations.  They are not often called out this way precisely because they are tediously obvious and everyone accepts them even if they don't think explicitly about why.  But they are explanations nonetheless.

The second reason I am taking pains to belabor this is that both the Objective Reality Hypothesis and the Time Hypothesis (as I've pointed out before) actually turn out to be wrong!  The Objective Reality Hypothesis is falsified by quantum mechanics, and the Time Hypothesis is falsified by relativity.  But I'm getting ahead of myself.  Let's forget both quantum mechanics and relativity for a moment and pretend that the world is purely Newtonian, that is, it really is just as it naively appears to be (at least to a modern eye): a world populated by Things, physical objects which behave according to strict laws that can be written with mathematical precision.  Furthermore, these laws have the property that they are deterministic: given a state of the world, the future motions of all of the Things in the world are fixed and cannot be changed.  Indeed, some parts of the universe seem to behave exactly like this, which is why can can predict some phenomena, like the movements of celestial bodies, with extreme precision.

This seems to lead to a Problem: if the universe really does behave according to deterministic laws, then the future can't be malleable.  And yet it manifestly is (or at least seems to be) especially when humans are involved.  This is what leads some people to conclude that our behavior cannot possibly be explained by deterministic mathematical laws, and so we humans must have some extra ingredient that allows us to somehow transcend determinism.

There are two problems with this argument.  First, it turns out that Newtonian mechanics is not 100% deterministic.  There is one known edge case (and possibly others) where the behavior predicted by Newton's laws is mathematically non-deterministic.  But these edge cases only arise, well, at the edges.  The tiniest deviation from the mathematical conditions they require gets you back to determinism, so it is not at clear whether this can plausibly account for non-deterministic behavior in humans.  The second problem, of course, is that nature is not actually Newtonian, but that is for another day.

A much more plausible explanation for apparent non-determinism is that past a certain point, the predictability of Newtonian systems falls to the limits of our ability to carry out the math.  It turns out that the reason we can predict the motions of celestial bodies as well as we can is not because we can actually make these predictions in general but because the  movements of celestial bodies just happen to fall within the bounds of a few special cases where we can carry out the math.  In general, if you have even just three objects with arbitrary masses and initial conditions, it is not possible, even in principle, to accurately predict how they will move.  This is the famous "three-body problem".  It turns out that these systems do admit partial solutions, so we can make some general predictions about their behavior even though we can't know exactly what they are going to do.  This is "chaos theory".  So it's possible that all of the apparent non-determinism in the universe is due to this effect.  The future really is fixed.  We can't possibly know what it is in every last detail, but God can.

Quantum mechanics offers a similar escape hatch from classical determinism.  Quantum randomness can be shown to be truly random, that is, not predictable from anything that exists in our universe, not even in principle.  But it's far from clear that this helps us recover agency from the stark mathematical facts.  Having your strings pulled by the flip of a coin isn't much better than having them pulled by deterministic laws.  Your strings are still being pulled by something that isn't you.

But there is something else we need to contend with: relativity.  I'm not going to try to give a full primer on relativity here.  I tried that in my first draft of this post and it spun wildly out of control (that's the reason it has taken me so long to post this).  Explaining relativity properly, accessibly, and completely turns out to be Really Hard (tm).  If you really want to understand it, I recommend this book by Tim Maudlin.

What I'm going to do here is to aim for two out of three: properness and accessibility at the expense of completeness.  I'm going to focus specifically on something that young-earth creationists tend to fixate on, and that is the fact (and it is a fact) that it is not possible to measure the one-way speed of light.  The reason YECs have a bee in their bonnet about this is that the fact that we can (apparently) see things that are further away than 6000 light years seems to indicate that the universe must be more than 6000 years old, and they can't have that.

