Thought Experiment: How Einstein Solved Difficult Problems

Thought experiments are a classic tool used by many great thinkers, which enable us to explore often impossible situations and predict their implications and outcomes. Mastering thought experiments can help you to stretch your mind by confronting difficult questions.

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The Basics of Thought Experiment

Imagine a small town with a hard-working barber. The barber shaves everyone in the town who does not shave themselves. He does not shave anyone who shaves themselves. So, who shaves the barber?

The ‘impossible barber’ is one classic example of a thought experiment — a means of exploring a concept, hypothesis or idea through extensive thought. When finding empirical evidence is impossible, we turn to thought experiments to unspool complex concepts.

In the case of the impossible barber, setting up an experiment to figure out who shaves him would not be feasible or even desirable. After all, the barber cannot exist. Thought experiments are usually rhetorical. No particular answer can or should be found.

The purpose is to encourage speculation, logical thinking and to change paradigms. Thought experiments push us outside our comfort zone by forcing us to confront questions we cannot answer with ease. They reveal that we do not know everything and some things cannot be known.

“All truly wise thoughts have been thought already thousands of times; but to make them truly ours, we must think them over again honestly, until they take root in our personal experience.” — Johann Wolfgang von Goethe

In a paper entitled Thought Experimentation of Presocratic Philosophy, Nicholas Rescher writes:

Homo sapiens is an amphibian who can live and function in two very different realms- the domain of actual facts which we can investigate in observational inquiry, and the domain of the imaginative projection which we can explore in thought through reasoning…A thought experiment is an attempt to draw instruction from a process of hypothetical reasoning that proceeding by eliciting the consequences of a hypothesis which, for anything that one actually knows to the contrary, may be false. It consists in reasoning from a supposition that is not accepted as true- perhaps even known to be false but is assumed provisionally in the interests of making a point or resolving a conclusion.

As we know from the narrative fallacy, complex information is best digested in the form of narratives and analogies. Many thought experiments make use of this fact to make them more accessible. Even those who are not knowledgeable about a particular field can build an understanding through thought experiments. The aim is to condense first principles into a form which can be understood through analysis and reflection. Some incorporate empirical evidence, looking at it from an alternative perspective.

The benefit of thought experiments (as opposed to aimless rumination) is their structure. In an organized manner, thought experiments allow us to challenge intellectual norms, move beyond the boundaries of ingrained facts, comprehend history, make logical decisions, foster innovative ideas, and widen our sphere of reference.

Despite being improbable or impractical, thought experiments should be possible, in theory.

The History of Thought Experiments

Thought experiments have a rich and complex history, stretching back to the ancient Greeks and Romans. As a mental model, they have enriched many of our greatest intellectual advances, from philosophy to quantum mechanics.

An early example of a thought experiment is Zeno’s narrative of Achilles and the tortoise, dating to around 430 BC. Zeno’s thought experiments aimed to deduce first principles through the elimination of untrue concepts.

In one instance, the Greek philosopher used it to ‘prove’ motion is an illusion. Known as the dichotomy paradox, it involves Achilles racing a tortoise. Out of generosity, Achilles gives the tortoise a 100m head start. Once Achilles begins running, he soon catches up on the head start. However, by that point, the tortoise has moved another 10m. By the time he catches up again, the tortoise will have moved further. Zeno claimed Achilles could never win the race as the distance between the pair would constantly increase.

In the 17th century, Galileo further developed the concept by using thought experiments to affirm his theories. One example is his thought experiment involving two balls (one heavy, one light) which are dropped from the Leaning Tower of Pisa. Prior philosophers had theorized the heavy ball would land first. Galileo claimed this was untrue, as mass does not influence acceleration. We will look at Galileo’s thought experiments in more detail later on in this post.

In 1814, Pierre Laplace explored determinism through ‘Laplace’s demon.’ This is a theoretical ‘demon’ which has an acute awareness of the location and movement of every single particle in existence. Would Laplace’s demon know the future? If the answer is yes, the universe must be linear and deterministic. If no, the universe is nonlinear and free will exists.

In 1897, the German term ‘Gedankenexperiment’ passed into English and a cohesive picture of how thought experiments are used worldwide began to form.

