Acoustic ‘beats’ from mismatched frequencies

Here’s a piece I wrote explaining the mathematics behind the peculiar phenomenon of acoustic ‘beats’.

https://www.physicsforums.com/insights/acoustic-beats-mismatched-musical-frequencies/

It’s a bit maths-y. But for those that don’t love maths quite as much as I do, it also has some interesting graphics and a few rather strange sound clips.

Andrew Kirk

Bondi Junction, August 2016

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More weird and wonderful observations about the expansion of the universe

In high school physics we are taught the Galilean principle of relativity (named after Galileo Galilei, not the biblical land of Galilee). That is that there is no such thing as absolute velocity. All velocity is relative. Consequently we cannot ‘feel’ velocity. If we are inside a closed compartment with no windows, we cannot tell whether it is ‘moving’. We might think we can, and compare the feeling of being in a moving railway carriage or aeroplane to being in that same vehicle when it is ‘stopped’. However, as our high school science teachers explained to us, the feeling of motion we experience there is not the general forward (horizontal) motion but the minor vertical or horizontal accelerations that happen when the vehicle goes over bumps or gaps in the rails (train) or air (plane). If the rails or air were perfectly smooth, and we were travelling straight – not turning – we would not be able to tell whether we were ‘moving’.

If our science teachers were especially good, they might have summed this up for us as ‘you can feel acceleration, but you can’t feel motion’.

In my earlier essay ‘Expansion of the Universe’ I pointed out that it is quite possible for two items to be moving apart at a rate faster than the speed of light, and that that does not contradict Einstein’s theory of relativity, as long as those two items are sufficiently far apart. The ‘acceleration’ that arises from the cosmological constant (or ‘dark energy’, if you prefer spooky mystical terms) is so mild that we cannot not feel it.

It has since occurred to me though, that, because that rate of relative acceleration increases with the distance between two objects, there must be objects immensely far from us, relative to which our acceleration will be very large – greater than the several ‘g’s needed to make a person black out. This observation follows from Hubble’s equation.

Those that don’t like maths had better close their eyes for a few seconds. Don’t worry, it won’t last long.

Hubble’s equation tells us that dr/dt = H r, where r is the ‘comoving’ distance from us to a distant galaxy, dr/dt is the rate of change of that distance (using ‘cosmological time’ for t) and H is the Hubble parameter (2.2 x 10-21 s-1). The solution to this differential equation is r = r0 eHt. where r0 is the current distance from us to the distant galaxy. Hence the relative acceleration between us and the distant galaxy is d2r/dt2 = r0H2eHt = H2r.

Almost any human will lose consciousness under an acceleration (‘g force’) of 10 ‘g’s, (‘g’ is the acceleration due to gravity at the Earth’s surface). That is 98ms-2. To find the distance away of objects that are accelerating relative to us at that rate we set 98ms-2 = H2r. So r = 98ms-2/H2 = 98ms-2/ (2.2 x 10-21 s-1)2 = 2 x 1043m = 2 x 1018 billion light years. That is almost 5 x 1016 (50 quadrillion) times the radius of the observable universe.

You can open your eyes again now.

We have just worked out that the relative acceleration between ourselves and galaxies that are 2 x 1018 billion light years away is around 10 ‘g’s. One or both of us and the distant galaxy must be accelerating away from the other, and the sum of those two accelerations must be 10g. The Cosmological Principle tells us that the universe is essentially the same all over (‘homogeneity’ and ‘isotropy’), so we should have the same acceleration as that galaxy, which means we should each be accelerating at 5g, in opposite directions. That would be enough to pull the Earth out of its orbit around the Sun, and to pull us off the Earth’s surface. Also, we should be able to feel it – in fact it should make us feel very sick, like a super-extreme roller coaster. Remember that science teacher that told us you can’t feel motion but you can feel acceleration? So why can’t we feel it?

The answer is actually at the heart of Einstein’s General Theory of Relativity and, fortunately, it’s not highly technical. What we can feel is acceleration, but in this context it is an oversimplification to say that acceleration is the rate of change of our speed. In General Relativity the acceleration of a body means the rate of divergence of the body’s path through spacetime (its ‘worldline’, in relativistic jargon), from the tangent geodesic. Let’s unpack this. A geodesic is the equivalent of a straight line on a curvy surface. For instance the ‘great circles’ that are lines of longitude on the Earth are geodesics, and so is the Equator (but not lines of latitude like the two tropics or the Arctic and Antarctic circles). At any point in your life, the tangent geodesic to your body is the path it would travel through four-dimensional spacetime if there were no non-gravitational forces acting on it. Fortunately for you, there are non-gravitational forces acting on you (principally from the ground, pushing up on the soles of your feet), otherwise you would follow your tangent geodesic which is a headlong plummet towards the centre of the Earth. The floor is constantly pushing you away from your tangent geodesic in four dimensions, just as a turned steering wheel pushes a car away from the line of the beam of its headlights (which is the tangent in three-dimensional space to the car’s curved 3D path).

The acceleration you feel, which is just your feeling of weight, arises because your body, by not plummeting, is constantly diverging from its tangent geodesic. You might think you don’t feel any acceleration when you’re just standing still on the floor, or lying in bed, but that’s because you have felt it ever since you were first snuggled up in your mother’s womb, so you don’t notice it. What you do notice is if the acceleration suddenly stops. That’s what that ‘stomach in your mouth’ feeling is that you get when you jump off a pier into the sea and are briefly ‘weightless’. For a second or two you are following your tangent geodesic, and it feels odd because you’re not used to it.

The science teacher’s advice, that you can feel acceleration, needs to be refined to specify that you only feel ‘General Relativistic acceleration’, which is not rate of change of speed (as it is described in Year 11 Physics) but divergence from our tangent geodesic*. The two are only the same in non-accelerated (‘inertial’) frames of reference. In high school Physics we usually only deal with inertial frames of reference so the distinction doesn’t matter. But it does when we are considering our position relative to very distant galaxies.

Now how does this explain our inability to feel our acceleration relative to that distant galaxy? The answer is that the Earth is following its tangent geodesic which, as well as orbiting the Sun, is heading increasingly rapidly away from the distant galaxy. So we can’t feel that change in motion because it is not diverging from the tangent geodesic. The only thing we can feel is the divergence that arises from the floor pushing us up in opposition to the Earth’s gravity.

In summary, the only ‘acceleration’ you can feel is that which pushes you away from your tangent geodesic. It doesn’t matter how wildly that geodesic may be curving through spacetime – you will never be able to feel it.

* Note: Technically speaking, the strength of acceleration that you feel – the ’g force’ – is the magnitude of the Covariant Derivative of your four-velocity in its current direction.

Cosmic Inflation

A much more extreme example of this is in the ‘inflationary era’ of the universe, which is believed to have occupied the first unimaginably tiny fraction of a second after the Big Bang (10-32 seconds). In that sliver of time every part of the universe is thought to have expanded by a linear factor of around 1026. Imagine if you had just put your pencil down for an instant, then looked to find it and noticed that it was now a billion light years away. Wouldn’t that be annoying? Fortunately, there were no humans around at the time to be annoyed by such nuisances, and not even any pencils either. I do sometimes wonder though whether a brief, local, reappearance of cosmological inflation might be the reason for the occasional mysterious disappearance of my socks.

Describing that period of inflation in terms of acceleration, the g forces are mind-boggling. If we take a brutally crude and almost certainly horribly wrong (but reasonable enough to make the point) assumption that the acceleration was constant, we can use the high school formula to calculate the average acceleration during this split second.

Mathophobes look away again briefly:

The formula is distance (r) = ½ a t2 where a is the acceleration, t is the time period and r is how far away the object is at the end of the period. This gives a = 2r / t2. If I put my pencil down one metre away and 10-32 seconds it is 1026 metres away we have a = 2 x 1026m / (10-32s)2 = 2 x 1090ms-2.