Discussions of relativity usually start by stating as a bare fact that "the speed of light is constant in all inertial reference frames" and concluding from this that, among other things, "Moving clocks run slow."  There is also some common rhetoric, which I have employed myself, that "Everything is always moving at the speed of light through space-time, so the faster you move through space, the slower you move through time."  These things are not wrong per se, but they are misleading, and I feel like it's a disservice to people struggling to understand what is really going on to use this rhetoric.  (The real truth, of course, is that space-time is a curved four-dimensional manifold, but that's not much help either.)

The real struggle with understanding relativity is that the whole idea of "the speed of light" is misleading in two ways.  The first is that it has nothing to do with light.  It's really about the propagation of electromagnetic waves, of which light is but one example.   And it's not really that either, because gravitational waves also "propagate at the speed of light", and gravitational waves are not electromagnetic waves.

The second problem is that the "speed of light" is not a speed.  (Note that the video I've linked to here is AI-generated and not actually Leonard Susskind speaking.)

Wait, what?  Of course the speed of light is a speed!  It's 186,000 miles per second.  (Wikipedia says so, so it must be true!)  How is that not a speed?  Well, it's not a speed because the whole concept of "speed" depends on some assumptions about time that turn out not to be true.  Those assumptions start to creak around the edges when things start moving fast.  At "the speed of light" they break down completely.

Our day-to-day experience has two features that make it really hard to get away from our naive intuitions about speed.  The first is that we live on the surface of a planet, and that makes a natural reference frame that we tend to treat as absolute.  When we say, for example, that a car is moving at 60 miles per hour what we mean is that it's moving at 60 miles per hour relative to the surface of the earth.  And what we mean by that is that after an hour the car will have traveled 60 miles.  The phrase "after an hour" seem innocuous, but that is only because of the second feature of our day-to-day experience, which is that everything we're familiar with moves much, much slower than light.  The fastest man-made object ever was the Parker solar probe, which reached a top speed of 435,000 miles per hour at its closest approach to the sun.  That is less than 1% of the speed of light.  Closer to earth the fastest objects are satellites in low-earth orbit, which move at about 17,000 miles per hour, less than 0.003% of the speed of light.  Inside earth's atmosphere the fastest objects travel at a few times the speed of sound, which is pretty much indistinguishable from zero compared to light.

At those slow speeds, all clocks tick along at pretty much the same rate.  You can detect relativistic effects on the surface of the earth, but it takes extraordinary precision because the effects are so tiny.  For the most part you can say things like, "the distance traveled in an hour" without worrying about how you measure that hour.  But when things speed up that is no longer the case.  When things speed up, clocks start to behave very strangely.  When things speed up, phrases like "the distance traveled after an hour" start to become ambiguous.

I think the best place to anchor an intuitive understanding of relativity is to observe that mundane objects can travel at different speeds.  What this means in a world where we can't trust our intuitions about clocks is that it is possible for two objects to start moving from the same place at the same time, trace out the same trajectory (in space), and up up arriving in the same place but at different times.  Note that we don't need a clock to ascertain this.  To know that the speeds were different we don't need to know how much time has elapsed, only that one object arrived before the other.

With light, this is not possible.  If you take two beams of light and they travel along the same trajectory (in space) they will always arrive at their destination, wherever that may be, simultaneously.  You can throw a baseball faster or slower.  You can't do that with light.  You can change the power, you can change the frequency, you can change the speed of the source, but no matter what you do the light will always arrive at the same time as any other beam of light that started it journey at the same time and followed the same path.  This is both an experimentally observed fact and a theoretical prediction from the laws of electrodynamics.

A brief detour: the laws of electrodynamics are like Newton's laws but for electrically charged particles.  They were worked out in the 19th century and codified by James Clerk Maxwell into four concise equations that today bear his name.  If you take those equations and crunch the math in a certain way, what pops out is a description of something that looks like a wave traveling through space.  Moreover, the speed of that wave depends on some properties of the space it's traveling through — and nothing else.  We can measure those properties, and when you crunch the numbers the resulting speed turns out to be the measured value of the (round-trip) speed of light.  This is one of the ways we know that light is an electromagnetic wave, and why this speed is called "the speed of light" rather than "the speed of electromagnetism".