Albert Einstein used thought experiments for some of his most important discoveries. The most famous of this thought experiments was on a beam of light, which was made into a brilliant children’s book. What would happen if you could catch up to a beam of light as it moved he asked himself? The answers led him down a different path toward time, which led to the special theory of relativity.

In On Thought Experiments, 19th-century Philosopher and physicist Ernst Mach writes that curiosity is an inherent human quality. We see this in babies, as they test the world around them and learn the principle of cause and effect. With time, our exploration of the world becomes more and more in depth. We reach a point where we can no longer experiment through our hands alone. At that point, we move into the realm of thought experiments.

Thought experiments are a structured manifestation of our natural curiosity about the world.

Mach writes:

Our own ideas are more easily and readily at our disposal than physical facts. We experiment with thought, so as to say, at little expense. This it shouldn’t surprise us that, oftentime, the thought experiment precedes the physical experiment and prepares the way for it… A thought experiment is also a necessary precondition for a physical experiment. Every inventor and every experimenter must have in his mind the detailed order before he actualizes it. Even if Stephenson knew the train, the rails and the steam engine from experience, he must have, nonetheless, have preconceived in his thoughts the combination of a train on wheels, driven by a steam engine, before he could have proceeded to the realization. No less did Galileo have to envisage, in his imagination, the arrangements for the investigation of gravity, before these were actualized. Even the beginner learns in experimenting than as insufficient preliminary estimate, or nonobservance of sources of error has for him no less tragic comic results than the proverbial ‘look before you leap’ does in practical life.

Mach compares thought experiments to the plans and images we form in our minds before commencing an endeavor. We all do this — rehearsing a conversation before having it, planning a piece of work before starting it, figuring out every detail of a meal before cooking it. Mach views this as an integral part of our ability to engage in complex tasks and to innovate creatively.

According to Mach, the results of some thought experiments can be so certain that it is unnecessary to physically perform it. Regardless of the accuracy of the result, the desired purpose has been achieved.

We will look at some key examples of thought experiments throughout this post, which will show why Mach’s words are so important. He adds:

It can be seen that the basic method of the thought experiment is just like that of a physical experiment, namely, the method of variation. By varying the circumstances (continuously, if possible) the range of validity of an idea (expectation) related to these circumstances is increased.

Although some people view thought experiments as pseudo-science, Mach saw them as valid and important for experimentation.

“Can’t you give me brains?” asked the Scarecrow.

You do not need them. You are learning something every day. A baby has brains, but it does not know much. Experience is the only thing that brings knowledge, and the longer you are on earth the more experience you are sure to get.” — L. Frank Baum, The Wonderful Wizard of Oz

Types of Thought Experiment

Several key types of thought experiment have been identified:

Prefactual – Involving potential future outcomes. E.g. ‘What will X cause to happen?’

– Involving potential future outcomes. E.g. ‘What will X cause to happen?’ Counterfactual – Contradicting known facts. E.g. ‘If Y happened instead of X, what would be the outcome?’

– Contradicting known facts. E.g. ‘If Y happened instead of X, what would be the outcome?’ Semi-factual – Contemplating how a different past could have lead to the same present. E.g. ‘If Y had happened instead of X, would the outcome be the same?’

– Contemplating how a different past could have lead to the same present. E.g. ‘If Y had happened instead of X, would the outcome be the same?’ Prediction – Theorising future outcomes based on existing data. Predictions may involve mental or computational models. E.g. ‘If X continues to happen, what will the outcome be in one year?’

– Theorising future outcomes based on existing data. Predictions may involve mental or computational models. E.g. ‘If X continues to happen, what will the outcome be in one year?’ Hindcasting – Running a prediction in reverse to see if it forecasts an event which has already happened. E.g. ‘X happened, could Y have predicted it?’

– Running a prediction in reverse to see if it forecasts an event which has already happened. E.g. ‘X happened, could Y have predicted it?’ Retrodiction – Moving backwards from an event to discover the root cause. Retrodiction is often used for problem solving and prevention purposes. E.g. ‘What caused X? How can we prevent it from happening again?’

– Moving backwards from an event to discover the root cause. Retrodiction is often used for problem solving and prevention purposes. E.g. ‘What caused X? How can we prevent it from happening again?’ Backcasting – Considering a specific future outcome, then working forwards from the present to deduce its causes. E.g. ‘If X happens in one year, what would have caused it?’