The average acceleration during the inflationary split-second was 2 x 1089 ‘g’s! That would give you a bit of a tummy ache.

Only it wouldn’t, for the same reason as why we can’t feel our acceleration relative to the distant galaxy. We would just be following our tangent geodesic, and although that geodesic was horrendously curved, we wouldn’t feel a thing, because we only feel divergence from that path.

Only it would be rather uncomfortable because our bodies take up space, and so have many different geodesics going through them. Those geodesics, although initially very close, would diverge from one another so rapidly that a moment later they would be separated by squillions of light years. So your body would be ripped apart by the expansion of space, and the molecules that were neatly collected together to make your kidney one instant would be spread over an unimaginably vast expanse of space an instant later. Fortunately, it would happen so fast that you’d never notice. Even more fortunately, we can be fairly sure that nobody reading this was born before 1879, and hence they probably weren’t around to experience such a cosmic evisceration.

Another odd thing is that none of the molecules in your kidney would break the cosmic speed limit of c (the speed of light), even though they become separated by many light years in a tiny fraction of a second. This relies on the (somewhat technical) fact that the molecules not only follow their geodesics, but that those geodesics are timelike. It is only non-timelike geodesics that are forbidden. The concept of timelike things is partially explained in my essay ‘Expansion of the Universe’ . For a full explanation, see any good relativity textbook such as Bernard Schutz’s ‘A First Course in General Relativity’.

Summary

We can only feel acceleration, not constant motion. But the acceleration we can feel is only ‘General Relativistic acceleration’ that is divergence of our path from our tangent geodesic. Acceleration that is just an increasing rate of separation between us and something else will not be felt if it is solely because of our following geodesics of spacetime. Such non-General Relativistic Acceleration exists between us and immensely distant galaxies. For sufficiently distant galaxies, that acceleration is greater than that of a ferocious roller-coaster, yet we cannot feel it at all.

In the first split second of our universe, non-General Relativistic acceleration is believed to have occurred – called ‘cosmic inflation’ – that is unimaginably huge in terms of ‘g forces’. The acceleration could not have been felt if we had been there at the time, but it would have instantly ripped apart any object that was there, because of the extreme divergence of temporarily nearby geodesics.

Andrew Kirk

Bondi Junction, Australia

September 2013


What is a cause? Trying to distil clarity from a very muddy concept

Preface

Scene 1: So! – shrieked the evil monocled Gestapo officer. Eef you do not tell me ze name off ze leader off your resistance group, I vill shoot zis prisoner. Make your choice keffully! Do you vish to be ze cause off ze death off zis poor eenocent civilian?

Fade to scene 2: And now m’lud, intoned the imposing barrister, as you have just heard, if the defendant had correctly diagnosed the plaintiff’s stomach pain as a torsion of the testicle rather than prescribing antacid tablets, the testicle could have been saved by a simple operation that would have enabled the plaintiff to live the happy, fulfilled sexual life that he so richly deserves. I ask the court to award damages of five million dollars against the defendant for causing this poor man’s loss of sexual function.

Fade to scene 3: Have you found out why my car won’t start asked Jedediah. Well, I’m not sure, mister, said the mechanic, with a sarcastic look on her face, but it might have something to do with this snake that’s gotten its tail wedged in your starter motor. Mind your hands there, it looks a bit annoyed. Well golly, said Jedediah, who’d’ve thought that a little ol’ critter like that could cause so much trouble?

Three stories, three problems, three causes. Or are they?

If our heroine refuses to name the resistance leader to the Gestapo, will she have caused the civilian’s death? Or will the Gestapo officer have caused it? Or both? Or something else?

Did the doctor really cause the loss of the plaintiff’s testicle, or was it the fact that it managed to twist so as to strangulate the blood supply, or perhaps it was the plaintiff’s genes that gave them a particular anatomy that made them vulnerable to such an occurrence? If the latter then were the plaintiff’s parents the cause of the loss, or should we perhaps blame the person that introduced the parents to one another?

And was the snake really the cause of Jedediah’s car problems, or was it that he’d parked his car in the bush while camping overnight, providing an enticing warm place for any passing snakes to nestle in the warm engine?

Defining a ’cause’

The idea of cause and effect is an ingrained part of our language. We all feel that we know what the terms mean. But do we really? The above examples show how it’s not usually possible to point to one thing and say that is the cause of this. We might feel however that, with more care and thought, we will be able to precisely describe what really caused any given event.

The amazing answer is that No, actually we can’t. There is no such thing as a single cause of an event in the way it is traditionally thought of. The purpose of this essay is to examine the idea of cause (and effect) and work out what, if any, meaning we can give to this vague and rubbery, yet ubiquitous concept.

Is a cause necessary? Is it sufficient?

A natural place to start looking for a meaning seems to be to ask whether a cause is a necessary or sufficient condition, or both, for its effect to occur.

None of the suggested causes in the preface are necessary conditions. There are plenty of other ways the civilian could have died, the testicle been lost or the car failed to start. So we can dismiss necessity as a feature of causes straight away.

What about sufficiency? Neither of the suggested causes in the first two stories in the preface are sufficient conditions. The prisoner could have refused to snitch but the Gestapo officer relented and didn’t shoot the civilian. The undiagnosed twisted testicle could have untwisted by itself, or another doctor passing five minutes after the defendant misdiagnosed it could have had a look and diagnosed it correctly. The snake is another story though. Having a snake’s tail wedged in your starter motor effectively guarantees that your car will not start. So perhaps some causes are sufficient conditions for their claimed effects. We’ll come back to that later.

Cause as a difference between alternative prior scenarios

If I go to the dentist and ask why my lower right incisor aches, she may find decay in it and say “the cause of your ache is decay in the tooth”. The decay is neither a necessary nor a sufficient condition for the ache. The ache could be psychosomatic with no decay, or there could be decay but a dead nerve, in which case I’d feel no ache.

Yet I know what she means. So what is it that I, and any other dental patient, understands from the dentist’s statement?

I think it is that the situation I am experiencing while sitting in the dentist’s chair, call it situation S1, may be compared with another situation S2, that is identical to S1 in every respect except that there is no decay in the tooth. In neither case do I suffer psychosomatic hallucinations, nor is the tooth’s nerve dead. The only physical differences between the two situations is the decay. If a message takes a nanosecond to travel along a nerve from the tooth to my brain then in the situations one nanosecond later than S1 and S2, call them S1a and S2a, S1a will have me experiencing toothache and S2a will not.

Now the dentist has not explicitly mentioned an alternative situation, but that’s because it’s implied. I naturally interpret her statement as meaning “According to my observations and the biology they taught me at dental school, the key difference, in the toothy-brainy part of your body, between you and somebody very like you that does not have a toothache is that you have decay and they do not”.

We can formalise this idea of a cause with a precise definition:

‘If:

    1. S1 and S2 are descriptions of alternative possible states of a system at time t, and
    2. the difference between S1 and S2 is C, and
    3. theory T requires that event E occurs at time t+dt if the system state at time t is S1, and
    4. theory T requires that event E does not occur at time t+dt if the system state at time t is S2,

then C is the cause of E in system state S1 with respect to system state S2, according to theory T.’

Note that lines 3 and 4 use the concept of sufficiency, raised in the previous section. S1 is sufficient reason for E to occur and S2 is sufficient reason for E to not occur.

People rarely, if ever, refer to two alternative system states when saying something is a cause. Usually, as with the dentist, the natural choice for S2 is evident and need not be stated. But it is useful to remember that there is nearly always an implied comparison state S2 when we talk about causes. Whenever controversial or confusing claims are made about causality, as happens so often in litigation, politics and philosophy in particular, it can help enormously if we analyse the claim by trying to identify what the implied comparison state is.

Do we really need to say ‘according to theory T’?

