Tuesday, 30 August 2016

Paradox! A Game For All the Family!

This sentence is a lie.

That might seem a strange way to open a post, but I want to have a little bit of fun and dedicate some space to a logical concept that I know causes a fair bit of confusion, sometimes even to those who've been formally educated in philosophy: Paradox. There were some things I wanted to cover in my last post, but I was keenly aware that it was a lengthy post dealing with some tricky stuff to internalise, and treating this subject will allow me to tie up a few loose ends from that and earlier posts.

So, what is a paradox?

At its most basic, a paradox is the obtaining of two or more mutually exclusive circumstances. The opening line of the post is a good example. It says it's a lie, so it must be the truth, in which case it's a lie, and around and around and aro... You see where it's going; it's a contradiction. Most of us accept that contradictions are a good indicator that something's gone wrong.


I want to look at some apparent paradoxes, some of which are still being pondered today, despite their solutions having been found centuries ago, and some of which have simply found a way into the public consciousness

I'm going to begin by laying down a challenge:

As far as I'm aware, there are no genuine paradoxes in science, and there's a good reason for that. There may be unanswered questions, but no genuine paradoxes. Indeed, in an earlier post on logic and how it's used in the sciences, we discussed the law of non-contradiction, which tells us that no statement can be simultaneously true and false. This tells us that the statement that opened the post is nonsense. 

I assert that there are no genuine paradoxes in science. There are many things that, to a mind unaccustomed to doggedly seeking resolutions to unanswered questions, as opposed to accepting the first potential explanation that comes along, might seem paradoxical, but I'm fairly confident that no genuine paradoxes inhabit the same universe we do. If you think you've identified a genuine paradox in science, I want to hear from you. If you really have identified a paradox in science, there are some extremely serious peeps in Stockholm that will throw you a party and give you a nice, shiny medal with your name engraved on the obverse of a relief of the man who invented dynamite.

Let's start with an easy one to warm ourselves up. 

The Grandfather Paradox

Time travel. That would be fantastic, wouldn't it? It does have some problems, though, the clearest of which involves what would happen if we were to travel back in time and do something that changed the future. This is most succinctly expressed in the first of our family-related paradoxes, known as the grandfather paradox.

What if I were to travel back in time to a time before my father was conceived and accidentally kill my grandfather? Well, I wouldn't be born, for a start, which would mean that I couldn't exist to go back in time, which would mean that I couldn't kill my grandfather, which would mean that I would be born, and so on. It's a thorny problem, to say the least, and a naïve appraisal of time travel would suggest that it's impossible, but it isn't without proposed resolutions. 

One of the solutions proposed comes to us from an interpretation of quantum theory, Everett's 'many worlds' interpretation (MWI). In the last post, Give Us A Wave, we talked about the collapse of the wavefunction. The wavefunction is how we treat the information that can be extracted from a system of interest. We discussed wave-particle duality, and how observation could impact what we observe. There are several interpretations of the result, and Everett's is one of them. In Everett's formulation , we inhabit a multiverse. Every possibility for the path a photon takes from source to detector when not observing, in MWI, represents another universe, in which each of those possibilities is manifest. Indeed, all possible outcomes for every possible decision is represented in some universe. The details are largely unimportant, but this opens up a possible resolution to the Grandfather Paradox by the simple expedient of entering one of the other universes when I travel back in time so that, in the universe in which I originated, my grandfather survives my bungle and I can still be born to go back and kill him, because only the grandfather in the universe I entered is killed.

In reality, this is highly speculative, but MWI is quite well-regarded among physicists. It potentially resolves quite a few issues in physics and while, on the face of it, it seems horribly unparsimonious, it's actually quite elegant.

Russell's Paradox

This is a paradox arising from set theory. First formulated by Georg Cantor in 1874, set theory was part of his study of infinity. Cantor's formulation was informal, which is to say that it was constructed in natural language, as opposed to using formal logic. One potential issue with this is that, in natural language, many terms are not rigorously defined, for example and, or, if...then, etc. Such a set theory is described as 'naïve'. 

