Shades of Darker

We'll develop this line of thought later on with more examples but for now let's make a pause to realize what we just analysed. We described the act of performing an experiment as the interaction of one item with the many, as a mechanical interplay of a bigger, blunter agent with a more susceptible one. The study of this simple model with dramatically unorthodox outcomes proved that the idea is more than just a philosophical concept but one of real world significance and, as such, it showed us the importance of understanding the implications of being ourselves a natural phenomenon. Too often do we study the stuff we're made of as a dead substance, subject only to the simple rules we make of its behaviour, without realising it also turns us into mere happenings, general tendencies in the motions of atoms, cultural trends in the behaviour of molecules. Our thoughts, our emotions, everything we trust to set us apart from all that oblivious stuff are like moving shades in the wheat on a windy day, pressed between opposing forces.
As scientists, we know that perception, measurement and decision must be natural phenomena and therefore we can only to a limited extent approximate them by an external agent, independent of the whole system. While that simpler view can help us understand simple things, it can also blind us to the greater implications of our true nature.
To bear this in mind is a crucial ingredient if we are to study the mind of other phenomena. It's an essential sanity aid if we are to accept and explore a world filled with rich subjective experiences, of conscious perception at every corner of existence. In the next section we will describe how concepts, learning, pleasure and suffering come about naturally in many systems.

The curse of the attractive observer

Imagine you have a magnet and you want to measure its orientation. You design an apparatus with two polarizations along a solid rod. At any instant it can be in state + with a probability p and in state - with a probability 1-p. Picture yourself tossing a weighted coin labeled + and - on each side every second. A probability p means after an hour or so, having tossed the coin N times, you will have obtained close to p*N times + and (1-p)*N times  - .
When left alone, the detector relaxes to p = 0.5, i.e., it's positively and negatively polarized an equal number of times during any large enough interval of time. If the polarization is something like a electric tension and we read it from a voltmeter (which measures the average voltage over some short time), we'll view on its screen the value V = (+1)*p + (-1)*(1-p) = 2 * 0.5 -1 = 0.
Let's now say the detector is placed in such a way that it makes an angle x with the direction along which the magnet is pointing. Then p will tend to p = 1 (V = 1) if the detector is aligned with the magnet and to p = 0 (V = -1) if they are anti-aligned, and to p = (cos x +1) / 2 in general, so that

  V = (cos x +1) / 2 - [1 - (cos x +1) / 2] = cos x

which is the component of the magnet along the rod. Thanks to this, our detector allows us to fully measure the spatial orientation of the magnet: you first measure V along one direction, then V along its perpendicular and you combine the results. This figure sums it up:

We have plotted the evolution of V and p over time, so you can track what's going on in the detector as it performs a measurement at an angle of 45º. As you can see, V tends to cos 45º = 0.71 :

Now consider that you're not measuring a big magnet but a tiny one. So tiny, that actually the influence of your detector on the magnet is no longer negligible. Whenever you turn on your detector and it polarizes positively or negatively, the magnet will try to point in the same direction. Then what can we expect from this complicated interaction? Well, if you measure the direction of the magnet long enough so that you can tell its value with certainty (that is, V has reached a stable value) then so will the magnet have aligned itself with your detector. That means only two well defined outcomes are possible: V =1 and V = -1.

We ran a simulation with a very simple set of equations satisfying the properties described above. For each angle we ran the experiment a 1000 times and plotted the percentage of times the detector converged to a positive polarization. Here's our result:

The plotted curve is (cos x +1) / 2 . The result may appear at first as being the same as before but after close examination it is profoundly different. An observer trying to determine the orientation of a large magnet will indeed measure at the end of each experiment a value V that follows the curve above. However, an observer trying to measure a very small magnet will always measure a polarization V = 1 or -1 at the end of each experiment, even though an average of V over many experiments performed in the same conditions will result in the drawn curve. A very important consequence is that if you try to determine the spatial direction of your magnet by doing two measurements along two perpendicular directions you won't come to any conclusion at all, since the first measurement will ruin it for the second one.

A subject using such a tool to perceive the world around him will come to the puzzling conclusion that microscopic magnets have an inherent randomness to them, and that they can only be in two possible states. It is a curious fact that the peculiar properties described above happen to be exactly the ones found when you try to measure the spin of an electron. In this case, the weird quantum mechanical behaviour of particles can be fully understood using this classic analog we just studied. One might even wonder if the microscopic behaviour we see in electrons has something to do with the limits of how very delicate things can create discernible structures in very bulky ones.