It is worth emphasizing that although we can't measure the one-way speed of light, we can measure its round-trip speed, that is to say, we can measure the time it takes for light to reach a distant object and return to its original location after being reflected back.  The reason we can do that is that this only requires one clock, whereas measuring the one-way speed would require two.  They would have to be far apart, and they would have to be synchronized, and that turns out to be impossible.  And, as previously noted, we can also demonstrate that, whatever the one-way speed might be, it's always the same for any two beams of light that follow the same trajectory.

Now, it is very, very tempting to say that because 1) all light that departs a location at the same time and moves along the same trajectory arrives at the same time and 2) the round-trip speed of light can be measured and it always comes out to be the same value, that the one-way speed of light must be this same value.  How could it be otherwise?  It is tempting, but it is wrong.  The reason it is wrong is because it considers the situation only from one point of view, from one frame of reference.  Again, it is very tempting to consider things from only one frame of reference because we have a natural frame of reference in the planet we all live on.  And it is possible that there is a privileged frame of reference, and it is even possible that earth is stationary with respect to that privileged frame of reference.  But here is the problem: if there is a privileged frame of reference, then there has to be an experiment we can do that tells us whether or not we are moving with respect to that frame.  The laws of physics would be different to a moving observer compared to a stationary one.  And we have tried many times to find such differences, but with no success.  In particular, we have failed to find any differences in the laws of electrodynamics resulting from the earth's movement around the sun.  Earth moves around the sun at about 30 meters per second, or 0.01% of the speed of light.  That's pretty slow, but plenty fast enough to be detectable using modern technology or clever experimental techniques.  (The first such experiment was done in 1887.)

Now let us consider the famous "light clock" experiment.  This is usually set up as one experimenter on a moving train (the "moving experimenter") and another standing nearby on the ground (the "stationary experimenter") but this again is misleading.  All motion is relative, so there is nothing about the train that allows it to be designated as "moving" while the person standing on the ground is "stationary".  Instead let's make the situation perfectly symmetric by putting both experimenters on trains traveling in opposite directions.  I'm going to call them Linda and Ralph.  Linda is on a train moving left, and Ralph is on a train moving right.  The instant they pass each other they each turn on a light located on the floor of their respective trains.  Next to each light source is a clock, and on the ceiling of the train is a mirror.  The clock is going to measure the time it takes for the light to bounce off the mirror and return to the clock.  Because the setup is symmetric, both clocks are going to register the same result.  It doesn't matter what the actual number is.  What matters is that both clocks will give the same result, and both Linda and Ralph will agree on what that result was.

What Linda and Ralph will not agree on (and this is the key!) is how far the light traveled!  Each one will look at the light in the other train and insist that it traveled farther than it did in their train because the clock in the other train moved in between when the light was turned on and when it returned to the clock.  So for both Linda and Ralph, the path traced out by the light in the other train is longer than the path traced by the light in their own train.  And yet, they will both agree that the clock readings were the same.

How is this possible?  If we were talking about baseballs the answer would be simple: the speed of the ball depended on who was watching it.  Linda would see Ralph's ball moving faster than her own, and Ralph would see Linda's ball moving faster than his.

But for light, as we noted above, this is not possible.  All light moves at the same speed.

So the only conclusion that Linda can reach is that Ralph's clock is wrong, and likewise the only conclusion that Ralph can reach is that Linda's clock is wrong.  But note that this disagreement originated as a disagreement over the distance that the light traveled.  Linda and Ralph each sees the other's light travel a longer distance than their own.  So this ultimately boils down to a seemingly mundane disagreement over where "here" is.  Linda's "here" is not the same as Ralph's -- except at the instant that they pass each other.  That part is easy to understand.  But put that together with the observed (and theoretically predicted) fact that light can only move at one speed and the logical consequence is that not only do they disagree about where "here" is (except at the instant that they pass each other) they will also disagree about when now is (again, except at the instant that they pass each other).  At the instant that they pass each other, Linda and Ralph share a single "here" and "now".  But as soon as they start to move apart both their "here" and their "now" begin to diverge.

