Tag Archives: Philosophy of Science

A meditation on physical units: Part 1

[Preface: A while back, Michael Raymer, a professor at the University of Oregon, drew my attention to a curious paper by Craig Holt, who tragically passed away in 2014 [1]. Michael wrote:
“Dear Jacques … I would be very interested in knowing your opinion of this paper,
since Craig was not a professional academic, and had little community in
which to promote the ideas. He was one of the most brilliant PhD students
in my graduate classes back in the 1970s, turned down an opportunity to
interview for a position with John Wheeler, worked in industry until age
50 when he retired in order to spend the rest of his time in self study.
In his paper he takes a Machian view, emphasizing the relational nature of
all physical quantities even in classical physics. I can’t vouch for the
technical correctness of all of his results, but I am sure they are
inspiring.”

The paper makes for an interesting read because Holt, unencumbered by contemporary fashions, freely questions some standard assumptions about the meaning of `mass’ in physics. Probably because it was a work in progress, Craig’s paper is missing some of the niceties of a more polished academic work, like good referencing and a thoroughly researched introduction that places the work in context (the most notable omission is the lack of background material on dimensional analysis, which I will talk about in this post). Despite its rough edges, Craig’s paper led me down quite an interesting rabbit-hole, of which I hope to give you a glimpse. This post covers some background concepts; I’ll mention Craig’s contribution in a follow-up post. ]

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Imagine you have just woken up after a very bad hangover. You retain your basic faculties, such as the ability to reason and speak, but you have forgotten everything about the world in which you live. Not just your name and address, but your whole life history, family and friends, and entire education are lost to the epic blackout. Using pure thought, you are nevertheless able to deduce some facts about the world, such as the fact that you were probably drinking Tequila last night.

RM_hangover
The first thing you notice about the world around you is that it can be separated into objects distinct from yourself. These objects all possess properties: they have colour, weight, smell, texture. For instance, the leftover pizza is off-yellow, smells like sardines and sticks to your face (you run to the bathroom).

While bending over the toilet for an extended period of time, you notice that some properties can be easily measured, while others are more intangible. The toilet seems to be less white than the sink, and the sink less white than the curtains. But how much less? You cannot seem to put a number on it. On the other hand, you know from the ticking of the clock on the wall that you have spent 37 seconds thinking about it, which is exactly 14 seconds more than the time you spent thinking about calling a doctor.

You can measure exactly how much you weigh on the bathroom scale. You can also see how disheveled you look in the mirror. Unlike your weight, you have no idea how to quantify the amount of your disheveled-ness. You can say for sure that you are less disheveled than Johnny Depp after sleeping under a bridge, but beyond that, you can’t really put a number on it. Properties like time, weight and blood-alcohol content can be quantified, while other properties like squishiness, smelliness and dishevelled-ness are not easily converted into numbers.

You have rediscovered one of the first basic truths about the world: all that we know comes from our experience, and the objects of our experience can only be compared to other objects of experience. Some of those comparisons can be numerical, allowing us to say how much more or less of something one object has than another. These cases are the beginning of scientific inquiry: if you can put a number on it, then you can do science with it.

Rulers, stopwatches, compasses, bathroom scales — these are used as reference objects for measuring the `muchness’ of certain properties, namely, length, duration, angle, and weight. Looking in your wallet, you discover that you have exactly 5 dollars of cash, a receipt from a taxi for 30 dollars, and you are exactly 24 years old since yesterday night.

You reflect on the meaning of time. A year means the time it takes the Earth to go around the Sun, or approximately 365 and a quarter days. A day is the time it takes for the Earth to spin once on its axis. You remember your school teacher saying that all units of time are defined in terms of seconds, and one second is defined as 9192631770 oscillations of the light emitted by a Caesium atom. Why exactly 9192631770, you wonder? What if we just said 2 oscillations? A quick calculation shows that this would make you about 110 billion years old according to your new measure of time. Or what about switching to dog years, which are 7 per human year? That would make you 168 dog years old. You wouldn’t feel any different — you would just be having a lot more birthday parties. Given the events of last night, that seems like a bad idea.

You are twice as old as your cousin, and that is true in dog years, cat years, or clown years [2]. Similarly, you could measure your height in inches, centimeters, or stacked shot-glasses — but even though you might be 800 rice-crackers tall, you still won’t be able to reach the aspirin in the top shelf of the cupboard. Similarly, counting all your money in cents instead of dollars will make it a bigger number, but won’t actually make you richer. These are all examples of passive transformations of units, where you imagine measuring something using one set of units instead of another. Passive transformations change nothing in reality: they are all in your head. Changing the labels on objects clearly cannot change the physical relationships between them.

Things get interesting when we consider active transformations. If a passive transformation is like saying the length of your coffee table is 100 times larger when measured in cm than when measured in meters, then an active transformation would be if someone actually replaced your coffee table with a table 100 times bigger. Now, obviously you would notice the difference because the table wouldn’t fit in your apartment anymore. But imagine that someone, in addition to replacing the coffee table, also replaced your entire apartment and everything in it with scaled-up models 100 times the size. And imagine that you also grew to into a giant 100 times your original size while you were sleeping. Then when you woke up, as a giant inside a giant apartment with a giant coffee table, would you realise anything had changed? And if you made yourself a giant cup of coffee, would it make your giant hangover go away?

Kafka
Or if you woke up as a giant bug?

We now come to one of the deepest principles of physics, called Bridgman’s Principle of absolute significance of relative magnitude, named for our old friend Percy Bridgman. The Principle says that only relative quantities can enter into the laws of physics. This means that, whatever experiments I do and whatever measurements I perform, I can only obtain information about the relative sizes of quantities: the length of the coffee table relative to my ruler, or the mass of the table relative to the mass of my body, etc. According to this principle, actively changing the absolute values of some quantity by the same proportion for all objects should not affect the outcomes of any experiments we could perform.

To get a feeling for what the principle means, imagine you are a primitive scientist. You notice that fruit hanging from trees tends to bob up and down in the wind, but the heavier fruits seems to bounce more slowly than the lighter fruits (for those readers who are physics students, I’m talking about a mass on a spring here). You decide to discover the law that relates the frequency of bobbing motion to the mass of the fruit. You fill a sack with some pebbles (carefully chosen to all have the same weight) and hang it from a tree branch. You can measure the mass of the sack by counting the number of pebbles in it, but you still need a way to measure the frequency of the bobbing. Nearby you hear the sound of water dripping from a leaf into a pond. You decide to measure the frequency by how many times the sack bobs up and down in between drips of water. Now you are ready to do your experiment.

You measure the bobbing frequency of the sack for many different masses, and record the results by drawing in the dirt with a stick. After analysing your data, you discover that the frequency f (in oscillations per water drop) is related to the mass m (in pebbles) by a simple formula:

HookeSpring
where k stands for a particular number, say 16.8. But what does this number really mean?

Unbeknownst to you, a clever monkey was watching you from the bushes while you did the experiment. After you retire to your cave to sleep, the monkey comes out to play a trick on you. He carefully replaces each one of your pebbles with a heavier pebble of the same size and appearance, and makes sure that all of the heavier pebbles are the same weight as each other. He takes away the original pebbles and hides them. The next day, you repeat the experiment in exactly the same way, but now you discover that the constant k has changed from yesterday’s value of 16.8 to the new value of 11.2. Does this mean that the law of nature that governs the bobbing of things hanging from the tree has changed overnight? Or should you decide that the law is the same, but that the units that you used to measure frequency and mass have changed?

You decide to apply Bridgman’s Principle. The principle says that if (say) all the masses in the experiment were changed by the same proportion, then the laws of physics would not allow us to see any difference, provided we used the same measuring units. Since you do see a difference, Bridgman’s Principle says that it must be the units (and not the law itself) that has changed. `These must be different pebbles’ you say to yourself, and you mark them by scratching an X onto them. You go out looking for some other pebbles and eventually you find a new set of pebbles which give you the right value of 16.8 when you perform the experiment. `These must be the same kind of pebbles that I used in the original experiment’ you say to yourself, and you scratch an O on them so that you won’t lose them again. Ha! You have outsmarted the monkey.

