Brown, J. R. (1991). The laboratory of mind: Thought experiments in natural sciences. London/New York: Routledge.
Illustrations from the laboratory of mind. This is the title of Brown’s (1991) first chapter. In the preface, and again in the introduction to this chapter, Brown acknowledges – in an explicit manner, which I fancy – the fact that he hasn’t got a definition to work with.
Thought experiments are performed in the laboratory of the mind. Beyond that bit of metaphor it’s hard to say just what they are. We recognize them when we see them: they are visualizable; they involve mental manipulations; they are not the mere consequence of a theory-based calculation; they are often (but not always) impossible to implement as real experiments either because we lack the relevant technology or because they are simply impossible in principle. If we are ever lucky enough to come up with a sharp definition of thought experiment, it is likely to be at the end of a long investigation.
But as he advances, he adds some spice. First he brackets out what are not thought experiments, that is, psychological experiments on thought or other inky stuff. Then he explains the “experiment” component as follows:
As well as being sensory [i.e. they are visualizable or experiencable], thought experiments are like real experiments in that something often gets manipulated: the balls are joined together, the links are extended and joined under the inclined plane, the observer runs to catch up to the front of the light beam. (Brown, 1991, p. 17)
The thought experimenter does two things: imagining and manipulating.
1. Galileo on falling bodies
Galileo first notes Aristotle’s view that heavier bodies fall faster than light ones. This was the prevailing view of motion and it had the advantage of being very close to one’s intuition: if a feather falls on your toes, you’re fine. But Galileo reasoned as follows. Suppose we tie two objects together: a cannon ball and a musket ball.
First, the light ball will slow up the heavy one (acting as a kind of drag), so the speed of the combined system would be slower than the speed of the heavy ball falling alone (H > H+L).
This might sound fishy to begin with, but imagine someone hopping in a fast-moving carriage. In order for the whole system to keep the speed, the poor horses need more strength, therefore, with the same strength put into action they will be slowed down.
On the other hand, the combined system is heavier than the heavy ball alone, so it should fall faster (H+L > H).
This does not sound fishy at all. It is heavier, and it should fall faster – so intuition tells us. But, of course, this is just half of the story, because Galileo does not only expose the paradox of Aristotelian mechanics, but resolves it. The right equation, he says, is that H = L = H+L. That is, all things fall at the same speed. (Namely, some will add, gravitational speed).
2. Stevin on inclined planes
So we have three self-evident situations: if a weight is on an horizontal plane, it will remain at rest; if a weight is on a vertical plane, it will fall; and if the weight is on an inclined plane, it will either fall or rest depending on the inclination. Great, but how could we get more precise about these things? If we could calculate the point of echilibrium, that would be great: every smaller inclination of the plane means rest every greater inclination means fall.
Stevin come up with this thing:
From the first figure, we wouldn’t know what to say: are the balls in an equilibrium? From the second, however, we are compelled to say: they must be, otherwise it would be perpetual motion (and perpetual motion is an absurdity, as far as we can know). Therefore,
when we have inclined planes of equal height then equal weights will act inversely proportional to the lengths of the planes
Swell.
3. The flat planet with shrinking people
Now, Brown makes an interesting claim, which can only be understood under a very broad definition of the term “thought experiment”: “Consequently, [i.e. since Euclidean geometry is an a priori abstraction from everyday mechanics]we can see the results of Euclidean geometry (at least those produced before the rise of non-Euclidean geometry) as a vast collection of thought experiments” (p. 11). And further, “theorem of Euclidean geometry is then a kind of report of an actual construction carried out in the imagination.”
The thought experiment related to Euclidean geometry is one devised by H. Poincaré and H. Reichenbach. (The text is from Cohen, 2005, p. 45):
Imagine a planet made only of gases. At the centre the temperature is very high, and this is where all the gaseous people evolved and normally live. At the surface, however, the temperature is very, very low. In fact, M. Poincaré tells us, it is absolute zero. (The significance of this will become clear later.) As the gaseous people, let us call them ‘the Jeometers’, move around their planet, a small but subtle change takes place. Because of the change in temperature, the further they go from the centre, the smaller they become. And not just them, the smaller all the creatures and all the artefacts of the gaseous planet become. The most important thing is that everything changes at exactly the same rate, so nothing gets out of kilter.
One year, the Jeometers determine they must explore the upper reaches of their planet and construct a massive ladder which they stand upright with its top disappearing far into the clouds. One of the Jeometers’ geometers sets off up it, with the task of finding out how far the gaseous planet extends. There is great excitement, but it is dissipated somewhat when the geometer returns a few days later to say the ladder is nowhere near long enough.
For years and years sections of ladder are added, but it seems it is in vain. Each time the geometers return to say that the ladder is still not long enough.