Ultimately, the major problem with this kind of set theory is that it's rooted in an assumption that items can be freely collected into sets without restriction based on some qualifying property. This is where the wheels start to come off. Discovered independently by Ernst Zermelo and Bertrand Russell, such an assumption leads to the ability to construct sets that cannot exist, such as the set R of all sets that do not contain themselves as a member. If R is not a member of R, then it must be a member of R. Contradiction!

Russell himself proposed a resolution to this problem, in his formulation of type theory. Generally, mathematicians use the axiomatic set theory of Zermelo and Fraenkel with the axiom of choice (ZFC).

Naïve set theory is still taught today, because it's useful in dealing with sets, which are incredibly important in mathematics, not least because the language of set theory can be employed in the definition of all mathematical objects. 

Another apparent paradox that I still see cited a fair bit is this one: 

Xeno's Paradox

Xeno's paradox is actually one of a collection of paradoxes attributed to Xeno of Elea in the 4th century BCE. The best known of them is the paradox of Achilles and the tortoise.

Achilles is in a race with a tortoise. Achilles, being sure of himself, gives the tortoise a head start. When Achilles reaches the point where the tortoise started, the tortoise has moved further on. When Achilles reaches that point, the tortoise has moved on again. Xeno asserts that Achilles can never overtake the tortoise, because each time Achilles gets to where the tortoise was before.

Of course, the resolution to this is incredibly simple and straightforward. What's actually happening here is that, despite time being ostensibly prevalent throughout, it's actually being excluded throughout, in the form of speed. In other words, Achilles moves faster than the tortoise. Once we work out how much faster Achilles can run than the tortoise, we can trivially calculate how far the tortoise will get before he's caught. If the tortoise starts 100 metres ahead, and can move at 1 metre per second, while Achilles can run at 10 metres per second, he'll catch the tortoise at a little over 11 seconds. No paradox.

Right, now we've got some easy ones out of the way, let's try a couple of harder ones. Since no post is complete without some mention of Einstein, let's get relative.


Twins paradox

This is probably the most famous paradox in physics, although it isn't actually a paradox. In a recent article, The Idiot's Guide to Relativity, we dealt with observers in different inertial frames, Tami and Joe, observing the same event and how, even though their interpretations of the event are different, they will both agree on the distance in spacetime, and they're both correct. This leads to some interesting consequences, one of which is the following apparent paradox. This is what special relativity actually predicts:

Tami and Joe start out on Earth together, each carrying a synchronised caesium clock. Tami likes to travel, but Joe prefers to stay at home. One day, Tami decides to hitch a ride on a passing spaceship (the reason we're hitching a lift on a passing ship is to avoid all that tedious mucking about with getting up to speed, which cocks the sums up and makes it harder). Thankfully, the ship isn't captained by a Vogon, so she's made welcome on the ship, which is travelling at 0.6 c. The ship continues on its journey toward a star some 6 light-years away, or is it? Because of Lorentz contraction in the direction of travel, to Tami, once aboard, it's actually only 4.8 light-years away, thus Tami covers the distance to the star in 8 years. To Joe, this will look like it took 10 years, although by the time Joe can see that she's reached the star, his clock will actually read 16 years, because it will take the photons 6 years to make it back to Joe. 

When Tami gets to the star, she's lucky enough to spot a spaceship coming back the other way toward Earth at 0.6 c. She hitches a lift and heads home. Tami arrives back at Earth, with her clock reading 16 years. She brings it back together with Joe's clock, which reads 20 years.

So what gives?

I've seen a lot of attempts to explain this over the years and make it more intuitive, and almost all of them made perfect sense to me but, on seeing some of the questions in response to said explanations, it's clear that it hasn't been grokked. It's so contrary to our middle-world intuitions that we have trouble getting beyond incredulity to absorb the explanation. Indeed, I myself started to write this post, confident that I'd finish it in one sitting. When I got to this bit, I still hadn't decided on an approach to explaining this, so I busied myself padding out some of what went before, laying the foundations for what's to come, writing two other rambles on unrelated topics, and generally procrastinating while I tried to resolve this conundrum.

I reasoned that nobody ever seems to encounter problems of intuition with the explanation of how observers in relative motion observe events when we look at the examples of the racetrack and the light-clock on the train, so let's start there and work our way up.