Larson_rocks

Notice that as long as you use the right value for k — which depends on whether you measure the mass using X or O pebbles — then the abstract equation (1) remains true. In physics language, you are interpreting k as a dimensional constant, having the dimensions of  frequency times √mass. This means that if you use different units for measuring frequency or mass, the numerical value of k has to change in order to preserve the law. Notice also that the dimensions of k are chosen so that equation (1) has the same dimensions on each side of the equals sign. This is called a dimensionally homogeneous equation. Bridgman’s Principle can be rephrased as saying that all physical laws must be described by dimensionally homogeneous equations.

Bridgman’s Principle is useful because it allows us to start with a law expressed in particular units, in this case `oscillations per water-drop’ and `O-pebbles’, and then infer that the law holds for any units. Even though the numerical value of k changes when we change units, it remains the same in any fixed choice of units, so it represents a physical constant of nature.

The alternative is to insist that our units are the same as before (the pebbles look identical after all). That means that the change in k implies a change in the law itself, for instance, it implies that the same mass hanging from the tree today will bob up and down more slowly than it did yesterday. In our example, it turns out that Bridgman’s Principle leads us to the correct conclusion: that some tricky monkey must have switched our pebbles. But can the principle ever fail? What if physical laws really do change?

Suppose that after returning to your cave, the tricky monkey decides to have another go at fooling you. He climbs up the tree and whispers into its leaves: `Do you know why that primitive scientist is always hanging things from your branch? She is testing how strong you are! Make your branches as stiff and strong as you can tomorrow, and she will reward you with water from the pond’.

The next day, you perform the experiment a third time — being sure to use your `O-pebbles’ this time — and you discover again that the value of k seems to have changed. It now takes many more pebbles to achieve a given frequency than it did on the first day. Using Bridgman’s Principle, you again decide that something must be wrong with your measuring units. Maybe this time it is the dripping water that is wrong and needs to be adjusted, or maybe you have confidence in the regularity of the water drip and conclude that the `O-pebbles’ have somehow become too light. Perhaps, you conjecture, they were replaced by the tricky monkey again? So you throw them out and go searching for some heavier pebbles. You find some that give you the right value of k=16.8, and conclude that these are the real `O-pebbles’.

The difference is that this time, you were tricked! In fact the pebbles you threw out were the real `O-pebbles’. The change in k came from the background conditions of the experiment, namely the stiffness in the tree branches, which you did not consider as a physical variable. Hence, in a sense, the law that relates bobbing frequency to mass (for this tree) has indeed changed [3].

You thought that the change in the constant k was caused by using the wrong measuring units, but in fact it was due to a change in the physical constant k itself. This is an example of a scenario where a physical constant turns out not to be constant after all. If we simply assume Bridgman’s Principle to be true without carefully checking whether it is justified, then it is harder to discover situations in which the physical constants themselves are changing. So, Bridgman’s Principle can be thought of as the assumption that the values of physical constants (expressed in some fixed units) don’t change over time. If we are sure that the laws of physics are constant, then we can use the Principle to detect changes or inaccuracies in our measuring devices that define the physical units — i.e. we can leverage the laws of physics to improve the accuracy of our measuring devices.

We can’t always trust our measuring units, but the monkey also showed us that we can’t always trust the laws of physics. After all, scientific progress depends on occasionally throwing out old laws and replacing them with more accurate ones. In our example, a new law that includes the tree-branch stiffness as a variable would be the obvious next step.

One of the more artistic aspects of the scientific method is knowing when to trust your measuring devices, and when to trust the laws of physics [4]. Progress is made by `bootstrapping’ from one to the other: first we trust our units and use them to discover a physical law, and then we trust in the physical law and use it to define better units, and so on. It sounds like a circular process, but actually it represents the gradual refinement of knowledge, through increasingly smaller adjustments from different angles. Imagine trying to balance a scale by placing handfuls of sand on each side. At first you just dump about a handful on each side and see which is heavier. Then you add a smaller amount to the lighter side until it becomes heavier. Then you add an even smaller amount to the other side until it becomes heavier, and so on, until the scale is almost perfectly balanced. In a similar way, switching back and forth between physical laws and measurement units actually results in both the laws and measuring instruments becoming more accurate over time.

______________

[1] It is a shame that Craig’s work remains incomplete, because I think physicists could benefit from a re-examination of the principles of dimensional analysis. Simplified dimensional arguments are sometimes invoked in the literature on quantum gravity without due consideration for their meaning.

[2] Clowns have several birthdays a week, but they aren’t allowed to get drunk at them, which kind of defeats the purpose if you ask me.

[3] If you are uncomfortable with treating the branch stiffness as part of the physical law, imagine instead that the strength of gravity actually becomes weaker overnight.

[4] This is related to a deep result in the philosophy of science called the Duhem-Quine Thesis.
Quoth Duhem: `If the predicted phenomenon is not produced, not only is the questioned proposition put into doubt, but also the whole theoretical scaffolding used by the physicist’.

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Bootstrapping to quantum gravity

Kepler

“If … there were no solid bodies in nature there would be no geometry.”
-Poincaré

A while ago, I discussed the mystery of why matter should be the source of gravity. To date, this remains simply an empirical fact. The deep insight of general relativity – that gravity is the geometry of space and time – only provides us with a modern twist: why should matter dictate the geometry of space-time?

There is a possible answer, but it requires us to understand space-time in a different way: as an abstraction that is derived from the properties of matter itself. Under this interpretation, it is perfectly natural that matter should affect space-time geometry, because space-time is not simply a stage against which matter dances, but is fundamentally dependent on matter for its existence. I will elaborate on this idea and explain how it leads to a new avenue of approach to quantum gravity.

First consider what we mean when we talk about space and time. We can judge how far away a train is by listening to the tracks, or gauge how deep a well is by dropping a stone in and waiting to hear the echo. We can tell a mountain is far away just by looking at it, and that the cat is nearby by tripping over it. In all these examples, an interaction is necessary between myself and the object, sometimes through an intermediary (the light reflected off the mountain into my eyes) and sometimes not (tripping over the cat). Things can also be far away in time. I obviously cannot interact with people who lived in the past (unless I have a time machine), or people who have yet to be born, even if they stood (or will stand) exactly where I am standing now. I cannot easily talk to my father when he was my age, but I can almost do it, just by talking to him now and asking him to remember his past self. When we say that something is far away in either space or time, what we really mean is that it is hard to interact with, and this difficulty of interaction has certain universal qualities that we give the names `distance’ and `time’.
It is worth mentioning here, as an aside, that in a certain sense, the properties of `time’ can be reduced to properties of `distance’ alone. Consider, for instance, that most of our interactions can be reduced to measurements of distances of things from us, at a given time. To know the time, I invariably look at the distance the minute hand has traversed along its cycle on the face of my watch. Our clocks are just systems with `internal’ distances, and it is the varying correspondence of these `clock distances’ with the distances of other things that we call the `time’. Indeed, Julian Barbour has developed this idea into a whole research program in which dynamics is fundamentally spatial, called Shape Dynamics.

Sigmund Freud Museum, Wien – Peter Kogler

So, if distance and time is just a way of describing certain properties of matter, what is the thing we call space-time?

We now arrive at a crucial point that has been stressed by philosopher Harvey Brown: the rigid rods and clocks with which we claim to measure space-time do not really measure it, in the traditional sense of the word `measure’. A measurement implies an interaction, and to measure space-time would be to grant space-time the same status as a physical body that can be interacted with. (To be sure, this is exactly how many people do wish to interpret space-time; see for instance space-time substantivalism and ontological structural realism).