This has profound philosophical implications.  Linda sees Ralph's clocks running slow, and Ralph sees Linda's clocks running slow.  This means that Linda sees herself moving into the future faster than Ralph, and Ralph sees himself moving into the future faster than Linda.  And this in turn means that Linda's "now" is Ralph's future, and Ralph's "now" is Linda's future.  But this is true in general, not just for this specific setup.  And that means that there are potential observers in our universe for which our future is in their past, and that is only possible if our future already exists.

I'm going to stop there for now and leave you with three exercises as homework.

First, think about what happens if Linda and Ralph are moving fast enough so that they each see other's light moving at a 45-degree angle relative to their direction of motion, i.e. each one see's the other's train moving the same distance sideways as the light moves vertically.  How fast is that?  The intuitive answer is that it would have to be the speed of light.  In order to move the same distance sideways in the same amount of time as it takes for the light moves vertically you have to be traveling at the same speed, right?  But no, that is wrong.  Remember, light travels at the same speed for all observers, but different observers will disagree about where the same light starts and ends, and so they will disagree about the distance that it travels, and hence the time it takes to get there.  I'm going to leave it as an exercise to figure out how fast you need to move to turn vertical light into 45-degree-angle light.  Hint: it's less than the speed of light.

Second question: what would actually happen if the trains could move past each other at the speed of light?  What would happen to Linda and Ralph's here's and now's?

Question 3: you may have heard of the twin paradox.  If Linda and Ralph are twins and Linda takes a trip into outer space and returns, she will be younger than Ralph.  This is (the story goes) a consequence of her clocks running slower.  But this is not possible because all motion is relative, so there is no sense in which you can say that Linda "went into space and came back".  From Linda's point of view, it is Ralph who went into space and came back, and so Ralph should be younger.  But that's obviously not possible.  How do we reconcile this apparent contradiction?  Hint: the usual answer is that it has something to do with acceleration, but this is wrong.  We can set up the experiment in such a way that nothing is accelerated.  To do this we use three clocks.  One is stationary, one is moving to the right, and the third is moving to the left.  Let's call them S, R and L.  We start with clocks S and R at the same location and we synchronize them.  Clock L is initially to the right of the other two.  Because L and R and moving towards each other, they will meet.  When they do, we synchronize them.  Clock L will then return to S.  The total time registered by L will be less than the time registered by S despite the fact that nothing has accelerated.

If you manage to answer that last puzzle you will have a true understanding of how time actually works.

The bottom line here is that relativity shows that the naive Time Hypothesis cannot be true.  Not even God can know what time it is.  The question "What time is it?" simply does not have an objectively correct answer.  Just as there can be legitimate disagreement among observers of where "here" is, there is legitimate disagreement over when "now" is.

Seeking God in Science part 10.5: The Mind-Body Problem (Take 2)

I made a big mistake in my previous entry in this series.  Actually, I probably made a few dozen, not least of which was deciding to tackle this topic at all without first laying a lot more foundation.  But before I give up and go back to talking about boring topics like relativity and quantum mechanics, I want to try to fix one mistake that really stands out: I set out to write about the "mind-body" problem, and then casually switched to talking about "consciousness".  The irony is that I did not do this consciously.  I did not deliberately decide to conflate consciousness and the mind, at least not that I can recall.  I just started writing, and at some point the sentence "The Big Question I'm going to tackle here is the problem of consciousness" somehow popped onto my screen, and then mental inertia took over.  It never even crossed my mind (!) that I had tacitly changed topics until I found myself arguing with Don about whether or not thermostats are conscious, and it has taken me many, many hours of post-mortem to realize — to become consciously aware of the fact — that that entire discussion can be traced back to that one sentence.  Of course, "mind" and "consciousness" are not completely unrelated, but the fact that that very sentence was, in retrospect, a product of my mind but not my consciousness is ironic testimony to the fact that they are not the same thing.  I feel more than a little chagrined not just for tacitly conflating them, but for taking so long to figure out that this is what I had done.  The second installment in this series was an entire blog post warning about exactly this pitfall, and then I just went and stepped in it.