The appendage to the definition – ‘according to theory T’ – might seem superfluous and annoying to some. After all, people don’t usually quote a theory when they say that pricking the balloon with a needle caused it to burst. Nevertheless, just like the comparison state, a theory is always there. In the case of the balloon, the theory is Physics, as taught at modern universities. Training in Physics up to third-year university would provide all the understanding needed to explain the pop of the balloon.

Looking at the dentist example, we see that our interpretation of her diagnosis does include reference to a theory, viz: ‘according to … the biology they taught me at dental school’.

Now we might imagine that both Physics and Biology are just parts of a Grand Theory of Everything, of which science has so far only discovered a portion. If that were so, then we could leave off the appendage to our description of a cause, and just imply that the theory we mean is the Grand Theory of Everything.

But although some might find the Grand Theory of Everything a nice idea, and wish there really were one out there, we have no reason to suppose there is. I discuss this further in my essay ‘Some random thoughts on whether the world is random’. The conclusion is that, unless we are prepared to regard an enormous list of everything that ever happens in the universe as a theory of everything (which most people wouldn’t) there is no way to decide what sort of a collection of statements could qualify as such a theory. Is there a word limit? Does the collection have to be finite? Does it have to be expressible in English? Does it have to be comprehensible by an intelligent human?

In addition, as I argue in my essay ‘Replacing Truth with Reason’, there may not even be any ultimate description of the universe. Our scientific advances may lead to increasingly more complicated theories that, while intriguing, exciting and pragmatically useful, never converge to a final, stable, ultimate theory. Perhaps the universe is too complicated to be described by any theory.

So we will have to put up with the appendage for the time being. Devout Platonists may wish to assume that there is a Grand Theory of Everything, and omit the appendage, implying that T is that Grand Theory. But that is an act of faith that I do not feel inclined to emulate.

It does however seem reasonable to omit the appendage when conversing in the vernacular, if our implication is understood to be not that T is the Grand Theory of Everything, but that it is “Science as taught at universities, in the year in which we are speaking”. I will call this Science 2013, as that is the year in which I am writing. This ties the use of ‘cause’ to a sense of what the best scientists in the world currently understand about how the world works, and that seems to me to pretty accurately reflect how the person in the street would understand the term ‘cause’.

When discoursing philosophically though, as in this essay, it will be wise to retain the appendage specifying the reference theory, in order to be clear.

Can we define cause without a comparison state?

Some scenarios in which we might like to talk of causes do not naturally suggest comparison states. We might for instance consider the Cosmic Microwave Background Radiation (CMBR) that suffuses the sky, which is left over from the ‘last scattering surface’ of the Big Bang. We want to say that the Big Bang caused the CMBR. But we are stymied by the fact that we cannot think of an alternative situation with no CMBR. That situation would have to have no Big Bang, and hence possibly no spacetime, and hence no place in which to observe the lack of CMBR.

Here is an alternative definition of ‘cause’ that solves that problem.

‘If S is a description of a physical system at time t and theory T requires that event E occurs at time t+dt if the system was in state S at time t, then we say that S is the cause of E in system state S, according to theory T.’

In most situations this definition will be useless, because it requires a full description of the system state at the prior time. In order for E to be inevitable, that will have to be something like the location, momentum, type and spin of every particle within radius c.dt of the location of E (c is the speed of light) at time t. That is way too much information for everyday use. It’s a bit like saying ‘everything’ is the cause of E. But it may be useful to have this definition available as an alternative if we want to talk about causality in relation to situations that don’t have natural comparison scenarios.

In order to distinguish our two definitions of cause we’ll call the first one the Comparative Definition and the second one the Singular Definition. If we don’t specify, we’ll mean the Comparative definition because that’s likely to be most often the one we mean.

Looking back at the snake’s tail story, we can see that that meets the definition of a Singular cause of the engine not starting, if the tail is still wedged in the starter motor when the electric current unleashed by the ignition key hits the coils in the motor. If the time the current hits the coils is t, then we can say that the configuration of a spherical region of space with radius 10cm centred at the middle of the starter motor is the cause of the engine not commencing to fire at t+3.3×10-10 seconds, and that region includes the wedged snake’s tail.

A Singular cause is always sufficient for its effect, but the price we pay for that sufficiency is that the cause either has to be a complete description of the state of an enormous volume or, as is the case with the snake’s tail, the effect must occur a very tiny interval of time after the cause (a third of a nanosecond here).

Causes must be prior to their effects

The two definitions I have suggested require a cause to be earlier than its effect, which we call being ‘temporally prior’. Sometimes people talk of causes that are not temporally prior, so we should consider whether that can make sense. There are two common ways people do this.

‘Simultaneous’ causes

Some people give examples of what they think are physical causes that are simultaneous to their physical effects. They all turn out however, to be based on a misunderstanding of physics. There is a very simple reason why one physical event cannot cause another that happens at the same time, and that is the principle of relativity, which states that physical influences cannot travel faster than the speed of light. For event E1 at time t to affect event E2, also at time t, would require the influence of E1 to travel the distance between the two locations in no time at all, that is, at an infinite speed, which would break the speed limit and irritate the Great Cosmic Traffic Cop.

Examples offered of putative simultaneous causes are

  • a ball (cause) sitting on a pillow and causing a depression (effect), or
  • pushing one end of a lever down (cause) so the other end goes up (effect).

It is not the ball’s presence at time t that causes the depression in the pillow at time t, but the ball’s presence at earlier times. We can see this by imagining the ball suddenly magically pouffing out of existence. The pillow would not instantly regain shape. Rather it would start to spring back to its original, undepressed shape. If the ball were present on the pillow up to time t and instantly then disappeared, the shape of the pillow at time t would be exactly the same as if the ball were still there. The depression would gradually disappear as the pillow started to regain its usual shape after time t. In the real, non-Harry Potter world, change takes time.

Similarly, the footpath of a bridge does not stay up because its supporting beams are there, but because those beams were there an instant earlier.

When we push down one end of a lever, the other end does not instantly lift. Rather, a shock wave travels through the lever, deforming it in such a way that, a tiny instant of time later, the other end lifts. The shock wave travels at the speed of sound in the lever, which will be very fast indeed if it is made of a stiff substance like steel, but still much slower than light. Because the wave is so fast, we cannot perceive it without specialised equipment, so the effect seems instantaneous. If we had a fast enough camera, we might even be able to film the deformation of the lever as the shock-wave passes through. But we’d need an enormous enlargement of the frames to see the lever’s deformation in the film, because the shockwave of the initial push has probably reached the other end before the end we are pushing has moved a millimetre.

Readers who are familiar with the Quantum Mechanical phenomenon of entangled particles might hope for a loophole in the cosmic speed limit via the fact that, when one member of a pair of entangled particles is measured, the wave function collapses and the other member attains a definite value of the measured quantity.

This ‘spooky action at a distance’ as Einstein called it, does not however break the speed limit, because no physical influence is being transmitted. The wave function is simply a mathematical abstraction we use in Quantum Mechanics to make predictions and its collapse has no physical significance. In particular, there is no experiment we can do to find out whether the wave function of a particle has already collapsed. It will collapse when we make the measurement in the experiment, but that cannot tell us whether it had already collapsed before that.

So in summary, there is no escape from the cosmic speed limit, and hence there is no such thing as a simultaneous physical cause.

‘Logically prior’ causes

Another way people try to escape the need for temporal priority is to talk of a cause as something ‘non-physical’ that entails its effect via the laws of logic rather than of science. They could for instance say that the rules of arithmetic are the cause of 2+2 equalling 4, or that the fact that all men are mortal and Socrates is a man is the cause of Socrates being mortal.

This could be formalised by saying that if A→B where A and B are propositions and → denotes logical entailment (if the proposition before the arrow is true then the proposition after the arrow must be true) then A is the cause of B. Let’s call it a Logical Cause to distinguish it from the Comparative and Singular definitions of causes that we discussed above. In this context only, we will refer to causes meeting those definitions as ‘physical’ causes. Defining ‘physical’ is usually a controversial mess. But here all we mean by ‘physical cause’ is a cause that satisfies our Comparative or Singular Definitions.

