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.