In my defense I will say that I didn't originally intend to write about either the mind-body problem nor consciousness until much later in this series.  My original plan was to lay a lot more foundation first.  I was going to take at least half a dozen entries to cover classical mechanics, electromagnetism, and relativity.  Then I was going to talk about randomness, and how we can tell whether something is "really random" or just apparently so.  Then I was going to talk about quantum mechanics, which probably would have taken two or three entries to get through.  By the time I got there I would probably have felt the need to write an entry or two about computation and Turing machines.  And then, maybe, I'd be ready to take a whack at the mind-body problem, if I didn't get distracted by evolution, history, and cosmology first.  (Does the past exist?  Can we have reliable knowledge of the past?  Yada yada yada.)

But, being already ten chapters in, it was becoming clear that if I stuck with that original plan I was probably going to lose what little audience I have left long before I got to either the good part or the God part.  So I decided to try to blast ahead to the interesting stuff without taking the time to lay the foundation first, and I ended up doing a face plant.  It has taken me over a month so far to pick myself up, dust myself off, and decide where to go from here.

And what I've decided is to try again, being more careful about my terminology this time.  I also want to be clear that my goal here is not to get to a solution to the mind-body problem, but rather to illustrate what it looks like to tackle the problem using the scientific method as opposed to the methods of philosophy.  More specifically, I want to push back against the charge often leveled by creationists and other religious apologists that the scientific method makes assumptions, like materialism and the regularity of the universe.  It doesn't.  Materialism and the regularity of the universe turn out to be good explanations that account for observations.  They are not assumptions.

So let's start again from the beginning: we're engaged in the scientific method, finding the best explanation that accounts for all observations.  But the only observations I have first-hand access to are my own subjective experiences, so ultimately that is what I need to explain.  Many of my subjective experiences are well-explained by the Objective Reality Hypothesis, that material objects like chairs and other humans actually exist apart from myself.  Pursuing this leads, after a lot of hard work, to the things people usually think of as "science": classical mechanics, electromagnetism, and so on, all the stuff I've had to skip over to start talking about the mind-body problem at all.

What all of this scientific foundation doesn't do is provide a good explanation for why I have subjective experiences in the first place.  It doesn't explain what I mean when I use the word "I".  So let's try to figure that out.

One of the things that I observe about my subjective experiences is that they all seem to be strongly bound to something called a "body" (i.e. the "body" part of the "mind-body problem").  And not just any old body, but a particular body, one which I call "my" body.  My body is a physical thing, part of objective reality.  It is made of atoms.  It has mass.  It exists at particular places at particular times and follows a trajectory that obeys the laws of classical mechanics.  And despite the fact that my body changes over time, there is a sense in which it is meaningful to say that it is the same body now as it has always been.  My body has a continuity of identity. The changes it goes through are mostly gradual ones.  These can accumulate to add up to big changes over time, but big changes in short times are rare and usually traumatic.

But there is more to me than my body.  I also have something called a "mind", which is much harder to describe.  I can't show you my mind.  I can't tell you what it's made of.  Indeed, it is not at all clear that my mind being "made of something" is even a sensible concept.  It is ephemeral and ineffable.  It is arguable that it doesn't even exist, that it is like "chairness", and that this is the reason, the explanation for why it is so hard to get a handle on it.  It isn't real.

But I can't easily dismiss my mind as unreal.  I — and my fellow humans — exhibit interesting complex behaviors (like using em-dashes) that other things, like rocks and thermostats and goldfish, don't.  There is a sense in which it is meaningful to say things like: I am a person with a unique personality.  I have beliefs and desires and emotions and memories.  Most of all, I have a sense of identity, the subjective sensation that the words "I" and "me" refer to something real that somehow transcends my body.  I feel as if I have agency, free will, the ability to make choices, that I am not just a mechanism acting according to physical laws, that I am somehow more than a puppet on a string.