There’s nothing incoherent about defining logical causes this way. No contradictions or ambiguities arise. The trouble is just that it’s a completely different use of the term cause from how it is used in relation to everyday physical things, so one cannot apply any conclusions drawn about physical causes to logical causes, or vice versa.

Further, there is already a perfectly good word in use within the field of symbolic logic for a logical cause. It’s called an antecedent. And the thing coming after the arrow is called a consequent.

So all we achieve by using ‘cause’ in this context is confusion, by applying a word that has a meaning in a different, completely unrelated field (the physical) to a concept that already has a perfectly clear label in this field.

Readers should beware of arguments that try to use logical causes. Such arguments might use words like ‘now consider causes that are logically prior rather than temporally prior to their effects’. The only reason I can think of to use the word ‘cause’ for a logical antecedent is to try to smuggle in some of the properties of physical causes and apply them to logical causes, without the validity of doing that being challenged. As logical and physical causes have no relation to one another, other than in a vague, touchy-feely sort of way, it is invalid to apply any properties of physical causes to logical causes.

Sorting out which event is the cause and which is the effect

Another problem of not requiring causes to be temporally prior is that it creates ambiguity as to which of the two events is the cause and which is the effect. In the physical case, this is clearly resolved by requiring a cause to be earlier than its effect. We lose that capacity if we don’t require temporal priority.

In the logical case, if we have A→B but not B→A then we can say, if we wish, that A is a Logical Cause and B is its logical effect. But if we have both A→B and B→A then there is no basis for saying one of A and B is the cause and the other is the effect. We will see in the next section how this can lead to grief.

Examples of the use of our definitions

Let’s try out our two definitions – Comparative Cause and Singular Cause – in a few situations where the word ‘cause’ is key to the thinking processes, to see how they fare.

Causation in philosophy

More than 2000 years ago Aristotle thought and wrote about causation, in a way that has been adopted by many philosophers since then. He listed four types of cause, of which only one, the Efficient Cause, is close to the way the term is typically used now. Unfortunately, even the notion of an Efficient Cause is bound up with Aristotle’s ideas about physics which, being pre-Newtonian, are incompatible with the way we now understand the world to work.

Nevertheless, philosophers still blithely make arguments using the word ‘cause’, only rarely pausing to consider what if anything the word actually means, and whether it really belongs in their arguments. A notable exception is Bertrand Russell in his marvellous 1912 essay ‘On the notion of cause’.

Here are a couple of examples of how ‘cause’ is used in philosophical arguments, and how we can use the considerations above to understand them better.

First Cause arguments for the existence of God

There is a very old and venerable argument that there must be a being (God) that is the cause of the universe’s existence. There are a number of versions, including a popular one that has been revived recently, based on a medieval Islamic argument from the Kalam school. All versions of the argument rely on God being a Cause for the universe. An obstacle to all these arguments is that there can be no ‘before’ the universe, as time is itself a feature of the universe, not something that applies outside it. So there cannot be a cause that temporally precedes the universe. Devotees of the First Cause argument sometimes respond that God is logically prior, rather than temporally prior to the universe. That is, God→Universe.

There are two problems with this argument.

Firstly it relies on a premise that every object of a certain type must have a cause. It tries to generate support for that premise by appealing to our experience, and all the examples used are of physical causes. Hence the premise is restricted to physical causes and tells us nothing about non-physical causes, which is what the argument wishes to argue God is. This is a smuggling attempt, of the kind discussed above.

Secondly, what the argument actually does is to reason from the existence of the universe to the existence of God. That is, Universe→God.

But now we have a situation that is logically symmetrical between God and the Universe, which a logician would denote as God↔Universe. Each implies the other, so we cannot say that one is logically prior. One might be tempted to say that there was a time, before the creation of the universe, when there was only God and no Universe, which makes God prior, and hence the cause. But that route is forbidden because it relies on the existence of time, which is part of the Universe.

So the philosopher that pursues this route is committed to saying that, if there is a God, then it is caused by the Universe as much as it causes the Universe.

Such a conclusion is likely to satisfy neither theist nor atheist, and demonstrates quite nicely the futility of trying to reason about causes that do not temporally precede their effects.

The Epiphenomenal hypothesis of consciousness

Epiphenomenalism is a hypothesis that says mental events (consciousness) are caused by physical events in the brain, but have no effects upon any physical events. In other words, brain activity causes consciousness, but consciousness does not cause any brain activity.

For this to be the case, given our definition of cause, a mental event must occur after the physical (presumably brain) event to which it relates. Hence the brain event can be a cause of the mental event, but not vice versa.

Importantly, if the mental event occurs simultaneously with the related brain event then we cannot say that either causes the other, because neither precedes the other. This is a crucial observation because sometimes people talk about Epiphenomenalism as if it is a simultaneous occurrence caused by the contemporary brain activity. However, as we have seen above, for simultaneous events there is no way to identify which is cause and which is effect. So a mind-body model that involves simultaneous processes is not Epiphenomenalism.

Causation in Science

Does all science rest on the assumption that everything has a cause? It might seem so, and this claim is often made, but it’s wrong. Science doesn’t need everything to have a cause, to be useful. Science rests on the observation that there are patterns in nature, such that systems appear to evolve in regular, repeatable ways that can be described by natural laws. If we can discover such a law, by inventing theories based on experimentation, and then testing the theory’s predictions using further experiments, then we may be able to predict future events, and shape the course of those events.

So science is best described not as a search for causes, but as a search for laws that describe how physical systems evolve.

We don’t even need to believe that everything is governed by natural laws. For instance, some interpretations of Quantum Mechanics hold that there is no law determining the precise time at which a radioactive particle will decay. The apparent absence of a cause for that particular aspect of reality does not however prevent us from making very precise predictions about the behaviour of physical systems using Quantum Mechanics.

In science we don’t need to have causes for everything, or even to believe they exist. At most we need causes for the important features of the system we are evaluating.

Causation in Physics

Light cones

An important concept in physics is that of the light cone. For a given point P in spacetime, the past light cone is the set of all spacetime points from which a particle could have travelled prior to passing through P. There is also a future light cone, which is the set of all spacetime points that can be reached by a particle that first passes through P. The particles in question may be photons, which travel at the speed of light, or slower particles with mass, like electrons or cricket balls.

Physicists talk about two spacetime points as being ‘causally connected’ if one is in the other’s past light cone. This means that the later point can be affected by something that happens at the earlier point. Events at points that are not causally connected cannot affect one another. That is, changing what happens at one point will have no impact on the other. Such points are called space-like separated points.

For point P, the future light cone marks out the limits of the points P can causally influence, and the past light cone marks out the limits of what points can causally influence P. Hence the light cones are regarding as showing the limits of causality.

This usage harmonises with both our Comparative and Singular definitions. In the Singular definition, the cause (according to Science 2013) of an event E at spacetime point P, with time coordinate t, is the state of the set R of all points in P’s past light cone that have time coordinate t-h for some positive h. In the Comparative definition, if S1 and S2 are alternative possible states of R, such that E happens at P if R has state S1 but not if R has state S2, then the difference C between S1 and S2 is the cause of E in S1 with respect to S2, according to Science 2013.

It might seem that the light cone perspective adds an additional constraint to causality above the constraint in our definitions that causes must precede effects. For not only must the cause precede the effect, but it must also lie in the effect’s past light cone.

It turns out that, because of the theory of relativity, this is not an additional constraint at all. We can only say unambiguously that C precedes E if C is in E’s past light cone, because then the time of C will be earlier than that of E in every possible reference frame. If C is in E’s future light cone we can say unambiguously that E precedes C, so C cannot be a cause of E. That much is obvious. But if C is in neither the future nor the past light cone of E, it will be later than E in some reference frames and earlier than E in others. Einstein’s theory of relativity tells us that no reference frame is any more valid than any other, so C cannot be a cause of E if there is even just one reference frame in which it occurs after E (in fact if there is one such frame then there will be infinitely many).