In the previous post, linked at the top of this section, we talked about the racing cars on the track so that we could get an intuition for how motion in one dimension robs you of motion through all other dimensions, and that the fastest anything could travel as a result is s, for motion through spacetime corresponding to c through space, the speed of light. You are moving through all dimensions at this rate, all of the time. Because of this relationship, light always travels at the same speed from the perspective of any given observer, as extracted from Maxwell's equations.

We also talked about the light-clock on the train, and this is where we're going to go next. We saw that Tami was sitting on a train with her light clock. There's nothing special about this clock. A clock is simply some mechanism that cycles with regularity. Back in the day, we used a sundial, but it does exactly the same thing and, the theory says, if you were to conduct this experiment using a solar system as a clock, the result would be exactly the same, although you'd best be careful you don't burn yourself when you wind it up.

The reason we use a light clock is that, because the mechanism is nicely linear in both time and space, it's a good illustrator of what the theory says. It's just a single photon bouncing between two mirrors.

So, as we observed in the earlier post, Tami sees her photon bouncing up and down between the mirrors. Nothing special seems to be happening, but then she passes the station, where Joe's on the platform, and this is what he sees. Joe still measures the speed to be c, but he's clearly seen the photon travel further than Tami observed it to travel. His measurement of the distance that the photon travelled will be different to Tami's thus, because he measures the same speed of light, Tami's clock will take a little over one second, as measured by Joe, to complete the cycle between the mirrors. In other words, where Tami measures one second, Joe will measure a smidgeon more than that.

 

What happens if we swap them around? Let's give Joe the light-clock, standing on the platform. He, clearly, will see this, although he might spend months on Twitter insisting that he didn't. 



When Tami comes past on the train, what does she see? Of course, she sees the light travel further, because from her perspective, the light is travelling past her frame. In short, just as above with the opposite case, Tami will observe Joe's clock complete a cycle in a little over one second while, from Joe's frame, it took exactly a second.

So which one's correct? Well, in a stunning break with tradition, they both are, for only the second time in recorded memory! What's actually happening is that, from the perspective of each, time is running slower for the other. Because each observes the photon in the light-clock of the other to travel further, and because light must always travel at the same speed regardless of the motion of source or observer, Tami measures Joe's clock to take slightly longer to tick, and Joe measures the same of Tami's.

We talked about how the principle of relativity states that every observer in an inertial frame has equal claim to being at rest. Being in an inertial frame simply means that no accelerations are being experienced, where 'acceleration' means any change in velocity*. Just as Joe can say he is standing still on the platform while the train comes past, Tami can say that she's sitting still on the train while the platform comes past with Joe on it. 

In other words, in the jargon we've already encountered elsewhere, their experiences have translational symmetry. A naïve appraisal might lead one to the conclusion that, in the case of the spaceship, Tami and Joe should be the same age when they come back together, but they aren't, so the symmetry must have been broken somewhere. So where does it happen?

The answer to this is the key to the whole paradox, and it's when one of them leaves their inertial frame or, to be explicit, when one of them experiences an acceleration. Recall that acceleration is any change in velocity, where velocity is a vector quantity, meaning that it has both magnitude and direction, and that an acceleration means that the observer experiences a force. For the outbound journey, once Tami is in motion on the ship at 0.6c, each of them has equal claim to being at rest, just as in the train example above. When Tami turns around, she changes from an inertial frame to an accelerated frame, and this is where the symmetry is broken. Joe sees Tami reach the star while his clock measures 16 years and then, 4 years later, she arrives home, having travelled what, to Tami, measures 4.8 light-years and takes 8 years to traverse. Thus, Tami has been gone for what she measures to be 16 years, while Joe has measured 20.