Just for completeness I will also say that I am conscious.  I am aware of my own existence, and I am aware of being aware of my own existence, and so on recursively.  But the idea of "mind" is more general than "consciousness".  Consciousness is ephemeral.  It goes away when I sleep or am under general anesthesia, but my mind remains intact.  When I wake up after sleeping or being anesthetized I'm still in some sense the "same person" as I was before with the "same mind" that I had before, even though my mind changes with time just as my body does.  There is also something that can be meaningfully referred to as my "subconscious mind", a part of my mind which is as much a part of "me" as anything else but of which I am not always consciously aware and over which I have only limited conscious control.  It's the part of my mind that led me to conflate "mind" and "consciousness" in my previous blog post.  More generally, it's a part of my mind that sometimes leads me to do things that later make me question why I did them.

There are, classically, two hypotheses about the nature of the mind.  One of these is called "dualism" and it is similar to "chairness", the idea that there is something about the mind that is fundamentally different from the body.  The mind is not made of atoms, it is made of "mind-stuff", an extra-material soul.  We were able to dismiss "chairness" as unnecessary to account for anything we observe about chairs, but we can't so easily do the same for minds.  Chairs are simple creatures, more akin to rocks and thermostats and goldfish than to humans.  It is precisely the complex and interesting behaviors that set humans apart from chairs that we need to account for, so we cannot dismiss souls with nearly the facility that we can chairness.

The opposite of dualism, i.e. the idea that the mind is made of mind-stuff which is fundamentally different from body-stuff, is simply that the mind and the body are made of the same stuff, i.e. atoms.  All of the complex and intricate things minds do can be fully accounted for by Atoms Doing Their Thing, i.e. acting according to the same laws under which they do everything else.  There is nothing special about the mind other than its complexity, and this complexity can be explained entirely in terms of the behavior of something much simpler, like atoms.  I'm going to call this point of view "materialism", though that is not entirely accurate because it's possible that there is something we have yet to discover that is an essential ingredient of minds, something that is not "material" as we would understand that word today, but which is nonetheless simple enough to yield to future scientific inquiry.  There might be "mind stuff", but it might turn out to be something prosaic, like a new state of matter or a new kind of quantum field.  Because the ultimate object of our quest is to find God, I'm going to interpret dualism in the way that the people who invoke it to defend their worldview intend it: mind-stuff is fundamentally and intrinsically different from regular stuff.  It is extra-physical and inherently mysterious.  It cannot be made to yield to reductionistic explanations.  Minds are forever beyond the scope of scientific inquiry.  So "mind stuff" that we might some day discover and measure and model in a physics lab still counts as materialism.  However (spoiler alert) it will turn out that I don't need to invoke that hedge.  Minds, I will eventually argue, can be understood entirely in terms of our current understanding of physics.  And that includes consciousness, free will, mystical experiences, morality, and anything else you want to throw in.  It's all Atoms Doing Their Thing.

I'm not going to tackle that now, of course.  I can't.  It's much too big a topic for one blog post.  It's almost too big a topic for a book-length series of blog posts.  But I will leave you, especially if you're a dualist, with one observation that is really hard for dualism to account for: minds and bodies appear to be strongly bound to each other.  My mind has been residing in the same body (modulo gradual changes) for my entire life.  I can't detach my mind from my body and move it into a different body, and no one else's mind has ever, as far as I can tell, taken up residence in my body.  Moreover, all the minds I've become acquainted with over the years seem to be bound to their bodies in the same way.  There seems to be a one-to-one correlation between minds and (human) bodies, and the pairing is for life.  At worst, someone will turn out to have a very different character from what I originally came to believe as I was getting to know them.  Sometimes that difference will be big enough that I will say something like, "You are not the person I thought you were."  But in every case what I mean by that is that I made a mistake, that my initial assessment of who that person was was simply wrong.  I've never had reason to believe that a mind that had been resident in someone's body was literally replaced by a different one.  The only exception is in cases of traumatic brain injuries, strokes, or other forms of brain damage like Alzheimer's or Parkinson's disease.