This last consideration tells us that, if we ever discovered particles or other influences that could travel faster than light, it would destroy our notion of causality entirely. Because then we would have pairs of events that we thought were cause and effect, for instance the beginning and end of a path followed by one of these particles, but for which in some perfectly valid reference frames the effect preceded the cause. We would have to either jettison the notion of causality entirely, or develop a completely new one, that may only have very slight similarities to the existing one.

It is fortunate for us then that the superluminal neutrino speeds observed in experiments in 2011-12 turned out to be experimental errors.

Quantum indeterminacy

In both our definitions of Cause we say theory T ‘requires that’ the effect occurs after the cause. However quantum mechanics tells us that nothing is certain to happen. Things we think of as inevitable are really only very, very likely. How then can we meaningfully talk of an effect being required to occur after its cause?

One solution is to replace statements of certainty by probability statements. We could replace ‘theory T requires that’ by ‘under theory T there is a greater than 99.9% probability that’. Here T is of course Quantum Mechanics. If we make this substitution in the Comparative Definition (twice, for the two instances of ‘theory T requires’) and the Singular Definition (once) then these definitions are ship-shape and ready to be used in the Quantum Mechanical world.

We might wish to go further and call C a cause if the probability of E occurring after S1 is lower than 99.9%, say 50%, and the probability of E not occurring after S2 is still 99.9%. In that case C is a sort of enabling condition for E to occur, but it does not guarantee it. If we wanted to go down that route it would be better to give this type of relationship a slightly different name like ‘probabilistic cause’, to avoid confusion with the cases where C makes E almost certain to occur.

Correlation does not imply causation

A famous dictum that is often used in both science and social studies is ‘correlation does not imply causation’. Let’s put our Comparative Definition to the test to see if it supports this uncontroversial dictum. But because medical and social sciences are quite complex, we’ll use an example involving something simple instead – bowling alleys.

Imagine that a bowling alley has an easily depressed light switch placed in the middle of the alley, 20cm away from the central lead skittle. When depressed, the switch closes an electric circuit that illuminates a light above the skittles. After watching a few matches we notice that the light goes on for a fraction of a second and then off, immediately prior to every strike (knocking down all ten skittles). We have observed a correlation between illumination and strikes, and we wonder whether the light causes a strike.

First we compare two situations, describing the region R around the bowling alley, at the time a ball that has been bowled passes the switch. The situations S1 and S2 are identical except that in S1 the ball is on the switch and illuminating the light, while in S2 the ball is to the left of the switch, too far left for a strike to occur, and the light is not illuminated. The region R is large enough that nothing that is outside R when the ball passes the switch can change whether a strike occurs.

In S1, Science 2013 requires that a strike will shortly occur and in S2 it requires that a strike will not occur. So our definition of cause is satisfied. We can say that the difference between S1 and S2 caused the strike after S1. But what is the cause we have identified? It is everything in S1 that is different from S2.

That includes the light being on but it also includes the ball being in the middle of the lane. We could if we wish say that B caused the strike where B is ‘the light being on and the ball being in the middle of the lane’. The latter is consistent with what a lay person would think of as being the cause, so that’s a good start. It is reasonable to describe B as the cause. The bit about the light seems superfluous though. Can we get rid of it?

Yes we can, as follows. We add a new situation, S3, which is the same as S2 except that someone stands on the lane, avoiding the ball, and briefly depresses the light switch as the ball passes, if the ball does not itself roll over the switch. Now let’s compare S2 and S3. In both cases there is no strike. They are identical except for the man standing on the lane and the light being on. So it appears that the light being on is not a cause of a strike. The light illumination is correlated with, but not causative of, strikes.

This confirms that the Comparative Definition can, at least in this case, reproduce results that accord with our intuitions about causation.

Conclusion

We have developed a definition of cause – the Comparative Definition – that captures the everyday meaning of the term while removing ambiguity. The price of the additional accuracy was having to specify a comparison scenario S2 and a reference theory T.

For cases where a comparison scenario is not readily imaginable, we have an alternative definition – the Singular Definition – that still captures the commonly understood meaning. The price of this additional power is having to specify the prior scenario – the ‘cause’ – either over an enormous volume of space or a tiny period of time prior to the effect.

We have seen that an essential feature of any useful, unambiguous notion of cause is that it requires causes to precede effects in time. We observe that invocation of simultaneous causes or logical causes is usually a symptom of a flawed argument.

We have identified a way to generalise the notion of cause to handle the uncertainty that comes from Quantum Mechanics, by including probabilities in the description of a cause.

We have observed how these definitions of cause can be used in practice in a variety of fields of inquiry.

Finally, if we can take any ‘moral’ from this rather prolonged meditation, it is that in any argument that relies on notions of cause we should examine closely how the term ‘cause’ is used and what properties are ascribed to it in the argument. If this is not clearly set out, the argument may well have hidden flaws or, in some cases, be incoherent, no matter how plausible it may sound.

Andrew Kirk. Bondi Junction, 8 June 2013


Expansion of the universe

I wonder if I am the only person who has found the various explanations available in the public domain, of the expansion of the universe, confusing. For about two years after learning the essentials of general relativity I struggled with the apparent contradictions in the idea of cosmological expansion. I asked several questions on internet forums but never got an answer that helped me resolve the confusion. That was probably because I was asking the wrong questions, but it is so hard to know what are the right questions to ask.

I am fortunate to have finally been able to discover the right questions and, together with answers to those, and some useful papers by cosmologists, to piece together a coherent understanding of what is actually going on. This essay is an attempt to explain that for anybody else who finds the concepts puzzling.

I’ll start by describing why the concept of an expanding universe is confusing. The usual explanation involves an analogy with a balloon that is being inflated, on which dots have been marked with a felt pen. As the balloon inflates, the dots – which correspond to galaxies – become further apart. There are three key problems with this analogy, and with the idea of expansion in general:

  1. The balloon analogy fails because it seems to require the existence of a physical object that is space, corresponding to the rubber of the balloon. But the Michelson Morley experiments showed that there is no ‘absolute space’, which was referred to at the time as the ‘luminiferous ether’. The balloon analogy implies the existence of a ‘privileged reference frame’, and they are not supposed to exist.
  2. We are told that faraway galaxies are receding from us faster than the speed of light. This appears to contradict the rule that nothing can travel faster than light.
  3. If the universe’s expansion is accelerating then that seems to imply that energy is being created, as the kinetic energy of the galaxies will be ever increasing, without any compensatory reduction in potential energy.

The Cosmological Principle – a Privileged Reference Frame

To start on trying to resolve these objections, we need to first state the fundamental principle on which this sort of analysis is based – the Cosmological Principle. This principle effectively says that there exists a system of coordinates for spacetime under which a snapshot of the universe at any ‘cosmic’ time coordinate is isotropic and homogeneous at the large scale.

A system of coordinates (also known as a ‘coordinate system‘ or ‘reference frame‘) is a scheme that assigns a unique set of four numbers to every point in spacetime. This enables any point in spacetime to be referred to by those numbers – its ‘coordinates’. One of those coordinates is a time coordinate and the other three are spatial – denoting a location in space.

Homogeneous means it is the same everywhere. This rules out possibilities such as there being a ‘centre of the universe’ where stars occur most frequently, with the distribution of stars getting ever sparser as you travel further from that centre.

Isotropic means it is the same in every direction. So an observer that is stationary with respect to this coordinate system would not for instance observe stars on her left moving towards her on average and stars on her right moving away from her.