The limits of reality

Everyone loves to tweet that one sentence that's going to be retweeted hundreds of times. That's why facebook keeps track of likes and why we stare at that little like counter whenever we post something valuable, and wait for it to boil. But unless you're some kind of celebrity, or a hot girl, chances are your stuff won't cause much stir. And the humility this experience teaches us should make us wonder about the limits of our own perception.

Electrons interact with each other and with protons through little tweets we call photons. Light, turns out, is simply a large number of tweets between a large number of electrons. They can be seen with a photomultiplier. When one free electron tweets with a specific energy another electron in the photomultiplier, the latter will retweet the tweet to his neighbours and so on, creating a cascade of interactions and a movement in the trillionfold electron population strong enough to generate a macroscopic current. If you think of it, this little electron's tweet is a tweet only akin to Snowden publishing his data on internet surveillance, only akin to Russell Brand tweeting his latest outrageous say or to a syrian journalist posting pictures of a chemical massacre. It is the tweet that starts a revolution.

Knowing just how difficult and rare it is to tweet anything popular, one might sympathize, feel proud and even awe at this electron's achievement. It could also lead us to question our own experience. Can anything in the microscopic world create macroscopic structures, signals in our detectors and memories in our brains? Next comes a thought experiment which draws stringent limits on reality at the large scale and exhibit behaviour that reminds of quantum mechanics.

The word of a century

    His body shook with an excitement he perceived as revolt. The little facts that had brought him to his present situation no longer fiddled with each other and were now blatantly yelling in his mind. A clashing of thoughts at the root of his pain that was starting to show the dynamic enthusiasm of troops lining up for battle. Soon they would agree. Soon they would drive his body to action and soon the crowds would cascade under the knowledge still known to them as silence, and justice would flow in a rage of turmoil. Then he paused. And said:

Big thermometers (clarification)

Here is a mathematical description of the temperature measurement experiment described in the previous post:

On the distinction of subject and object

We stated in the last post how artificial it is to consider the observer and the observed object as two separate entities, but so far it sounds more like a philosophical statement tailored for an LSD trip than an interesting fact to be explored. We will now present a few examples to show just how important it can be to consider it. Some parts of this post might be a bit mathematical, but we will try to juice out the most important features and leave the details aside.

Consider a very simple measurement: the temperature of a body. The simplest tool for it is a mercury-in-glass thermometer. When the tip of the thermometer is put in contact with a hot body, heat from the body is transfered to the mercury which expands inside the bulb of glass. Once thermal equilibrium is reached the column of mercury will have a stable length. The Celsius perscription to calibrate a thermometer is to dip the tip in ice and make a mark on it indicating 0ºC and then dip it in boiling water and mark 100ºC where the mercury column stabilizes. In between draw a hundred marks which correspond to a Celsius degree each. Then you just have to place the tip on the body whose temperature you wish to measure, wait for the end of the mercury column to stop moving and write down the temperature of the corresponding mark.
This procedure works because the volume of mercury is very small and therefore the heat released by a large body will generate a large change in the length of the column, whereas the heat poured into the body by the thermometer will be irrelevant.
Let's however imagine we are studying the temperature of a collection of delicate objects, of very small volume themselves. Their temperatures are uniformly distributed between 0ºC and 100ºC. That means there are just as many bodies between 1ºC and 2ºC as between 78ºC and 79ºC; in fact, between any two temperatures separated by the same amount.

If these bodies were heavy, performing 10 000 measurements of the temperature of these bodies would yield a histogram corresponding exactly to that distribution. But as the mass of the bodies gets smaller and smaller, the distribution of the measured temperatures shrinks (because the temperature of the body and of the thermometer will try to reach some middle ground), until it reaches a single value when the thermometer is much heavier than the bodies, which is the temperature of the thermometer. For this extreme case, picture yourself measuring the temperature of a small drop of water with a big thermometer: the drop of water heats up to the temperature of the thermometer, not the opposite.

This means a macroscopic observer who uses this thermometer to describe the microscopic world around him will see a very different world from another microscopic observer with a better (i.e., smaller) thermometer. The macroscopic observer cannot measure the temperature of small things without changing it and therefore will come to the conclusion that all these bodies are at temperatures which lie in a small range, which is true, but only because he looked. His knowledge is constrained by physical limits which have nothing to do with the precision of his apparatus, which in theory may be infinitely precise. The constraint lies in the fact that his perception is an interaction with the object, a logical loop which he's forced to step into: if he doesn't look he can't perceive the object, if he does he can only perceive the object in a limited range of states.
Of course, such small objects are short lived in our world for you to test this conclusion at home. Since we're all immersed in an atmospheric thermal bath, any small object reaches thermal equilibrium with the air long before you get any chance to measure its temperature with your oversized thermometer.
Which is why in the next post we will present a much more interesting system where the effects of a measurement are much less artificial and far more intriguing.