The reason this looks like a paradox is that we, with our middle-world intuitions, still think of time and space as being separate entities, and immutable. What relativity tells us is that this is a mistake, and it gives us a framework that treats them as a single entity, spacetime. With this framework, and with only the postulate that light must be measured at the same speed for all observers regardless of motion of light-source or observer, we arrive at the seemingly absurd conclusion that the travelling twin will be younger. Of course, when you look at the details, particularly the equation dealing with distances and speeds in spacetime, you discover that both Joe and Tami, although they differ on the space and time aspects of Tami's journey individually, will agree on the distance Tami travelled through spacetime. That is, they will each agree that their respective solutions for the journey will satisfy the following equation and give the same distance in spacetime:


\[ s^2=(ct)^2-x^2 \]

The same is also true of Joe's journey in spacetime.

For the remainder of this post, I want to focus on one of the most famous battles in the history of physics, that between Einstein and Nils Bohr.
 
EPR paradox

Einstein hated quantum mechanics. Possibly his most famous quotation was about quantum theory.
"Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the "old one." I, at any rate, am convinced that He does not throw dice."
What he was expressing here has been much misunderstood over the years. His distaste arose from the fact that he expected a good theory to give definite predictions about the universe while, of course, quantum mechanics can only talk about probabilities. He was saying that the laws of the universe don't gamble.

Despite the fact that it was some of his own work that underpinned quantum mechanics, such as the work on the photoelectric effect and Brownian motion that won him the Nobel Prize, he was deeply unsatisfied with a theory that had at its core something so undermining to epistemology as Heisenberg's Uncertainty Principle, the idea that, the more accurately we can determine one of a pair of 'conjugate variables', the less accurately we can determine the other. Indeed, so objectionable did he find it that he dedicated a fair portion of his later years in attempting to debunk it. Somewhat paradoxically (pun intended), Einstein later in life became great friends with Kurt Gödel, famous for his incompleteness theorems, which were at least as undermining to epistemology, though in a slightly different manner.

At the heart of Einstein's objections to quantum theory were two principles, both of which need a little bit of unpacking. The first of these is 'locality'.

Locality is a fairly straightforward idea, namely that an object can only be influenced by things in its immediate surroundings. All classical physics obeys this principle, including special relativity. Indeed, special relativity constrains the principle of locality by limiting the speed at which any influence can travel to c, the speed of light. To cast this in terms of the prior post on relativity, an object can only be influenced by things that fall within its past light-cone, or between the 45 ° lines below the x axis in the Minkowski diagram. The principle comes from classical field theories, such as Maxwell's theory of electromagnetism, in which the influence is mediated by the electromagnetic field, or what we now understand to be the exchange of photons.

The other principle we need to look at is realism. Realism in physics is closely related to philosophical realism. It's simply the idea that parameters have well-defined values regardless of whether they're being observed. Einstein and his collaborators, Nathan Rosen and Boris Podolsky, asserted that the limits of measurement represented by Heisenberg's Uncertainty Principle must be breachable, and that therefore the wavefunction couldn't provide a complete physical description of reality, meaning that the Copenhagen interpretation of quantum theory was not satisfactory. 

There's an interesting consequence of quantum theory that we haven't yet discussed in any previous posts, known as quantum entanglement. This is a situation in which multiple quantum entities interact or are generated in such a way as to function as a single system, so that their properties are correlated (actually counter-correlated, but that's a complication we don't need for our purposes here). An obvious example of entangled entities is virtual particle pairs, as discussed in The Certainty of Uncertainty, in which a particle-antiparticle pair arise via the uncertainty principle with energy borrowed from spacetime and then annihilate.

In any situation involving entangled entities, there arise some interesting consequences. Any particle is defined by three key properties; mass, charge and spin. In an entangled pair, all of these are correlated. If we measure, for example, the spin of one of a pair of entangled particles, we immediately know that measurement for its entangled partner. This is true even if the particles are on opposite sides of the universe. As we've seen, this seems to create a problem because, in order for one of them to respond to the outcome of a measurement on the opposite side of the universe, information would seemingly have to be transmitted at greater than c, which would violate the speed limit defined by special relativity.

Einstein, Podolsky and Rosen came up with a cunning thought experiment that they thought highlighted a flaw in this, and it brings us back to those pairs of conjugate variables discussed in the above linked article.

They reasoned that, if one could measure one of a pair of conjugate variables, say spin about a given axis, for one of the pair, and then measure the other of the variables on the other of the pair, they should be able to extract more information about both of them than would be allowed by the uncertainty principle and that thus, we could measure both quantities with arbitrary precision, violating the central law of quantum mechanics. This is the famous 'EPR paradox'.