At the large scale means we ignore local variations. Of course stars are more frequent within a galaxy than outside one, but once we zoom our perspective out enough to incorporate many clusters of galaxies, the density of matter should appear roughly constant. Similarly, we may have more objects within our galaxy moving towards us from the left than from the right – meaning local isotropy does not hold – but if we zoom out enough and take enough distant galaxies into our view, their motion relative to us will be the same in all directions.

The homogeneity assumption is easy enough to grasp. It simply says there is nothing special about our part of the universe, that what we can see is a fair sample of the whole thing.

The isotropy assumption is the really powerful one, because it establishes the privileged reference frame that we can associate with the rubber in the balloon analogy. Given any point P in spacetime, we can define a location in space as being the unique worldline through P for which the universe appears isotropic from every point in that worldline. In the analogy, that means that the worldline follows the path of a dot marked on the balloon’s surface, as it is being blown up.

By the way, an isotropic universe must be homogeneous, but the reverse is not true.  A homogeneous but non-isotropic universe could be one in which the speed of light is different depending on which direction the light is travelling.

We can now satisfy the first objection. There is indeed a privileged reference frame, being the one that generates the required conditions of homogeneity and isotropy. It does not require the existence of a substance like the luminiferous ether, whose existence was disproven by the Michelson Morley experiments. All it requires is the existence of matter throughout the universe, and the application of the Cosmological Principle. The existence of this frame does not contradict the principles of Galilean and Special Relativity. Those principles state that there is no privileged ‘inertial reference frame’, which means a frame that is not accelerated or subject to gravitational forces. The frame identified by the Cosmological Principle is not inertial and hence does not contradict those theories. In one sense it is similar to the ‘laboratory frame’ on Earth, which is the frame in which the laboratory in which experiments are being conducted is stationary. That frame is privileged in a sense, but like the cosmological frame it is not inertial, because of the effect of the Earth’s gravity.

The spatial coordinates of a point that is stationary with respect to the cosmic reference frame are frequently called ‘co-moving coordinates’. In the balloon analogy this means that a point with those coordinates is moving in the same way as the rubber, i.e. it is stationary with respect to the rubber.

Given this special reference frame, we can now make sense of another commonly used aspect of the balloon analogy, that of ants crawling on the balloon. In the balloon analogy, there is a clear difference between the relative motion of dots marked on the balloon’s surface and the relative motion of ants crawling on that surface. In the former case, the motion is caused solely by the expansion, deflation or other deformation of the balloon. In the latter, the motion is a combination of that with the motion of the ant relative to the rubber.

We can use the same concept in our universe model. The coordinates of a stationary section of the ‘rubber’ are determined by application of the cosmological principle. Objects that are stationary in those coordinates may still be moving relative to one another due to the expansion of the universe. On top of that, an object may be moving locally, relative to that coordinate system. For instance, the Earth is orbiting the sun, which involves constant changes of direction, incorporating a full 180 degree change of direction of motion every six months. This motion is relative to the cosmic coordinates, and we call it ‘peculiar motion’. It is analogous to an ant walking around in circles on the rubber of the balloon.

The Andromeda galaxy is moving towards ours under gravitational attraction, and they seem destined to collide. Those relative motions are also peculiar motion, like two ants rushing towards one another along the surface of an expanding balloon.

One final note on this special cosmic reference frame. The standard way to determine it is by reference to the Cosmic Microwave Background Radiation (CMBR), which is the radiation left over from the Big Bang, that fills the sky in every direction. We can identify the co-moving coordinates of our location by identifying the velocity we would need to have for the wavelength of the CMBR to be the same in every direction. An observer that is not ‘co-moving’ with the cosmic reference frame will see longer wavelengths in one direction than in its opposite, because of the Doppler effect.

But how can galaxies travel faster than light?

The resolution to the second objection comes from the realisation that the prohibition on faster than light (‘superluminal’) travel, precisely stated, is not the same as how it is often popularly described. The popular description is either:

  1. an object cannot travel faster than light, or
  2. two objects cannot have a relative speed that exceeds the speed of light

Neither of these is strictly correct. They should be really stated as:

1a. There is no inertial reference frame in which an object’s velocity is faster than light, and

2a. Two objects within the same inertial reference frame cannot have a relative speed that exceeds the speed of light

We can immediately see that 2a is not breached by the motion of distant galaxies, because there is no inertial reference frame that contains both us and such a distant galaxy. A reference frame is inertial if spacetime is very close to flat within that frame, and there is too much spacetime curvature between us and a distant galaxy for any reference frame containing both to be flat. This can be understood by comparison to the Earth. It is reasonable to assume the Earth is flat in measuring travelling distances between my house and the local shops. However, if I am planning a trip between from my house in Sydney to London, I have to take the curvature of the Earth into account.

1a also is not breached by the distant galaxy, because it is not travelling faster than light in any inertial frame containing it. Nor is it travelling faster than light relative to any object close enough to it to be in the same inertial frame.

This may seem all a little unsatisfactory, as it leaves a big grey area in between the local shops and the distant galaxy, in which we are uncertain how far we can extend an inertial reference frame. Fortunately, we can resolve this by expressing the prohibition on superluminal travel in a more precise way, as follows:

  1. No object can have a spacelike four-velocity.

A four-velocity is a vector that can be used to represent an object’s motion, which is independent of any reference frame (‘co-ordinate independent’). Like all vectors, it has a magnitude (‘size’) and a direction, but the direction is in spacetime, not space. In any given reference frame, a four-velocity can be denoted by four numbers, called components, of which one will be a ‘time’ component and the other three will be ‘spatial’ components. The values of the four components will differ between reference frames, but they will all refer to the same physical phenomenon, whose magnitude and direction does not differ between reference frames. There is a mathematical formula, involving something called the metric tensor, for determining the magnitude of any vector. In relativity, magnitudes can be positive, negative or zero. A vector with negative magnitude can be the velocity of an object with mass, and such vectors are called ‘timelike‘. Light rays, which have no mass, must have velocity vectors with zero magnitudes, and those vectors are called ‘lightlike‘*. Vectors with positive magnitudes are called ‘spacelike‘.

The law 3 is the most general and precise statement of the prohibition on superluminal travel. Prohibitions 1a and 2a are consequences of this rule and, given a local, inertial reference frame, prohibitions 1 and 2 follow.

In an expanding universe, no object, including no galaxy, can have a spacelike velocity, so the prohibition is respected. It will be the case that the distance between two galaxies far away from one another is increasing at more than 3 x 108 ms-1, but that does not breach the prohibition.

One last point on this objection. When we refer to distance in that last paragraph, we mean what is called the ‘proper distance’. That means the shortest distance between the two galaxies in the snapshot of the universe taken at an instant of cosmic time. In an expanding universe that shortest distance will be increasing with cosmic time.

Does the expansion violate conservation of energy?

If galaxies are rushing apart at ever increasing rates that appears to increase the net energy of the universe on two counts. Firstly, their kinetic energy is increasing with their increasing relative velocities. Secondly, their gravitational potential energy is also increasing as they get farther apart. This appears to violate the principle of conservation of energy.

The answer to this objection is that conservation of energy only applies locally, within an inertial reference frame. There is no coordinate-dependent global equivalent to that local principle. In fact there is not even any global, coordinate-independent definition of total energy. There are approaches using coordinate-dependent pseudo-tensors, but these are controversial, and seem unsatisfying given their coordinate-dependence.

Even so, it appears that these pseudo-tensor approaches can be used to derive a conclusion that the net energy of the universe is zero, and will remain zero. Hence, under such an analysis, there does not appear to be any violation of conservation of energy.

* Note: If you are familiar with vectors in two and three-dimensional Euclidean space, you might assume that any vector with zero magnitude must be a trivial vector whose components are all zero in any reference frame, and hence which has no direction. While that conclusion is true for Euclidean space, it is not true for spacetime, which is non-Euclidean. Lightlike vectors have zero magnitudes but they also have a well-defined direction. They will have non-zero components, which cancel one another out when we calculate the magnitude.