“With our limited senses and consciousness, we only glimpse a small portion of reality. Furthermore, everything in the universe is in a state of constant flux. Simple words and thoughts cannot capture this flux or complexity. The only solution for an enlightened person is to let the mind absorb itself in what it experiences, without having to form a judgment on what it all means. The mind must be able to feel doubt and uncertainty for as long as possible. As it remains in this state and probes deeply into the mysteries of the universe, ideas will come that are more dimensional and real than if we had jumped to conclusions and formed judgments early on.” — Robert Greene, Mastery

Thought Experiments in Philosophy

Thoughts experiments have been an integral part of philosophy since ancient times. This is in part due to philosophical hypotheses often being subjective and impossible to prove through empirical evidence.

Philosophers use thought experiments to convey theories in an accessible manner. With the aim of illustrating a particular concept (such as free will or mortality), philosophers explore imagined scenarios. The goal is not to uncover a ‘correct’ answer, but to spark new ideas.

An early example of a philosophical thought experiment is Plato’s Allegory of the Cave, which centers around a dialogue between Socrates and Glaucon (Plato’s brother.)

A group of people are born and live within a dark cave. Having spent their entire lives seeing nothing but shadows on the wall, they lack a conception of the world outside. Knowing nothing different, they do not even wish to leave the cave. At some point, they are lead outside and see a world consisting of much more than shadows.

“The frog in the well knows nothing of the mighty ocean.” — Japanese Proverb

Plato used this to illustrate the incomplete view of reality most of us have. Only by learning philosophy, Plato claimed, can we see more than shadows.

Upon leaving the cave, the people realize the outside world is far more interesting and fulfilling. If a solitary person left, they would want others to do the same. However, if they return to the cave, their old life will seem unsatisfactory. This discomfort would become misplaced, leading them to resent the outside world. Plato used this to convey his (almost compulsively) deep appreciation for the power of educating ourselves. To take up the mantle of your own education and begin seeking to understand the world is the first step on the way out of the cave.

Moving from caves to insects, let’s take a look at a fascinating thought experiment from 20th-century philosopher Ludwig Wittgenstein. Imagine

Imagine a world where each person has a beetle in a box. In this world, the only time anyone can see a beetle is when they look in their own box. As a consequence, the conception of a beetle each individual has is based on their own. It could be that everyone has something different, or that the boxes are empty, or even that the contents are amorphous.

Wittgenstein uses the ‘Beetle in a Box’ thought experiment to convey his work on the subjective nature of pain. We can each only know what pain is to us, and we cannot feel another person’s agony. If people in the hypothetical world were to have a discussion on the topic of beetles, each would only be able to share their individual perspective. The conversation would have little purpose because each person can only convey what they see as a beetle. In the same way, it is useless for us to describe our pain using analogies (‘it feels like a red hot poker is stabbing me in the back’) or scales (‘the pain is 7/10.’)

Thought Experiments in Science

Although empirical evidence is usually necessary for science, thought experiments may be used to develop a hypothesis or to prepare for experimentation. Some hypotheses cannot be tested (e.g string theory) – at least, not given our current capabilities.

Theoretical scientists may turn to thought experiments to develop a provisional answer, often informed by Occam’s razor.

Nicholas Rescher writes:

In natural science, thought experiments are common. Think, for example, of Einstein’s pondering the question of what the world would look like if one were to travel along a ray of light. Think too of physicists’ assumption of a frictionlessly rolling body or the economists’ assumption of a perfectly efficient market in the interests of establishing the laws of descent or the principles of exchange, respectively…Ernst Mach [mentioned in the introduction] made the sound point that any sensibly designed real experiment should be preceded by a thought experiment that anticipates at any rate the possibility of its outcome.

In a paper entitled Thought Experiments in Scientific Reasoning, Andrew D. Irvine explains that thought experiments are a key part of science. They are in the same realm as physical experiments. Thought experiments require all assumptions to be supported by empirical evidence. The context must be believable, and it must provide useful answers to complex questions. A thought experiment must have the potential to be falsified.

Irvine writes:

Just as a physical experiment often has repercussions for its background theory in terms of confirmation, falsification or the like, so too will a thought experiment. Of course, the parallel is not exact; thought experiments…no do not include actual interventions within the physical environment.