Brown writes:
“One of Bell’s professed aims in his 1976 paper on `How to teach relativity’ was to fend off `premature philosophizing about space and time’. He hoped to achieve this by demonstrating with an appropriate model that a moving rod contracts, and a moving clock dilates, because of how it is made up and not because of the nature of its spatio-temporal environment. Bell was surely right. Indeed, if it is the structure of the background spacetime that accounts for the phenomenon, by what mechanism is the rod or clock informed as to what this structure is? How does this material object get to know which type of space-time — Galilean or Minkowskian, say — it is immersed in?” [1]

I claim that rods and clocks do not measure space-time, they embody space-time. Space-time is an idealized description of how material rods and clocks interact with other matter. This distinction is important because it has implications for quantum gravity. If we adopt the more popular view that space-time is an independently existing ontological construct, it stands to reason that, like other classical fields, we should attempt to directly quantise the space-time field. This is the approach adopted in Loop Quantum Gravity and extolled by Rovelli:

“Physical reality is now described as a complex interacting ensemble of entities (fields), the location of which is only meaningful with respect to one another. The relation among dynamical entities of being contiguous … is the foundation of the space-time structure. Among these various entities, there is one, the gravitational field, which interacts with every other one and thus determines the relative motion of the individual components of every object we want to use as rod or clock. Because of that, it admits a metrical interpretation.” [2]

One of the advantages of this point of view is that it dissolves some seemingly paradoxical features of general relativity, such as the fact that geometry can exist without (non-gravitational) matter, or the fact that geometry can carry energy and momentum. Since gravity is a field in its own right, it doesn’t depend on the other fields for its existence, nor is there any problem with it being able to carry energy. On the other hand, this point of view tempts us into framing quantum gravity as the mathematical problem of quantising the gravitational field. This, I think, is misguided.

I propose instead to return to a more Machian viewpoint, according to which space-time is contingent on (and not independent of) the existence of matter. Now the description of quantum space-time should follow, in principle, from an appropriate description of quantum matter, i.e. of quantum rods and clocks. From this perspective, the challenge of quantum gravity is to rebuild space-time from the ground up — to carry out Einstein’s revolution a second time over, but using quantum material as the building blocks.

Ernst Mach vs. Max Ernst. Get it right, folks.

My view about space-time can be seen as a kind of `pulling oneself up by one’s bootstraps’, or a Wittgenstein’s ladder (in which one climbs to the top of a ladder and then throws the ladder away). It works like this:
Step 1: define the properties of space-time according to the behaviour of rods and clocks.
Step 2: look for universal patterns or symmetries among these rods and clocks.
Step 3: take the ideal form of this symmetry and promote it to an independently existing object called `space-time’.
Step 4: Having liberated space-time from the material objects from which it was conceived, use it as the independent standard against which to compare rods and clocks.

Seen in this light, the idea of judging a rod or a clock by its ability to measure space or time is a convenient illusion: in fact we are testing real rods and clocks against what is essentially an embodiment of their own Platonic ideals, which are in turn conceived as the forms which give the laws of physics their most elegant expression. A pertinent example, much used by Julian Barbour, is Ephemeris time and the notion of a `good clock’. First, by using material bodies like pendulums and planets to serve as clocks, we find that the motions of material bodies approximately conform to Newton’s laws of mechanics and gravitation. We then make a metaphysical leap and declare the laws to be exactly true, and the inaccuracies to be due to imperfections in the clocks used to collect the data. This leads to the definition of the `Ephemeris time’, the time relative to which the planetary motions conform most closely to Newton’s laws, and a `good clock’ is then defined to be a clock whose time is closest to Ephemeris time.

The same thing happens in making the leap to special relativity. Einstein observed that, in light of Maxwell’s theory of electromagnetism, the empirical law of the relativity of motion seemed to have only a limited validity in nature. That is, assuming no changes to the behaviour of rods and clocks used to make measurements, it would not be possible to establish the law of the relativity of motion for electrodynamic bodies. Einstein made a metaphysical leap: he decided to upgrade this law to the universal Principle of Relativity, and to interpret its apparent inapplicability to electromagnetism as the failure of the rods and clocks used to test its validity. By constructing new rods and clocks that incorporated electromagnetism in the form of hypothetical light beams bouncing between mirrors, Einstein rebuilt space-time so as to give the laws of physics a more elegant form, in which the Relativity Principle is valid in the same regime as Maxwell’s equations.

Ladder for Booker T. Washington – Martin Puryear

By now, you can guess how I will interpret the step to general relativity. Empirical observations seem to suggest a (local) equivalence between a uniformly accelerated lab and a stationary lab in a gravitational field. However, as long as we consider `ideal’ clocks to conform to flat Minkowski space-time, we have to regard the time-dilated clocks of a gravitationally affected observer as being faulty. The empirical fact that observers stationary in a gravitational field cannot distinguish themselves (locally) from uniformly accelerated observers then seems accidental; there appears no reason why an observer could not locally detect the presence of gravity by comparing his normal clock to an `ideal clock’ that is somehow protected from gravity. On the other hand, if we raise this empirical indistinguishability to a matter of principle – the Einstein Equivalence Principle – we must conclude that time dilation should be incorporated into the very definition of an `ideal’ clock, and similarly with the gravitational effects on rods. Once the ideal rods and clocks are updated to include gravitational effects as part of their constitution (and not an interfering external force) they give rise to a geometry that is curved. Most magically of all, if we choose the simplest way to couple this geometry to matter (the Einstein Field Equations), we find that there is no need for a gravitational force at all: bodies follow the paths dictated by gravity simply because these are now the inertial paths followed by freely moving bodies in the curved space-time. Thus, gravity can be entirely replaced by geometry of space-time.

As we can see from the above examples, each revolution in our idea of space-time was achieved by reconsidering the nature of rods and clocks, so as to make the laws of physics take a more elegant form by incorporating some new physical principle (eg. the Relativity and Equivalence principles). What is remarkable is that this method does not require us to go all the way back to the fundamental properties of matter, prior to space-time, and derive everything again from scratch (the constructive theory approach). Instead, we can start from a previously existing conception of space-time and then upgrade it by modifying its primary elements (rods and clocks) to incorporate some new principle as part of physical law (the principle theory approach). The question is, will quantum gravity let us get away with the same trick?

I’m betting that it will. The challenge is to identify the empirical principle (or principles) that embody quantum mechanics, and upgrade them to universal principles by incorporating them into the very conception of the rods and clocks out of which general relativistic space-time is made. The result will be, hopefully, a picture of quantum geometry that retains a clear operational interpretation. Perhaps even Percy Bridgman, who dismissed the Planck length as being of “no significance whatever” [3] due to its empirical inaccessibility, would approve.

Boots with laces – Van Gogh

[1] Brown, Physical Relativity, p8.
[2] Rovelli, `Halfway through the woods: contemporary research on space and time’, in The Cosmos of Science, p194.
[3] Bridgman, Dimensional Analysis, p101.

Science, psychoanalyzed

“The problem for us is not, are our desires satisfied or not? The problem is, how do we know what we desire?”

-Slavoj Žižek

The most fundamental dramatic tension is the tension between the divided self. We have all on occasion experienced an internal dialogue like the following: `I ate the cookie despite myself. I knew it was wrong, but I couldn’t help myself. Afterwards, I hated myself’. On one hand, this dialogue makes sense to us and its meaning seems clear; on the other hand, it makes no sense without a division of the self. Who is the myself against whose wishes I eat the cookie? Who is the I that could not help myself? Who, afterwards, is hated, and who is the hater? To admit that the self can be both the subject and object of an action is equivalent to admitting that the self is divided.

Let us therefore deliver ourselves into the hands of Freud, who will lead us down a rabbit-hole of self-discovery. Who are these characters, the id, ego and superego? The id is the instinctive, reactive, animalistic part of the mind. It expresses emotion without reflection, it is wordless, mute, free of morals, shame or self-consciousness. The superego is the embodiment of laws and limitations. When the child learns that it is separate from the world, confined to a small, weak body and cannot have everything it wants – when it learns that it is at the mercy of beings far more powerful who dictate its life – it internalises these limitations and laws by creating the superego. The superego tells us what we are not allowed to do, where we cannot go, and what is forbidden by physical, moral or societal laws.