Some random thoughts on whether the world is random

In discussions about free will, consciousness or interpretations of quantum mechanics people talk a great deal about whether the world is deterministic, random or something else. Well I don’t know what the ‘something else’ bit could be but I’m also starting to wonder whether the idea of randomness makes any sense.

The general idea of the distinction between a deterministic and a random universe seems to be that, in the former, events are somehow ‘fixed’ before they occur, whereas in the latter they are not – there are multiple different possible events. That sounds clear enough if you don’t think too hard about it, but if we do then we come up against the question of what do we mean by fixed? Fixed how, and by whom?

The apparent answer is that they are fixed by the ‘laws of nature’. But what are the laws of nature? Isn’t this some fairly heavy duty reifying to posit that there exist some actual laws, perhaps dwelling in some Platonic kingdom of Forms or written on magical stone tablets? Sure we have useful laws like Schrodinger’s and Einstein’s equations, but all we know for sure is that these are ideas we use to make sense of what we see and make predictions about the future. Whether they have some mysterious metaphysical existence independent of human minds is an entirely separate matter. It seems quite unlikely to me, given that every century or so we have to tweak the laws we use when further experiments and theorising show they are not completely accurate.

A Platonist might argue that there really are laws of nature out there, which determine how the universe behaves. OK if that’s the case then what about the law that is simply a description of where every single particle in the universe will be at every instant in the history of time? This is essentially a very, very long shopping list, but it also happens to perfectly describe the behaviour of the universe. Let’s call it the M-law, as it is the ‘mother of all laws’. Is that a law of nature? If it is then the universe cannot be random because everything that ever happens is described by the M-law.

So what are our choices about universe types? If we deny that laws of nature have any existence independent of human brains then everything in the universe is random because there is nothing that fixes it. If we assert the separate existence of laws of nature then nothing is random because it is all predicted by the M-law. Or, we could try and be picky and assert the existence of laws but only count them if they meet certain criteria. But what would such criteria be?

One option is to say that a law only counts if it can be known prior to the event we want to use it to predict. That would certainly disqualify the M-law. The trouble is, it would also disqualify everything else, because we cannot prove any of the laws. All we can do is build up supportive evidence for them, and there is never enough to be certain.

Sure, says the law-enthusiast, you can’t ever be completely certain, but in practice, if we are 99% certain of something, we would consider that good enough. Alright then. That does seem a bit arbitrary, but let’s go with it and see where it leads. What can we say about the motions of the planets prior to the discoveries of Kepler, Galileo and Newton? Back then we knew nothing of the ‘laws’ we now use to describe those motions, so under this criterion those laws didn’t apply, which apparently makes the motions of planets in 1347 CE random. Is that what we want?

‘Ah yes’ replies the enthusiast, ‘but you are limiting yourself to what we managed to work out based on our imperfect interpretation of the available information and our limited ability to make observations. The validity of a law should be based on all the information that was available up to that time, to an omniscient – but not future-seeing – observer, who was able to develop the best possible theories with the available information.’

Well I have to say this is getting even weirder and more implausible! We now have an ideal scientist-observer that is our yardstick for what constitutes a law of nature. If we go with that, and accept some arbitrary threshold of confidence – say 99% – on the validity of a law (leaving aside the very difficult question of how we would try to implement that threshold and whether it would be possible to validly calculate probabilities against it) then maybe we have arrived at a definition that could be pressed into use for laws of nature while excluding the M-law.

But – haven’t we ended up with a definition of randomness that is entirely epistemological? We have effectively defined a random event as one for which our ideal observer could not have 99% confidence beforehand of what the outcome would be. And the trouble with that is that we can no longer make the distinction that metaphysicists like to make between epistemological uncertainty over deterministic but chaotic events such as a coin toss, and ‘genuine’ randomness such as the decay of a radioactive isotope. With our new definition both of these types of uncertainty are ‘merely’ epistemological, and there is no such thing as metaphysical or ontological randomness. If we take this path we have to conclude that all randomness is epistemological, and there is not distinction from ‘metaphysical randomness’.

One last point to wrap up with. I went searching for a mathematical definition of randomness and came up with a complete blank. There are definitions of random variable, random (stochastic) process, probability space and various other related objects. But none of them have anything in them that captures the idea of ‘metaphysical undeterminedness’ that lurks under the popular conception of randomness. In fact, rather oddly, of the various interpretations of quantum physics, the only one that has close parallels to any of those mathematical objects in the field of probability theory is the ‘many worlds interpretation’, which looks very like the peculiar object that is a ‘stochastic process in continuous time’. That is ironic, as the many-worlds interpretation is regarded as a ‘deterministic’ interpretation, standing in contrast to the most popular ‘Copenhagen interpretation’ which is regarded as indeterministic, ie random.

Andrew Kirk
Bondi Junction
July 2012

Hilbert’s Hotel

Theologian William Lane Craig, in his resuscitation of the Kalam Cosmological Argument, relies on a proposition that ‘an actual infinite cannot exist’. His primary argument in favour of this proposition is a reference to Hilbert’s Hotel and the apparent paradoxes it creates.
Hilbert’s hotel is a thought experiment devised by the early 20th century German mathematician David Hilbert. It is a hotel with an infinite number of rooms, all of which are full. A new guest arrives wanting to stay and is told all the rooms are full. But the hotel manager makes room for the new guest by moving the guest in room N to room N+1, for N=1 to infinity, and putting the new guest in room 1. Craig adduces the apparent absurdity of this as a compelling reason why an actual infinite, by which we presume he means an infinite set of objects each comprised of a finite, nonzero amount of matter, cannot exist.
But why is the outcome absurd? It is unusual, but there are many things in the universe that are strange and unusual. Is the sticking point that an empty room appears to have been created where none previously existed? If so then that problem is resolvable, by considering the process of transition to the state in which all the guests, including the newcomer, are accommodated. To do so we need to consider the process by which the hotel manager effects the transition, bearing in mind that we are dealing with a hypothetical physical situation here, so the laws of physics must be obeyed.
One possible such process is as follows: the manager writes an order, addressed to all guests, stating that, on receipt of the written order, each guest should:

  1. pack up and move out of their room
  2. go to the room numbered one greater than theirs and deliver the letter to the guest in that room
  3. when the guest in that next room has moved out, move into that room.