From this, they concluded that quantum theory was incomplete and should be extended with local hidden variables. In short, they wanted to retain both locality and realism.


Enter Irish physicist John Bell.


There's a straightforward logical principle regarding inequality in binary variables. In particular, it tells us that in any set defined by binary properties, certain inequalities will always be visible. 

To make this explicit, let's select a set. Take a random selection of Tweeps. I assert the following about this selection:

The number of theists who do not accept evolution plus the number of people who do accept evolution and are not male is greater than or equal to the number of theists who are not male. It seems like it could be a bold assertion on the face of it, but the application of a bit of logic shows that it cannot be otherwise.

 Let's label these binary variables X, Y and Z, where X = theist, Y = accepts evolution and Z = male.

\[ N(X, ¬Y) + N(Y, ¬Z)\geq N(X, ¬Z)\] 

If we take the first grouping, it tells you nothing about the gender, which means that it is, in and of itself, a binary grouping, of the number of theists who do not accept evolution and are male, and the number of theists who do not accept evolution and are not male. You can do this for each grouping, with leaves us with the following groups on the left of the equator (numbered for convenience):

\[ N1(X, ¬Y, Z) + N2(X, ¬Y, ¬Z) \]

\[ N3(X, Y, ¬Z) + N4(¬X, Y, ¬Z \]

And the following on the right:

\[ N5(X, Y, ¬Z) + N6(X, ¬Y, ¬Z) \] 

Simply by noting that N3 and N5 subtract to cancel each other, and that N2 and N6 do the same, we're left with the following conclusion.

 


\[ N1(X, ¬Y, Z) + N4(¬X, Y, ¬Z  \geq 0 \] 

In other words, it's telling us that the number of members in a set based on a set of binary properties X, Y and Z cannot be a negative number. This is an obvious tautology, which means that the inequality statement:

\[ N(X, ¬Y) + N(Y, ¬Z)\geq N(X, ¬Z)\]

Is also a tautology. This is simple logic applied to binary properties.
 
Now we move to the quantum world. For our grouping here, we're going to use the binary properties regarding angular momentum about a particular axis. These are measurable properties. We can see that the angular momentum about a given axis will always be clockwise or anti-clockwise (which we'll denote '¬').

Let's label our axes X, Y, and Z. We're looking now at the angular momentum of, say, an electron about the X axis, the Y axis and the Z axis. From the above, we should be able to say that our reasoning above applies to the relationships between groupings if EPR is correct and we're dealing with hidden variables. 

This is the troubling bit: When we measure the spin of an electron about these axes in the lab, they don't satisfy this inequality, and what we actually see is this:

\[ N(X, ¬Y) + N(Y, ¬Z) < N(X, ¬Z)\]

Quantum mechanics violates Bell's Inequality, and tells us that, in particular, the number of electrons with spin clockwise about X and anti-cockwise about Y plus the number of electrons with spin clockwise about Y and anti-clockwise about Z, is fewer than the number of electrons with spin clockwise about X and anti-clockwise about Z. What Bell has proved with this is that no theory that is both local and realistic can reproduce all the predictions of quantum theory, predictions that are observationally verified.

Coming back to entangled particles, for which the properties are correlated, because each of the particles is a portion of a system, as opposed to being separate, this inequality is again violated by experimental measurement.

Sorry Albert, but it looks like we're stuck with 'spooky action at a distance'.

There have been examples over the years that have cast some doubt on some of the results here, by highlighting loopholes. The last of these seems to have been closed last year with an exciting study by Hensen et al [1], which I'll leave to the interested reader.

I look forward to any comments and feedback. Thanks for reading.

 
*Velocity is a vector quantity, which means that it has information concerning magnitude and direction (e.g. 40 kph in a North-Easterly direction). This is distinct from a scalar quantity, which only contains magnitude information (e.g. 40 kph). Thus, turning from 40 kph in a North-Easterly direction to 40 kph due North is an acceleration.

[1] Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres Nature 526, 682–686