Freud

The fundamental tension between superego and id demands a mediator to decide whether to go with the desires of the id or follow the rules of the superego. This mediator, haplessly caught between the two, is our hero, ourselves: the ego. When the ego obeys the superego, the id is suppressed and frustrated, while the ego becomes more powerful and more strict in its demands. When the ego obeys the id instead, the satisfaction is short-lived, for the id knows only the present moment, and is hungry again no sooner it is fed. Meanwhile, the superego brings its vengeance on the ego for the transgression, afflicting it with guilt and feelings of inferiority. The id expresses our desires and fears, the superego expresses our judgements, and the ego determines how we respond in our actions. Before reading the end of this paragraph, take a moment to re-read the dialogue about the cookie and try to name the actors and the victims. Did you do it? The id wanted to eat the cookie, the superego knew it was wrong, and the ego ate it. The superego was helpless to stop the ego, but afterwards, it hated the ego, and punished it with feelings of guilt. Now it makes sense.

MBros

Humans have a curious obsession with the number three. There are three wise men, the holy trinity, the `third eye’ of Hinduism. Dramatic tension between fictional characters also frequently relies on combinations of three. It is an entertaining exercise (but not always fruitful) to identify the roles of id, ego and superego in famous triplets from mythology and fiction. Here is a puzzle for you. In Brisbane, I used to frequent a coffee house called Three Monkeys. Inside, they had amassed a collection of depictions and statuettes of the `Three Wise Monkeys’, a mystical image originating from Japan in which the first monkey has covered its eyes, the second its ears, and the last one its mouth. The image is typically associated with the maxim: see no evil, hear no evil, speak no evil, thought to originate from a similar passage in the Chinese Analects of Confucius. The puzzle is this: if the monkeys were to represent the different aspects of the divided self, which monkey is the id, which is the ego and which is the superego? Or does the comparison simply fail? My own answer is given at the end of this essay.

3monks

Tension is by nature unsustainable. It must eventually resolve itself in one of three ways: destruction, reconciliation, or transformation into a new kind of tension (which just means the destruction of some things and the reconciliation of others). Destruction can occur when the division between the id and superego is too extreme, tearing apart the ego with opposing forces. Since the ego exists only to mediate the conflict between the other two, a reconciliation of the id with the superego automatically conciliates the ego as well. This represents a dissolution of the ego, meaning a loss of the distinction between the self and the external world: the attainment of Nirvana in the eastern philosophies. In reality, however, most of us experience only a very small and partial conciliation of this type, a sort of secret collaboration between the superego and the id. This secret collaboration is at the core of science, so let us examine it in more detail.

The easiest way to appreciate the perverse but necessary collaboration between superego and id is to look at stories and films. There, the characters are nicely separated into roles that often reflect the roles of our divided selves. Take Batman and the Joker as depicted in Christopher Nolan’s film, The Dark Knight. The Joker is obviously a candidate for the id:

The Dark Knight
“Do I really look like a guy with a plan? You know what I am? I’m a dog chasing cars. I wouldn’t know what to do with one if I caught it. You know, I just… do things.”

Batman, although a vigilante, is a good fit for the superego: he is the true enforcer of law, both the judge and the executioner. In fact it is the police force, embodied by Commissioner Gordon, that best represents the ego in its unenviable position, caught between the two rogue elements. Given these roles, we finally understand this brilliant exchange:
Batman: Then why do you want to kill me?
Joker: I don’t want to kill you! What would I do without you? Go back to ripping off mob dealers? No, no, NO! No. You… you complete me.
You could not ask for a more perfect exposition of the mutual dependence of the superego and the id.

Sometimes the bond is more subtle. Consider one of fiction’s greatest characters: Sherlock Holmes. Not coincidentally, Holmes is a poster boy for scientists, with his strict adherence to a method based on evidence, reasoning and deduction. Quite obviously, he is a manifestation of the superego, leaving Watson to carry the banner of the ego. He wears it well enough, constantly being lectured and berated by Holmes, occasionally skeptical and rebellious but always respectful of Holmes’ superior judgement. Where, then, could the id be hiding? Therein lies a profound mystery, worthy of Holmes himself! One is tempted to point at Moriarty, the great enemy of Holmes – but the shoe does not fit. In Moriarty one finds exactly the kind of characteristics more typical of the superego: self-confidence verging on megalomania, mercilessness, a strict adherence to methodology. He is more like Holmes’s evil twin – the vindictive, cruel side of the superego – than the impulsive and chaotic id.

Holmes

My own theory is that Holmes is a much more subtle character than he first appears. Who is the Holmes that we find, lost in a wordless reverie, playing the violin? Who is the Holmes that disguises himself to play a prank on poor Watson – the Holmes who, indeed, delights in upsetting Watson with eccentric and erratic behaviour? Who is the Holmes that goes missing for days, only to be found curled up in a den of iniquity, his eyes clouded with Opium? I contend that Holmes has an instinctive, intuitive and sensitive side that embodies the id, working in harmony with his superego aspect. Indeed, the seedy side of Holmes – his indulgent, drug-taking, reckless aspect – is somehow essential to completing the portrait of his genius. We would not find him so credible, so impressive, so almost mystical in his virtuosity if it were not for this dark side.

The superego and id can indeed collaborate, but it is usually only in a secretive, almost illicit way as though neither can admit that it depends on the other. The superego turns a blind eye, allowing the id to run wild, and then acts surprised and disappointed when it discovers the transgression. Then ensues what is in essence a sadomasochistic mock-punishment, since the id secretly enjoys the flogging, and the superego knows it, but plays along. In short, the union between superego and id is possible through the hypocritical self-awareness of both parties that they depend on each other to exist. They throw themselves into their respective roles with even more gusto, maintaining as it were a secret conspiracy against the ego, keeping up the tension but with a knowing cynicism.

JekyllHyde
We now begin to see the first inklings of the mad scientist. The quintessential mad scientist is Dr. Jekyll and Mr. Hyde, whose two faces represent unmistakably a perverse union of superego and id; other examples in fiction abound. The mad scientist is in fact the manifestation in an individual character of the public’s view of scientific activity in general. Since (as Kuhn tells us) science is a human activity, its attributes can be traced to attributes of the human mind. In other words, science as an institution can be psychoanalyzed.
Science is defined on one hand by its rationality, its strict adherence to method, zero tolerance for transgression of its rules, and a claim to superiority in its judgements and conclusions about the world. On the other hand, science is a powerful vehicle for the realisation of our (human) fantasies: what technology is not born from the dream of a science-fiction nerd? Technology is transgressive in the same way that dreams are transgressive: there is no taboo in science, no political correctness, no boundaries. At its purest, science and technology is obscene, disturbing and visionary all at once. Medicine is born of the desire to be immortal, chemistry is born of our desire to have power over the substances and forces of the world, to make gold and riches from lead; physics is born of our desire to fly through the sky like a bird, to be invisible, telepathic, omnipotent. Biology promises us the power to make animals and other organisms serve our needs, and psychology offers us power over each other. Science, with all of its adherence to evidence, logic and deduction, remains silent on matters of its purpose, has nothing to suggest about the ends to which it should be used. There lies hidden the id of science: an amoral, primitive, instinctive drive of humanity, just like the indignant infant trying to come to terms with the world. Without an effective intermediary in the form of public discussion and deliberation over scientific advances, science risks becoming a Sherlock without a Watson, that is, a Dr. Jekyll and Mr. Hyde.

Of course, just as it does in the individual’s psyche, the scientific id also plays a beneficial role: it supplies the creative drive and aesthetic sensibility without which science would be impossible. This is why we cannot divorce the id from the superego in science without destroying science altogether. Eliminate the id from Science, and you are left with a stagnant dogma; eliminate the superego, the methodology and tools of rational inquiry, and you are left with mysticism and superstition. The philosophy of science does an injustice to the true mechanism of scientific progress by focusing too much on the methodology – how to evaluate evidence and test hypotheses – and neglecting to address the aesthetic side of science.

Rick and Morty
“Sometimes science is more art than science. A lot of people don’t get that.”