In this scenario, there will always be one guest without a room, as they wait for the next guest to vacate. This situation persists forever because the manager’s message only proceeds at a finite – indeed a very slow – pace. So it would appear that no extra room has been created: we had one person without a room as soon as the new guest arrived, and we still do, in perpetuity. It’s just that the person without a room changes every few minutes.
OK then, but what if the manager asks all guests to move at the same time – does that remove the need for all this infinite waiting outside rooms? Maybe. Let’s see.
How can the manager ask all the guests to move at the same time, bearing in mind that this is a physical hotel in a physical world, and hence must obey the laws of physics? Well, he could transmit his order by email so that it flashes up on a large electronic screen prominently situated in each hotel room. Let us assume the message is transmitted at the speed of light – physics dictates that it cannot travel any faster – from the manager’s office adjacent to room 1. The message says ‘To all guests: on receipt of this message please immediately pack up and move to the room with number one greater than yours. The occupant of that room has been instructed to move to another room. Thank you for your cooperation. By order of the manager.’ Seeing the message, all the guests start packing up and moving. On arriving at the next room, they have to wait while it is vacated. The waiting time, including the time taken for the guest to travel form the door of her room to the door of the next room, will be a minimum of d/c where d is the distance between rooms and c is the speed of light. It will be longer if the guest in the next room is out, or if they weren’t paying attention to the screen and so did not see the message at the earliest possible moment. If the guest’s walking speed is v (which must be less than c) then the guest will spend time of at least d/v outside a room, longer if they have to wait to be admitted to the next room.
As in the previous scenario, the room shuffle will never be completed, as there will always be an infinite number of rooms at which the manager’s message – travelling at the fast but finite speed c – has not yet arrived. At any point in time there will be a very large number of people outside their rooms as they walk from one room to the next. As the typical time taken to move from one room to the next is d/v and the delay from one room receiving the message to the next room’s receiving it is d/c, there will typically be (d/v)/(d/c) = c/v people in the corridor at any time after the first few minutes. Based on a typical walking pace of 1m/s this means there will always be about 300,000,000 people in the corridor. Clearly this situation is much worse than with the hand-written message.
In fact, however you try to arrange the transition, it will always involve at least one person being without a room for an infinite amount of time.
Although it is possible for all the existing guests plus the newcomer to be accommodated in the hotel without having to create new rooms, the transition from the existing state to a new state where all are accommodated involves an infinite amount of homelessness. This seems unsurprising, given the hotel was full to start with. So, when we take account of the process of transition, there are no miraculous or absurd consequences that cause us to conclude that an actual infinity cannot exist.
Hilbert’s thought experiment can be extended to allow an infinite number of new guests to arrive, or even an infinite number of coaches each carrying an infinite number of new guests. As with the single new guest, these new guests can be accommodated in the existing set of rooms together with the existing guests. But, as in the above scenario, the transition will still involve an infinite amount of homelessness.
A further twist on the hotel ‘paradox’ is the situation where a guest leaves. In this case all the guests move down a room, thereby moving towards a state in which all the rooms are occupied again, despite the departure. This version is easily understood by inverting the logic above and thinking of a room without a guest in the same way as we previously thought about a guest without a room. Hence the transition will involve one or more rooms being empty for an infinite amount of time.
Craig’s assertion that an ‘actual infinity’ is impossible relies on being able to deduce an absurdity from the existence of such an actual infinity. That absurdity must be something other than our scepticism that a hotel with an infinite number of rooms could exist, otherwise the argument becomes circular: ‘an actual infinity cannot exist because I consider the existence of an actual infinity to be absurd.’ If, when the new guest arrived, the manager were able to wave a wand and make all the guests instantaneously and simultaneously teleport, with all their luggage, into the next room, then we would have an apparent absurdity via the immediate accommodation of the newcomer despite all the rooms being full (that is the absurdity adduced by Craig as proof that an actual infinity cannot exist). Such an action can be contemplated in mathematics, but Craig has no objection to mathematical infinities (‘potential infinities’). It is actual, physical infinities that he claims are impossible. But by considering the problem of transition, which only exists for physical infinities, we see that it is precisely the constraints imposed by physics that prevent the absurdity from arising.
One might object that this is a nitpicking point about transition and that, after the transition is complete – i.e. after an infinite period of time has passed – everyone including the newcomer will be accommodated and we will have a physical absurdity. This objection is easily dismissed on the grounds that:

  1. we will never get to the point when an infinite amount of time has passed
  2. after a finite period of time all the guests will have died (or disintegrated by slow radioactive decay even if the guests are androids and hence not vulnerable to death); and
  3. after an infinite period of time, nothing should surprise us – think monkeys typing Shakespeare

The conclusion is that, whatever arguments might be made against the possibility of an actual infinite, Hilbert’s Hotel cannot be one of them.


Finely Tuned Physical Constants

A popular argument used these days by those who seek to prove the existence of God is that a handful of key numbers that calibrate the laws of physics appear to be very ‘finely tuned’, as if a designer had specifically chosen the values that they have. ‘Finely tuned’ here means that if the numbers were significantly different, the universe would have been too chaotic for complex structures, intelligent life in particular, to evolve. The suggestion is that this is evidence that the numbers were chosen by some intelligent designer, ie God.

This argument may carry weight with some and if it makes them more secure and happy in their theistic beliefs I wish them much joy from it, as long as they don’t attempt to impose their beliefs on anybody else. But in my mind, far from providing the knockout blow some believe it possesses, it gains no traction at all. Such an observation is principally an observation of how one feels. Sometimes a new idea comes along that makes us think “oh… there could be something in that”, which leads to a questioning of our existing ideas and in some cases a revision of our beliefs. It’s the feeling that starts the process, and the rational inquiry occasioned by the feeling that completes it. But in this case I felt nothing at all, not even surprise about the ‘fine tuning’. Why?

I decided to think about this, given that some people find these constants such a compelling argument, and after a period of reflection, concluded that the ‘fine tuning’ just didn’t seem remarkable to me at all, for three principal reasons, which I discuss below.

Multiple Universes

The first reason is the possibility of multiple of universes. Perhaps there are a very large number, maybe even infinite number, of universes, each of which has different values of the physical constants. Only a very small proportion of these would be able to support intelligent life, and we find ourselves in such a universe because we could not have been in any other.

To some people, the existence of an enormous number of universes seems highly implausible and the postulation of a divine creator seems more plausible. Some assert that Occams Razor favours an explanation that the universe was designed by a god over one with no god but billions of universes. This is entirely a matter of feeling though, not logic. I cannot rebut their assertion that to them it seems implausible that there should be an enormous number of universes. However, to me it seems entirely plausible. In some of my moods, it seems inevitable, almost obvious. At other times it seems less likely. Why should there be only one universe? There’s no answer to that question. It’s like asking “what’s the best number: one, a billion or infinity?”.

The Intelligent Designer hypothesis that is proposed as an alternative to multiple universes, seems more plausible to some, but far less plausible to me. Neither hypothesis can be defended or attacked on logical grounds, as we are operating in the territory of fundamental axioms (what seems plausible or self-evident). It’s just a question of which you prefer.

From a mathematical perspective (I always have to have one of those), if you view the number of universes as a random variable from some a priori Poisson distribution then the argument just becomes one about what you consider to be a reasonable frequency parameter λ for the Poisson distribution. Theists and Deists find an extremely low value of λ most plausible, in which case there would be only one or at most a handful of universes. Atheists are more likely to be comfortable with very large frequency parameters. Again, which you find more plausible is just a matter of inclination and cannot be proved or disproved.

A deeper reason for the values

But we don’t need multiple universes to explain the constants. We may one day discover deeper scientific truths that explain why the constants have the values they do. To me it seems most likely that this will indeed turn out to be the case and that these truths will emerge as part of a grand unified theory of everything, or on the way to one. I can imagine that theory explaining why it would be impossible for the constants to have any other values. This is speculation but again it’s just a question of what appears plausible to you. To me, this seems not only plausible but highly likely. In the absence of evidence as to what is true I like to believe what is beautiful and to me a universe in which the physical constants had values that appeared to be entirely random yet necessary would be ugly. An underlying pattern that explained why they had to have the values they have could be an aesthete’s delight.

What does ‘fine tuning’ mean?

Finally, I’d like to ask: who says the constants are finely tuned? The fine tuning knob on an AM radio might make adjustments of only 10kHz per revolution, whereas the gross tuning knob might adjust by 200kHz per revolution, and the frequency range for the AM spectrum might be about 700kHz, or more than three full turns of the gross tuning knob. (It’s a long while since I’ve had a radio with tuning knobs on, so forgive me if these numbers are not entirely accurate). We give the fine tuning knob its name because it only changes the frequency by a small fraction of the overall range for each turn of the knob.

But what is the range of possible values for the physical constants? If one of them has a value of 5.6321 and values that differ from that by more than 0.0001 in either direction would not allow life to develop then that is fine tuning indeed if the range of possible values is anything from 1 to 10, but it is very coarse tuning if the range of possible values is only 5.6320009 to 5.6321001. So far as I am aware, nobody knows what the range of possible values for these constants is, or even whether there is a range, so how can we say whether they are finely or coarsely tuned?

So, in summary, if the ‘finely tuned constants’ argument makes you feel warm and secure in an enhanced belief in your Intelligent Designer, that’s great. Just don’t mistake that feeling for a logical argument.