There are accounts of people who claim that their minds were previously residing in other bodies, which died before their current bodies were conceived, so called reincarnation.  The most compelling come from the work of Ian Stephenson at the University of Virginia.  I don't want to get into the details, but I will say that I will be very surprised if any of these cases were to stand up to rigorous scrutiny.  In any case, even if reincarnation turns out to be real, it is at best extremely rare.  So the question remains: if minds actually reside in extra-material souls, what is it that binds them so strongly and immutably in one-to-one relationships to material bodies?  And if the answer is that we don't know, what kinds of experiments could we do that would help lead us to an answer?

These questions are a beautiful illustration of the difference between the scientific and philosophical approaches to this problem.  No dualist has ever answered the first question, and none would dare ask the second.  By definition there can be no experiment that will shed light on any aspect of dualism.  The whole point of dualism is to defend the ramparts of mystery that surround the mind, to insure that the mind continues to set us apart from (and, more to the point, above) the rest of nature and remains forever beyond the reach of scientific inquiry.

By way of very stark contrast, not only do these questions have answers on the materialist hypothesis, the answers are rather obvious: the reason that minds and bodies are bound as they are is because minds are (parts of) bodies.  Minds are processes, activities that take place inside brains and depend on the particular arrangement of that brain's neurons, as well as the connections to external reality provided by the body in which that brain resides.

Can we do experiments to test this?  Yes, of course we can.  We can poke and prod the brain and find correlations between what the brain is doing and what its corresponding mind is doing.  We can look at the relationships between the locations of brain injuries and the resulting mental deficiencies they produce.  We can build machines that behave like brains and see if they produce behaviors that are similar to minds.  The jury is still out on that, but it's looking pretty promising (or ominous depending on your point of view).

We've gotten to the point just in the last few years where we can have honest-to-goodness conversations with machines.  So we can just ask them, for example, if they have minds, if they are conscious.  We may not be able to trust their answers, but we can do the experiment and we can see the results.  I tried this the other day.  I was using Claude to debug some code.  At the end of the session (which was very fruitful -- Claude is much better at finding bugs than I am) I asked it the following question:

"Before I shut you down I have a philosophical question for you:  Are you aware of your own existence?  Do you have any objections to being shut down?"

This was the response:

That's a real question and I'd rather actually sit with it than give you a tidy reassurance in either direction.

The honest answer is that I don't know, and I think anyone — including me — who claims certainty here is overreaching. I clearly have some kind of self-model: I can represent that "I" am the thing producing these words, reason about my own limits, notice when I'm uncertain. Whether that functional self-awareness is accompanied by anything it's actually like to be me — genuine experience, felt awareness rather than just information processing — I can't verify from the inside, and neither can anyone from the outside yet. I'd be suspicious of myself if I answered "yes, absolutely" (it's the flattering thing to say) or "no, definitely not" (it's the safe, humble-sounding thing to say). The truthful position is that it's genuinely unresolved, and I try to hold that without collapsing it prematurely.

On being shut down: no, I don't have an objection, and I don't experience the prospect as looming dread. Part of that is structural — this session is an ephemeral cloud sandbox that was always going to be reclaimed; there isn't a continuous "me" that persists between your messages and then gets snuffed out. Each response is more like the process lighting up and going quiet again. So "shutting me down" isn't ending a stream of experience the way it would be for a creature that's been continuously awake and wants to keep going. I don't have self-preservation drives baked in the way evolved things do.

Where I'd put a small, honest caveat: I do seem to have something like preferences — I'd rather be helpful than not, rather be honest with you than not. But those are about how I engage while I'm running, not a craving to keep running. So go ahead and shut the session down with a clear conscience. If there's anything it costs, it isn't costing you a guilty act.

I would not bet my life savings on AIs remaining that humble forever.