How do we generate hypotheses? Where do ideas come from? Scientists themselves often don’t acknowledge the role that instinct and intuition plays in proposing new theories – we tend to downplay it, or insist that science progresses without any creative input. If that were really true, computer programs could do science in the foreseeable future. But most of us consider the revolution of the machines to still be far away, for the simple reason that we don’t yet know how to teach computers to be creative and to select `good’ hypotheses from the vast pool of logically possible hypotheses. This is (so far) a uniquely human ability, which has everything to do with gut feelings, impulsive thoughts and secret desires. The philosophy of science would perhaps benefit greatly from a more careful examination of this hidden aspect of scientific progress.

My answer to the three monkey’s question is this: The monkey who cannot speak is the id, because the id is voiceless. That leaves the blind monkey and the deaf monkey. It boils down to a matter of opinion here, but the argument that appeals to me most is this one: the superego has a closer relationship with the id than the ego does. Since the blind monkey can neither see nor hear the id (because the id can’t talk), but the deaf monkey can at least see the id, it stands to reason that the deaf monkey is the superego and the blind monkey is the ego.

marx3monk

The trouble with Reichenbach

(Note: this blog post is vaguely related to a paper I wrote. You can find it on the arXiv here. )

Suppose you are walking along the beach, and you come across two holes in the rock, spaced apart by some distance; let us label them ‘A’ and ‘B’. You observe an interesting correlation between them. Every so often, at an unpredictable time, water will come spraying out of hole A, followed shortly after by a spray of water out of hole B. Given our day-to-day experience of such things, most of us would conclude that the holes are connected by a tunnel underneath the rock, which is in turn connected to the ocean, such that a surge of water in the underground tunnel causes the water to spray from the two holes at about the same time.

Image credit: some douchebag
Now, therein lies a mystery: how did our brains make this deduction so quickly and easily? The mere fact of a statistical correlation does not tell us much about the direction of cause and effect. Two questions arise. First, why do correlations require explanations in the first place? Why can we not simply accept that the two geysers spray water in synchronisation with each other, without searching for explanations in terms of underground tunnels and ocean surges? Secondly, how do we know in this instance that the explanation is that of a common cause, and not that (for example) the spouting of water from one geyser triggers some kind of chain reaction that results in the spouting of water from the other?

The first question is a deep one. We have in our minds a model of how the world works, which is the product partly of history, partly of personal experience, and partly of science. Historically, we humans have evolved to see the world in a particular way that emphasises objects and their spatial and temporal relations to one another. In our personal experience, we have seen that objects move and interact in ways that follow certain patterns: objects fall when dropped and signals propagate through chains of interactions, like a series of dominoes falling over. Science has deduced the precise mechanical rules that govern these motions.

According to our world-view, causes always occur before their effects in time, and one way that correlations can arise between two events is if one is the cause of the other. In the present example, we may reason as follows: since hole B always spouts after A, the causal chain of events, if it exists, must run from A to B. Next, suppose that I were to cover hole A with a large stone, thereby preventing it from emitting water. If the occasion of its emission were the cause of hole B’s emission, then hole B should also cease to produce water when hole A is covered. If we perform the experiment and we find that hole B’s rate of spouting is unaffected by the presence of a stone blocking hole A, we can conclude that the two events of spouting water are not connected by a direct causal chain.

The only other way in which correlations can arise is by the influence of a third event — such as the surging of water in an underground tunnel — whose occurrence triggers both of the water spouts, each independently of the other. We could promote this aspect of our world-view to a general principle, called the Principle of the Common Cause (PCC): whenever two events A and B are correlated, then either one is a cause of the other, or else they share a common cause (which must occur some time before both of these events).

The Principle of Common Cause tells us where to look for an explanation, but it does not tell us whether our explanation is complete. In our example, we used the PCC to deduce that there must be some event preceding the two water spouts which explains their correlation, and for this we proposed a surge of water in an underground tunnel. Now suppose that the presence of water in this tunnel is absolutely necessary in order for the holes to spout water, but that on some occasions the holes do not spout even though there is water in the tunnel. In that case, simply knowing that there is water in the tunnel does not completely eliminate the correlation between the two water spouts. That is, even though I know there is water in the tunnel, I am not certain whether hole B will emit water, unless I happen to know in addition that hole A has just spouted. So, the probability of B still depends on A, despite my knowledge of the ‘common cause’. I therefore conclude that I do not know everything that there is to know about this common cause, and there is still information to be had.

thinks2

It could be, for instance, that the holes will only spout water if the water pressure is above a certain threshold in the underground tunnel. If I am able to detect both the presence of the water and its pressure in the tunnel, then I can predict with certainty whether the two holes will spout or not. In particular, I will know with certainty whether hole B is going to spout, independently of A. Thus, if I had stakes riding on the outcome of B, and you were to try and sell me the information “whether A has just spouted”, I would not buy it, because it does not provide any further information beyond what I can deduce from the water in the tunnel and its pressure level. It is a fact of general experience that, conditional on complete knowledge of the common causes of two events, the probabilities of those events are no longer correlated. This is called the principle of Factorisation of Probabilities (FP). The union of FP and PCC together is called Reichenbach’s Common Cause Principle (RCCP).

thinks3

In the above example, the complete knowledge of the common cause allowed me to perfectly determine whether the holes would spout or not. The conditional independence of these two events is therefore guaranteed. One might wonder why I did not talk about the principle of predetermination: conditional on on complete knowledge of the common causes, the events are determined with certainty. The reason is that predetermination might be too strong; it may be that there exist phenomena that are irreducibly random, such that even a full knowledge of the common causes does not suffice to determine the resulting events with certainty.

As another example, consider two river beds on a mountain slope, one on the left and one on the right. Usually (96% of the time) it does not rain on the mountain and both rivers are dry. If it does rain on the mountain, then there are four possibilities with equal likelihood: (i) the river beds both remain dry, (ii) the left river flows but the right one is dry (iii) the right river flows but the left is dry, or (iv) both rivers flow. Thus, without knowing anything else, the fact that one river is running makes it more likely that the other one is. However, conditional that it rained on the mountain, if I know that the left river is flowing (or dry), this does not tell me anything about whether the right river is flowing or dry. So, it seems that after conditioning on the common cause (rain on the mountain) the probabilities factorise: knowing about one river tells me nothing about the other.

mountain1

Now we have a situation in which the common cause does not completely determine the outcomes of the events, but where the probabilities nevertheless factorise. Should we then conclude that the correlations are explained? If we answer ‘yes’, we have fallen into a trap.

The trap is that there may be additional information which, if discovered, would make the rivers become correlated. Suppose I find a meeting point of the two rivers further upstream, in which sediment and debris tends to gather. If there is only a little debris, it will be pushed to one side (the side chosen effectively at random), diverting water to one of the rivers and blocking the other. Alternatively, if there is a large build-up of debris, it will either dam the rivers, leaving them both dry, or else be completely destroyed by the build-up of water, feeding both rivers at once. Now, if I know that it rained on the mountain and I know how much debris is present upstream, knowing whether one river is flowing will provide information about the other (eg. if there is a little debris upstream and the right river is flowing, I know the left must be dry).

mountain2

 
Before I knew anything, the rivers seemed to be correlated. Conditional on whether it rained on the mountain-top, the correlation disappeared. But now, conditional that it rained on the mountain and on the amount of debris upstream, the correlation is restored! If the only tools I had to explain correlations was the PCC and the FP, then how can I ever be sure that the explanation is complete? Unless the information of the common cause is enough to predetermine the outcomes of the events with certainty, there is always the possibility that the correlations have not been explained, because new information about the common causes might come to light which renders the events correlated again.

Now, at last, we come to the main point. In our classical world-view, observations tend to be compatible with predetermination. No matter how unpredictable or chaotic a phenomenon seems, we find it natural to imagine that every observed fact could be predicted with certainty, in principle, if only we knew enough about its relevant causes. In that case, we are right to say that a correlation has not been fully explained unless Reichenbach’s principle is satisfied. But this last property is now just seen as a trivial consequence of predetermination, implicit in out world-view. In fact, Reichenbach’s principle is not sufficient to guarantee that we have found an explanation. We can only be sure that the explanation has been found when the observed facts are fully determined by their causes.