Actually, as they ascend the ladder, both the Jeometers and the ladder itself are shrinking, shrinking so small that it is physically impossible for them to ever get to the outer surface. (At absolute zero, they will shrink to absolutely nothing.) Yet as they climb up, becoming colder and colder and at the same time smaller and smaller, the steps on the ladder, their measuring rods – everything – are also getting smaller and smaller, so they never realize the shrinking is happening. Eventually, the Jeometers decide their planet is infinitely large. Which it isn’t.
The substance of this thought experiment is to show that there is always a possibility to the effect that whatever geometry we invent, Euclidean or not, it may always fail to be the true one. In fact, “truth of geometry” is a strange expression since the statements of geometry are a matter of convention.
The geometrical axioms are therefore neither synthetic a priori intuitions nor experimental facts. They are conventions…. In other words, the axioms of geometry… are only definitions in disguise. What then are we to think of the question: Is Euclidean geometry true? It has no meaning. (Poincare, 1952)
4. Einstein vs. Maxwell
Classic electrodynamics told people this: light is a change (“oscillation”) in the electromagnetic field; and it works both ways. A change in the electric field gives rise to a magnetic field and a change in the magnetic field gives rise to an electric field. In response to this theory, (when, by the way, he was only sixteen supposedly…), Einstein thought of chasing a beam of light (oh, puberty…). The point of this chase – or “running along” – is to pinpoint impossibility: if change is essential for a light wave, then it should be so for any observer, be it a moving or at rest; but if you are moving with the exact speed c, then with respect to you, change is not coming about. Is like chasing a wave in the ocean. If your speeds are equal, then the hump of water does not change relative to you.
If I pursue a beam of light with the velocity c (velocity of light in a vacuum), I should observe such a beam of light as a spatially oscillatory electromagnetic field at rest. However, there seems to be no such thing, whether on the basis of experience or according to Maxwell’s equations.
Brilliant! (Quite literally.)
5. Heisenberg γ-ray microscope
The uncertainty principle was very much debated in the mid-twentieth century. Many philosophers knock-knock-knocking on physics door were appalled and tried their best to disprove its (a) validity, (b) importance.
For us now, it is strangely important in our taxonomy. As it happens, the thought experiment connected to this principle is not designed to refute any theory and it is subsequent to the principle itself. Could it be that it is a case of illustrative thought experiment – in Popper’s terminology?
From the first principles of quantum theory, Mr. Werner Heisenberg formally derived the principle that the product of uncertainties in present knowledge of a system is always greater or equal to a certain constant (known as Planck’s constant). “This may be expressed in concise and general terms”, Heisenberg adds, “by saying that every experiment destroys some of the knowledge of the system which was obtained by previous experiments.”
And then comes the thought experiment. Notice the new flavour: the principle had already been formally arrived at! Heisenberg himself calls this thought experiment an “example”.
As a first example of the destruction of the knowledge of a particle’s momentum by an apparatus determining its position, we consider the use of a microscope. Let the particle be moving at such a distance from the microscope that the cone of rays scattered from it through the objective has an angular opening ε. […] But, for any measurement to be possible at least one photon must be scattered from the electron and pass through the microscope to the eye of the observer. From this photon the electron receives a Compton recoil of order of magnitude h/λ. The recoil cannot be exactly known, since the direction of the scattered photon is undetermined within the bundle of rays entering the microscope. Thus there is an uncertainty of the recoil in the x-direction of amount…
Wonderful!
6. Schrodinger’s cat
Still indoors with quantum mechanics. The Copenhagen interpretation of the formalisms of quantum mechanics did not reject one (bizarre) possibility: namely that of a system being at two (superposed) states at the same time. Since we can only observe one – in other words, since for us a system is either one way or the other –this possibility remain only a theoretical one. On a philosophical level, sure, it said many things about our place in the universe: that we “create” reality, we restrict it to only one of its states, that we are doomed with only partial knowledge of the world, etc. But it remained a theoretical option nonetheless. This theoretical option however bugged many people, amongst whom Einstein of course, but als Mr. Schrödinger.
One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following diabolical device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of one hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay would have poisoned it. The ψ-function of the entire system would express this by having in it the living and the dead cat (pardon the expression) mixed or smeared out in equal parts.
This thought experiment seems no different from the first, classic ones. It thinks up a possible state of affairs which supposedly (a) was overlooked by quantum physicists and (b) gives rise to absurdity. It’s difficult to say here whether the absurdity is a matter of logical contradiction or otherwise “empirical”-ish. (But it’s the same with Galileo’s balls. It is mainly because of our language that we think of an object falling “both slower and faster” than another one as being a contradiction. It is, just the same, mainly because of our language that we think of a living object being “both dead and alive” as being a contradiction).
The people considered in philosophical thought experiments can get very weird: we are asked to imagine people splitting like amoebas, fusing like clouds, and so on. Stevin’s frictionless plane, or Einstein chasing a light beam are homely by comparison.
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