10.7 Thought Experiments
Once hypotheses have been generated it is time to start to evaluate and test them. The final test is often an actual experiment but many ideas can be evaluated and refuted just by thinking about them. Thought experiments are means for exploring the logical structure of a problem and can be very powerful tools for testing ideas. They use imaginary scenarios to understand features of the real world.
We often think that we are unable to make statements about a certain problem if we do not have basic knowledge of the specific problem area. For example, you may not think that you are able to find out what is wrong in the central heating system in your house if you are not a plumber. As a matter of fact we all have general knowledge about the world, both from experience and theoretical knowledge. Based on this we can exclude possibilities, since we also know how the world does not behave. You intuitively understand that pipes transporting water to the radiators of your heating system should be warm, for instance, and that pipes leading away from them should be slightly less warm. Such knowledge can be used for troubleshooting even without detailed knowledge of the internal workings of heat pumps, furnaces or what have you to heat your house. In the same manner, your general knowledge can be used for troubleshooting theoretical ideas. If an idea inevitably leads to a paradox, there is strong reason to believe that the idea is wrong. The cause-and-effect table is a very simple example of trying to foresee the consequences of ideas by reasoning. We have also seen more complex examples of elegant thought experiments in this book.
In Chapter 4 we saw how Galileo refuted Aristotle's idea that heavy objects fall faster than light ones in free fall using nothing but logical arguments. Say that a light object falls at a speed of four units, he said, and that a heavy object falls at a speed of eight. By attaching the two objects to each other, the heavy object will be retarded by the light one, and the light object will be sped up by the heavier. This means that the two objects together, though heavier than the parts, will fall at a speed less than eight. In other words, Aristotle's claim is not logically consistent. Or, as a sports commentator might put it: “Galileo–Aristotle, 1–0”.
Another thought experiment is described in Example 6.1, where William Harvey tested Galen's claim that blood is continuously formed from food in the liver and consumed in the limbs and organs of the body. Even a conservative estimate of the rate at which the heart pumps blood into the body would amount to far greater quantities than could possibly be provided by eating in a day. He also noted that a butcher could empty an animal of blood in less than fifteen minutes, which was logically inconsistent with the antique idea.
Thought experiments often employ specific scenarios or perspectives to test an idea. They can involve simple quantitative reasoning, as in the examples above, to see if a general statement applies to a specific situation. If the idea can be expressed in mathematical form it can also be tested using order-of-magnitude estimates, where powers of ten are put in the place of important variables to see if the resulting magnitude is consistent with the claim. This sort of estimation is often called a back-of-an-envelope calculation, a term that goes back to the physicist Enrico Fermi who was known for his ability to make good approximations using little or no data. They often involve guesstimates, a term coined by statisticians to describe that a quantity is very roughly approximated or even guessed for lack of accurate information. It is of course important to state the assumptions and approximations behind all estimates. Another common method is to use extremes: letting variables in a mathematical expression approach zero or infinity to see if paradoxes occur. The basic idea is that general claims must apply to specific situations. If they do not, they are not generally valid and may be discarded.
Let us look at how back-of-an-envelope calculations can be used in thought experiments. Most of us were able to refute the Santa Claus hypothesis at quite an early age. It claims that an outsized elf visits some hundred million homes in the western world on December 24–25 every year. It is a simple mathematical exercise to work out that, even if he spent only one second in each home, he would need several years to complete this trip. Without knowing anything about reindeer propulsion systems, the origin of elves or other such details, we are able to refute the statement on purely logical grounds. As Carl Sagan [7] put it, one second per house would at least explain why no one ever sees him much, but even with relativistic reindeer (travelling at the speed of light) the job is impossible. There is another, similar hypothesis that is held to be true by some people. It states that spacecraft from alien civilizations regularly come from outer space to visit the earth. Can we say anything about this claim? After all, we know very little about alien technology. Sagan [7] applies the Santa Claus analysis to refute this statement.
Example 10.1: Has the earth already been visited? During some periods there have been many reports of UFOs, so let us assume that we have at least one alien visit on earth every year. Though it is difficult, we could at least try to estimate the current number of advanced technical civilizations in our galaxy. It is usually done using a relationship called the Drake equation. This is a product of a series of numbers and probabilities; for example, the rate at which stars are formed in our galaxy, which is known quite well, the fraction of those stars that have planets, the fraction of planets that are suitable for life, and so on. Needless to say, the estimates become less certain the farther we go into the equation but, without entering into details, a very optimistic estimate of the number of civilizations is 106. Now, let us try to calculate how many interstellar spacecraft these civilizations must launch, on average, for one to end up on earth every year.
Say that each civilization sets out on Q interstellar expeditions per year and that each of them reaches one destination. This amounts to Q × 106 visits per year, somewhere in the galaxy. Since there are several times 1011 stars in the galaxy, there should be at least 1010 interesting places to visit. So, in order for one visit a year at any interesting place, such as the earth, we need Q = 10 000 launches per civilization every year. This seems excessive, especially considering the resources needed for such expeditions. Sagan mentions an estimate of the material needed for the spacecraft – they have to be larger than the US Apollo space capsule, let us say – which requires 1% of the stars in the galaxy to be mined.
A possible counter-argument is that we are the object of special attention. Maybe alien anthropologists think that we have just reached a very interesting stage in our technical development? But to imagine that our development is so fascinating is precisely contrary to the idea that there are so many civilizations out there. If there are, the development of technical civilizations must be quite common. If we are looking for an explanation for UFOs, alien visits is probably not the most promising hypothesis.
Apart from the order-of-magnitude analysis, this thought experiment builds on a useful analogy (comparing the problem to visits by Santa Claus). It also explores the problem from an alternative perspective. The “alien visits” hypothesis uses our perspective, assuming that we are unique enough to be worth visiting. The thought experiment takes the perspective of an alien civilization. They would have to know that we are here in order to visit us. How is that possible without regularly visiting promising worlds?
It should be noted that thought experiments are always based on current knowledge. If we overlook important possibilities, the validity of our conclusions may be affected. Interestingly, Sagan later wrote a novel where an alien civilization contacts us. To make it possible for them to find us, he envisaged a galactic radio surveillance system that could detect our civilization as soon as we started broadcasting television. It was connected throughout the galaxy in a sort of subway system of relativistic wormholes. Assuming the possibility of such a system, the conclusion of his thought experiment may be a different one.
The examples of thought experiments given here are perhaps more elegant than average and we ordinary researchers may think that it is futile to even try to apply the technique. It really is not. The key is to take a playful attitude to the research problems that we encounter. Conceptualize and visualize the claims that you analyze, explore them using different perspectives and scenarios, and ask critical questions. Does the claim lead to unreasonable results under some conditions? What would be required for the claimed effect to occur? As with any other technique, proficiency requires application, so here are a couple of exercises for practice.