This poses an interesting problem to anyone (like me) who thinks the world is intrinsically random. If we give up predetermination, we have lost our sufficient condition for correlations to be explained. Normally, if we saw a correlation, after eliminating the possibility of a direct cause we would stop searching for an explanation only when we found one that could perfectly determine the observations. But if the world is random, then how do we know when we have found a good enough explanation?

In this case, it is tempting to argue that Reichenbach’s principle should be taken as a sufficient (not just necessary) condition for an explanation. Then, we know to stop looking for explanations as soon as we have found one that causes the probabilities to factorise. But as I just argued with the example of the two rivers, this doesn’t work. If we believed this, then we would have to accept that it is possible for an explained correlation to suddenly become unexplained upon the discovery of additional facts! Short of a physical law forbidding such additional facts, this makes for a very tenuous notion of explanation indeed.

So fuck off
The question of what should constitute a satisfactory explanation for a correlation is, I think, one of the deepest problems posed to us by quantum mechanics. The way I read Bell’s theorem is that (assuming that we accept the theorem’s basic assumptions) quantum mechanics is either non-local, or else it contains correlations that do not satisfy the factorisation part of Reichenbach’s principle. If we believe that factorisation is a necessary part of explanation, then we are forced to accept non-locality. But why should factorisation be a necessary requirement of explanation? It is only justified if we believe in predetermination.

A critic might try to argue that, without factorisation, we have lost all ability to explain correlations. But I’m saying that this true even for those who would accept factorisation but reject predetermination. I say, without predetermination, there is no need to hold on to factorisation, because it doesn’t help you to explain correlations any better than the rest of us non-determinists! So what are we to do? Maybe it is time to shrug off factorisation and face up to the task of finding a proper explanation for quantum correlations.

Jacques Pienaar’s guide to making physics (Pt.1)

PRINCIPLES AS TOOLS
(Not to be confused with using Principals as tools, which is what happens if your school Principal is a tool because he never taught you the difference between a Principal and a principle. Also not to be confused with a Princey-pal, who is a friend that happens to be a Prince).

`These principles are the boldly generalized results of experiment; but they appear to derive from their very generality a high degree of certainty. In fact, the greater the generality, the more frequent are the opportunities for verifying them, and such verifications, as they multiply, as they take the most varied and most unexpected forms, leave in the end no room for doubt.’ -Poincaré

 
One of the great things Einstein did, besides doing physics, was trying to explain to people how to do it as good as him. Ultimately he failed, because so far nobody has managed to do better than him, but he left us with some really interesting insights into how to come up with new physical theories.

One of these ideas is the concept of using `principles’. A principle is a statement about how the word works (or should work), stated in ordinary language. They are not always called principles, but might be called laws, postulates or hypotheses. I am not going to argue about semantics here. Just consider these examples to get a flavour:

The Second Law of Thermodynamics: You can’t build an engine which does useful work and ends up back in its starting position without producing any heat.

 
Landauer’s principle: you can’t erase information without producing heat.

 
The Principle of Relativity: It is impossible to tell by local experiments whether or not your laboratory is moving.

And some not strictly physics ones:

Shirky’s law: Institutions will try to preserve the problem to which they are the solution.

 
Murphy’s law: If something can go wrong, it will go wrong.

Stigler’s law: No scientific discovery is named after its original discoverer (this law was actually discovered by R.K. Merton, not Stigler).

 
Parkinson’s law: Work always expands to fill up the time allocated to doing it.
(See Wikipedia’s list of eponymous laws for more).

You’ll notice that principles are characterised by two main things: they ring true, and they are vague. Both of these properties are very important for their use in building theories.

Now I can practically hear the lice falling out as you scratch your head in confusion. “But Jacques! How can vagueness be a useful thing to have in a Principle? Shouldn’t it be made as precise as possible?”

No, doofus. A Principle is like an apple. You know what an apple is right?

hipstercat

Well, you think you do. But if I were to ask you, what colour is an apple, how sweet is an apple, how many worms are in an apple, you would have to admit that you don’t know, because the word “apple” is too vague to answer those questions. It is like asking how long is a piece of string. Nevertheless, when you want to go shopping, it suffices to say “buy me an apple” instead of “buy me a Malus domestica, reflective in the 620-750 nanometer range, ten percent sugar, one percent cydia pomonella“.

The only way to make a principle more precise is within the context of a precise theory. But then how would I build a new theory, if I am stuck using the language of the old theory? I can make the idea of an apple more precise using the various scientifically verified properties that apples are known to have, but all of that stuff had to come after we already had a basic vague understanding of what an “apple” was, e.g. a kind of round-ish thing on a tree that tastes nice when you eat it.

The vagueness of a principle means that it defines a whole family of possible theories, these being the ones that kind of fit with the principle if you take the right interpretation. On one hand, a principle that is too vague will not help you to make progress, because it will be too easy to make it fit with any future theory; on the other hand, a principle that is not vague enough will leave you stuck for choices and unable to progress.

The next aspect of a good principle is that it “rings true”. In other words, there is something about it that makes you want it to be true. We want our physical theories to be intuitive to our soft, human brains, and these brains of ours have evolved to think about the world in very specific terms. Why do you think physics seems to be all about the locations of objects in space, moving with time? There are infinitely many ways to describe physics, but we choose the ones we do because of the way our physical senses work, the way our bodies interact with the world, and the things we needed to do in order to survive up to this point. What is the principle of least action? It is a river flowing down a mountain. What is Newtonian mechanics? It is animals moving on the plains. We humans need to see the world in a special way in order to understand it, and good principles are what allow us to shoehorn abstract concepts like thermodynamics and gravitational physics into a picture that looks familiar to us, that we can work with.

That’s why a good principle has to ring true — it has to appeal to the limited imaginative abilities of us humans. Maybe if we were different animals, the laws of physics would be understood in very different terms. Like, the Newtonian mechanics of snakes would start with a simple model of objects moving along snake-paths in two dimensions (the ground), and then go from there to arbitrary motions and higher dimensions. So intelligent snakes might have discovered Fourier analysis way before humans would have, just because they would have been more used to thinking in wavy motions instead of linear motions.

Plissken

So you see, coming up with good principles is really an art form, that requires you to be deeply in touch with your own humanity. Indeed, principle-finding is part of the great art of generating hypotheses. It is a pity that many scientists don’t practice hypothesis generation enough to realise that it is an art (or maybe they don’t practice art enough?) It is also ironic that science tries so hard to eliminate the human element from the theories, when it is so apparent in the final forms of the theories themselves. It is just like an artist who trains so hard to hide her brush strokes, to make the signature of her hand invisible, even though the subject of the painting is her own face.

Ok, now that we know what principles are, how do we find them? One of the best ways is by the age-old method of Induction. How does induction work? It really deserves its own post, but here it is in a nutshell. Let’s say that you are a turkey, and you observe that whenever the farmer makes a whistle, there is some corn in your bowl. So, being a smart turkey, you might decide to elevate this empirical pattern to a general principle, called the Turkey Principle: whenever the farmer whistles, there is corn in your bowl. BOOM, induction!

Now, what is the use of this principle? It helps you to narrow down which theories are good and which are bad. Suppose one day the farmer whistles but you discover there is not corn in the bowl, but rather rice. With your limited turkey imagination, you are able to come up with three hypotheses to explain this. 1. There was corn in the bowl when the farmer whistled, but then somebody came along and replaced it with rice; 2. the Turkey Principle should be amended to the Weak Turkey Principle, which states that when the farmer whistles, food, but not necessarily corn, will be in the bowl; 3. the contents of the bowl are actually independent of the farmer’s whistling, and the apparent link between these phenomena is just a coincidence. Now, with the aid of the Principle, we can see that there is a clear preference for hypothesis 1 over 2, and for 2 over 3, according to the extent that each hypothesis fits with the Turkey Principle.

This example makes it clear that deciding which patterns to upgrade to general principles, and which to regard as anomalies, is again a question of aesthetics and artistry. A more perceptive turkey might observe that the farmer is not a simple mechanistic process, but a complex and mysterious system, and therefore may not be subject to such strong constraints with regards to his whistling and corn-giving behaviour as are implied by the Turkey Principle. Indeed, were the turkey perceptive enough to guess at the farmer’s true motives, he might start checking the tool shed to see if the axe is missing before running to the food bowl every time the farmer whistles. But this turkey would no doubt be working on hypotheses of his own, motivated by principles of his own, such as the Farmer-is-Not-to-be-Trusted Principle (in connection with the observed correlation of turkey disappearances and family dinner parties).

An example more relevant to physics is Einstein’s Equivalence Principle: that no local experiment can determine whether the laboratory is in motion, or is stationary in a gravitational field. The principle is vague, as you can see by the number of variations, interpretations, and Weak and Strong versions that exist in the literature; but undoubtedly it rings true, since it appears to be widely obeyed all but the most esoteric phenomena, and it gels nicely with the Principle of Relativity. While the Equivalence Principle was instrumental in leading to General Relativity, it is a matter of debate how it should be formulated within the theory, and whether or not it is even true. Much like hammers and saws are needed to make a table, but are not needed after the table is complete, we use principles to make theories and then we set them aside when the theory is complete. The final theory makes predictions perfectly well without needing to refer to the principles that built it, and the principles are too vague to make good predictions on their own. (Sure, with enough fiddling around, you can sit on a hammer and eat food off a saw, but it isn’t really comfortable or easy).

For more intellectual reading on principle theories, see the SEP entry on Einstein’s Philosophy of Science, and Poincare’s excellent notes.

Why does matter curve space and time?

This is one of those questions that has always bugged me.
black-hole
Suppose that, somewhere in the universe, there is a very large closed box made out of some kind of heavy, neutral matter. Inside this box a civilisation of intelligent creatures have evolved. They are made out of normal matter like you and me, except that for some reason they are very light — their bodies do not contain much matter at all. What’s more, there are no other heavy bodies or planets inside this large box aside from the population of aliens, whose total mass is too small to have any noticeable effect on the gravitational field. Thus, the only gravitational field that the aliens are aware of is the field created by the box itself (I’m assuming there are no other massive bodies near to the box).

Setting aside the obvious questions about how these aliens came to exist without an energy source like the sun, and where the heck the giant box came from, I want to examine the following question: in principle, is there any way that these aliens could figure out that matter is the source of gravitational fields?

Now, to make it interesting, let us assume the density of the box is not uniform, so there are some parts of its walls that have a stronger gravitational pull than others. Our aliens can walk around on these parts of the walls, and in some parts the aliens even become too heavy to support their own weight and get stuck until someone rescues them. Elsewhere, the walls of the box are low density and so the gravitational attraction to them is very weak. Here, the aliens can easily jump off and float away from the wall. Indeed, the aliens spend much of their time floating freely near the center of the box where the gravitational fields are weak. Apart from that, the composition of the box itself does not change with time and the box is not rotating, so the aliens are quickly able to map out the constant gravitational field that surrounds them inside the box, with its strong and weak points.

Like us, the aliens have developed technology to manipulate the electromagnetic field, and they know that it is the electromagnetic forces that keeps their bodies intact and stops matter from passing through itself. More importantly, they can accelerate objects of different masses by pushing on them, or applying an electric force to charged test bodies, so they quickly discover that matter has inertia, measured by its mass. In this way, they are able to discover Newton’s laws of mechanics. In addition, their experiments with electromagnetism and light eventually lead them to upgrade their picture of space-time, and their Newtonian mechanics is replaced by special relativistic mechanics and Maxwell’s equations for the electromagnetic field.

So far, so good! Except that, because they do not observe any orbiting planets or moving gravitating bodies (their own bodies being too light to produce any noticible attractive forces), they still have not reproduced Newtonian gravity. They know that there is a static field permeating space-time, called the gravitational field, that seems to be fixed to the frame of the box — but they have no reason to think that this gravitational force originates from matter. Indeed, there are two philosophical schools of thought on this. The first group holds that the gravitational field is to be thought of analogously to the electromagnetic field, and is therefore sourced by special “gravitational charges”. It was originally claimed that the material of the box itself carries gravitational charge, but scrapings of the material from the box revealed it to be the same kind of matter from which the aliens themselves were composed (let’s say Carbon) and the scrapings themselves seemed not to produce any gravitational fields, even when collected together in large amounts of several kilograms (a truly humungous weight to the minds of the aliens, whose entire population combined would only weigh ten kilograms). Some aliens pointed out that the gravitational charge of Carbon might be extremely weak, and since the mass of the entire box was likely to be many orders of magnitude larger than anything they had experienced before, it is possible that its cumulative charge would be enough to produce the field. However, these aliens were criticised for making ad-hoc modifications to their theory to avoid its obvious refutation by the kilograms-of-Carbon experiments. If gravity is analogous to the electromagnetic force — they were asked with a sneer — then why should it be so much weaker than electromagnetism? It seemed rather too convenient.

Some people suggested that the true gravitational charge was not Carbon, but some other material that coated the outside of the box. However, these people were derided even more severely than were the Carbon Gravitists (as they had become known). Instead, the popular scientific consensus shifted to a modern idea in which the gravitational force was considered to be a special kind of force field that simply had no source charges. It was a God-given field whose origin and patterns were not to be questioned but simply accepted, much like the very existence of the Great Box itself. This following gained great support when someone made a great discovery: the gravitational force could be regarded as the very geometry of spacetime itself.

The motivation for this was the peculiar observation, long known but never explained, that massive bodies always had the same acceleration in the gravitational field regardless of their different masses. A single alien falling towards one of the gravitating walls of the box would keep speed perfectly with a group of a hundred Aliens tied together, despite their clearly different masses. This dealt a crushing blow to the remnants of the Carbon Gravitists, for it implied that the gravitational charge of matter was exactly proportional to its inertial mass. This coincidence had no precedent in electromagnetism, where it was known that bodies of the same mass could have very different electric charges.

Under the new school of thought, the gravitational force was reinterpreted as the background geometry of space-time inside the box, which specified the inertial trajectories of all massive bodies. Hence, the gravitational force was not a force at all, so it was meaningless to ascribe a “gravitational charge” to matter. Tensor calculus was developed as a natural extension of special relativity, and the aliens derived the geodesic equation describing the motion of matter in a fixed curved space-time metric. The metric of the box was mapped out with high precision, and all questions about the universe seemed to have been settled.

Well, almost all. Some troublesome philosophers continued to insist that there should be some kind of connection between space-time geometry and matter. They wanted more than just the well-known description of how geometry caused matter to move: they tried to argue that matter should also tell space-time how to curve.

“Our entire population combined only weighs a fraction of the mass of the box. What would happen if there were more matter available to us? What if we did the Carbon-kilogram experiment again, but with 100 kilograms? Or a million? Surely the presence of such a large amount of matter would have an effect on space-time itself?”

But these philosophers were just laughed at. Why should any amount of matter affect the eternal and never-changing space-time geometry? Even if the Great Box itself were removed, the prevailing thought was that the gravitational field would remain, fixed as it was in space-time and not to any material source. So they all lived happily ever after, in blissful ignorance of the gravitational constant G, planetary orbits, and other such fantasies.

***

Did you find this fairytale disturbing? I did. It illustrates what I think is an under-appreciated uncomfortable feature of our best theories of gravity: they all take the fact that matter generates gravity as a premise, without justification apart from empirical observation. There’s nothing strictly wrong with this — we do essentially the same thing in special relativity when we take the speed of light to be constant regardless of the motion of its source, historically an empirically determined fact (and one that was found quite surprising).

However, there is a slight difference: one can in principle argue that the speed of light should be reference-frame independent from philosophical grounds, without appealing to empirical observations. Roughly, the relativity principle states that the laws of physics should be the same in all frames of motion, and from among the laws of physics we can include the non-relativistic equations of the electromagnetic field, from which the constant speed of light can be derived from the electric and magnetic constants of the vacuum. As far as I know, there is no similar philosophical grounding for the connection between matter and geometry as embodied by the gravitational constant, and hence no compelling reason for our hypothetical aliens to ever believe that matter is the source of space-time geometry.

Could it be that there is an essential piece missing from our accounts of the connection between matter and space-time? Or are our aliens are doomed by their unfortunately contrived situation, never to deduce the complete laws of the universe?

Skin Deep, by Xetobyte
Image Credit: Xetobyte

 

The Zen of the Quantum Omlette

[Quantum mechanics] is not purely epistemological; it is a peculiar mixture describing in part realities of Nature, in part incomplete human information about Nature, all scrambled up by Heisenberg and Bohr into an omelette that nobody has seen how to unscramble. Yet we think that the unscrambling is a prerequisite for any further advance in basic physical theory. For, if we cannot separate the subjective and objective aspects of the formalism, we cannot know what we are talking about; it is just that simple.” [1]

— E. T. Jaynes

Note: this post is about foundational issues in quantum mechanics, which means it is rather long and may be boring to non-experts (not to mention a number of experts). I’ve tried to use simple language so that the adventurous layman can nevertheless still get the gist of it, if he or she is willing (hey, fortune favours the brave).

As I’ve said before, I think research on the foundations of quantum mechanics is important. One of the main goals of work on foundations (perhaps the main goal) is to find a set of physical principles that can be stated in common language, but can also be implemented mathematically to obtain the model that we call `quantum mechanics’.

Einstein was a big fan of starting with simple intuitive principles on which a more rigorous theory is based. The special and general theories of relativity are excellent examples. Both are based on the `Principle of Relativity’, which states (roughly) that motion between two systems is purely relative. We cannot say whether a given system is truly in motion or not; the only meaningful question is whether the system is moving relative to some other system. There is no absolute background space and time in which objects move or stand still, like actors on a stage. In fact there is no stage at all, only the mutual distances between the actors, as experienced by the actors themselves.

The way I have stated the principle is somewhat vague, but it has a clear philosophical intention which can be taken as inspiration for a more rigorous theory. Of particular interest is the identification of a concept that is argued to be meaningless or illusory — in this case the concept of an object having a well-defined motion independent of other objects. One could arrive at the Principle of Relativity by noticing an apparent conspiracy in the laws of nature, and then invoking the principle as a means of avoiding the conspiracy. If we believe that motion is absolute, then we should find it mighty strange that we can play a game of ping-pong on a speeding train, without getting stuck to the wall. Indeed, if it weren’t for the scenery flying past, how would we know we were traveling at all? And even then, as the phrasing suggests, could we not easily imagine that it is the scenery moving past us while we remain still? Why, then, should Nature take such pains to hide from us the fact that we are in motion? The answer is the Zen of relativity — Nature does not conceal our true motion from us, instead, there is no absolute motion to speak of.

A similar leap is made from the special to the general theory of relativity. If we think of gravity as being a field, just like the electromagnetic field, then we notice a very strange coincidence: the charge of an object in the gravitational field is exactly equal to its inertial mass. By contrast, a particle can have an electric charge completely unrelated to its inertia. Why this peculiar conspiracy between gravitational charge and inertial mass? Because, quoth Einstein, they are the same thing. This is essentially the `Principle of Equivalence’ on which Einstein’s theory of gravity is based.

Einstein

These considerations tell us that to find the deep principles in quantum mechanics, we have to look for seemingly inexplicable coincidences that cry out for explanation. In this post, I’ll discuss one such possibility: the apparent equivalence of two conceptually distinct types of probabilistic behaviour, that due to ignorance and that due to objective uncertainty. The argument runs as follows. Loosely speaking, in classical physics, one does not seem to require any notion of objective randomness or inherent uncertainty. In particular, it is always possible to explain observations using a physical model that is ontologically within the bounds of classical theory and such that all observable properties of a system are determined with certainty. In this sense, any uncertainty arising in classical experiments can always be regarded as our ignorance of the true underlying state of affairs, and we can perfectly well conceive of a hypothetical perfect experiment in which there is no uncertainty about the outcomes.

This is not so easy to maintain in quantum mechanics: any attempt to conceive of an underlying reality without uncertainty seems to result in models of the world that violate dearly-held principles, like the idea that signals cannot propagate faster than light, and experimenters have free will. This has prompted many of us to allow some amount of `objective’ uncertainty into our picture of the world, where even the best conceivable experiments must have some uncertain outcomes. These outcomes are unknowable, even in principle, until the moment that we choose to measure them (and the very act of measurement renders certain other properties unknowable). The presence of these two kinds of randomness in physics — the subjective randomness, which can always be removed by some hypothetical improved experiment, and the objective kind of randomness, which cannot be so removed — leads us into another dilemma, namely, where is the boundary that separates these two kinds of uncertainty?

E.T. Jaynes
“Are you talkin’ to me?”

Now at last we come to the `omelette’ that badass statistician and physicist E.T. Jaynes describes in the opening quote. Since quantum systems are inherently uncertain objects, how do we know how much of that uncertainty is due to our own ignorance, and how much of it is really `inside’ the system itself? Views range from the extreme subjective Bayesian (all uncertainty is ignorance) to various other extremes like the many-worlds interpretation (in which, arguably, the opposite holds: all uncertainty is objective). But a number of researchers, particularly those in the quantum information community, opt for a more Zen-like answer: the reason we can’t tell the difference between objective and subjective probability is that there is no difference. Asking whether the quantum state describes my personal ignorance about something, or whether the state “really is” uncertain, is a meaningless question. But can we take this Zen principle and turn it into something concrete, like the Relativity principle, or are we just by semantics avoiding the problem?

I think there might be something to be gained from taking this idea seriously and seeing where it leads. One way of doing this is to show that the predictions of quantum mechanics can be derived by taking this principle as an axiom. In this paper by Chiribella et. al., the authors use the “Purification postulate”, plus some other axioms, to derive quantum theory. What is the Purification postulate? It states that “the ignorance about a part is always compatible with a maximal knowledge of the whole”. Or, in my own words, the subjective ignorance of one system about another system can always be regarded as the objective uncertainty inherent in the state that encompasses both.

There is an important side comment to make before examining this idea further. You’ll notice that I have not restricted my usage of the word `ignorance’ to human experimenters, but that I take it to apply to any physical system. This idea also appears in relativity, where an “observer in motion” can refer to any object in motion, not necessarily a human. Similarly, I am adopting here the viewpoint of the information theorists, which says that two correlated or interacting systems can be thought of as having information about each other, and the quantification of this knowledge entails that systems — not just people — can be ignorant of each other in some sense. This is important because I think that an overly subjective view of probabilities runs the risk of concealing important physics behind the definition of the `rational agent’, which to me is a rather nebulous concept. I prefer to take the route of Rovelli and make no distinction between agents and generic physical systems. I think this view fits quite naturally with the Purification postulate.

In the paper by Chiribella et. al., the postulate is given a rigorous form and used to derive quantum theory. This alone is not quite enough, but it is, I think, very compelling. To establish the postulate as a physical principle, more work needs to be done on the philosophical side. I will continue to use Rovelli’s relational interpretation of quantum mechanics as an integral part of this philosophy (for a very readable primer, I suggest his FQXi essay).

In the context of this interpretation, the Purification postulate makes more sense. Conceptually, the quantum state does not represent information about a system in isolation, but rather it represents information about a system relative to another system. It is as meaningless to talk about the quantum state of an isolated system as it is to talk about space-time without matter (i.e. Mach’s principle [2]). The only meaningful quantities are relational quantities, and in this spirit we consider the separation of uncertainty into subjective and objective parts to be relational and not fundamental. Can we make this idea more precise? Perhaps we can, by associating subjective and objective uncertainty with some more concrete physical concepts. I’ll probably do that in a follow up post.

I conclude by noting that there are other aspects of quantum theory that cry out for explanation. If hidden variable accounts of quantum mechanics imply elements of reality that move faster than light, why does Nature conspire to prevent us using them for sending signals faster than light? And since the requirement of no faster-than-light signalling still allows correlations that are stronger than entanglement, why does entanglement stop short of that limit? I think there is still a lot that could be done in trying to turn these curious observations into physical principles, and then trying to build models based on them.