In Do All Rational Folks Think As We Do? Barbara D. Massey writes:

Often critique of thought experiments demands the fleshing out or concretizing of descriptions so that what would happen in a given situation becomes less a matter of guesswork or pontification. In thought experiments we tend to elaborate descriptions with the latest scientific models in mind…The thought experiment seems to be a close relative of the scientist’s laboratory experiment with the vital difference that observations may be made from perspectives which are in reality impossible, for example, from the perspective of moving at the speed of light…The thought experiment seems to discover facts about how things work within the laboratory of the mind.

One key example of a scientific thought experiment is Schrodinger’s cat.

Developed in 1935 by Edwin Schrodinger, Schrodinger’s cat seeks to illustrate the counterintuitive nature of quantum mechanics in a more understandable manner.

Although difficult to present in a simplified manner, the idea is that of a cat which is neither alive nor dead, encased within a box. Inside the box is a Geiger counter and a small quantity of decaying radioactive material. The amount of radioactive material is small, and over a period time, it is equally probable it will decay or not. If it does decay, a tube of acid will smash and poison the cat. Without opening the box, it is impossible to know if the cat is alive or not.

Let’s ignore the ethical implications and the fact that, if this were performed, the angry meowing of the cat would be a clue. Like most thought experiments, the details are arbitrary – it is irrelevant what animal it is, what kills it, or the time frame.

Schrodinger’s point was that quantum mechanics are indeterminate. When does a quantum system switch from one state to a different one? Can the cat be both alive and dead, and is that conditional on it being observed? What about the cat’s own observation of itself?

In Search of Schrodinger’s Cat, John Gribbin writes:

Nothing is real unless it is observed…there is no underlying reality to the world. “Reality,” in the everyday sense, is not a good way to think about the behavior of the fundamental particles that make up the universe; yet at the same time those particles seem to be inseparably connected into some invisible whole, each aware of what happens to the others.

Schrodinger himself wrote in Nature and The Greeks:

We do not belong to this material world that science constructs for us. We are not in it; we are outside. We are only spectators. The reason why we believe that we are in it that we belong to the picture, is that our bodies are in the picture. Our bodies belong to it. Not only my own body, but those of my friends, also of my dog and cat and horse, and of all the other people and animals. And this is my only means of communicating with them.

Another important early example of a scientific thought experiment is Galileo’s Leaning Tower of Pisa Experiment.

Galileo sought to disprove the prevailing belief that gravity is influenced by the mass of an object. Since the time of Aristotle, people had assumed that a 10g object would fall at 1/10th the speed of a 100g object. Oddly, no one is recorded as having tested this.

According to Galileo’s early biography (written in 1654), he dropped two objects from the Leaning Tower of Pisa to disprove the gravitational mass relation hypothesis. Both landed at the same time, ushering in a new understanding of gravity. It is unknown if Galileo performed the experiment itself, so it is regarded as a thought experiment, not a physical one. Galileo reached his conclusion through the use of other thought experiments.

“We live not only in a world of thoughts, but also in a world of things. Words without experience are meaningless.” — Vladimir Nabokov

Biologists use thought experiments, often of the counterfactual variety. In particular, evolutionary biologists question why organisms exist as they do today. For example, why are sheep not green? As surreal as the question is, it is a valid one. A green sheep would be better camouflaged from predators. Another thought experiment involves asking: why don’t organisms (aside from certain bacteria) have wheels? Again, the question is surreal but is still a serious one. We know from our vehicles that wheels are more efficient for moving at speed than legs, so why do they not naturally exist beyond the microscopic level?

Psychology and Ethics — The Trolley Problem

Picture the scene. You are a lone passerby in a street where a tram is running along a track. The driver has lost control of it. If the tram continues along its current path, the five passengers will die in the ensuing crash. You notice a switch which would allow the tram to move to a different track, where a man is standing. The collision would kill him but would save the five passengers. Do you press the switch?

This thought experiment has been discussed in various forms since the early 1900s. Psychologists and ethicists have discussed the trolley problem at length, often using it in research. It raises many questions, such as:

Is a casual observer required to intervene?

Is there a measurable value to human life? I.e. is one life less valuable than five?

How would the situation differ if the observer were required to actively push a man onto the tracks rather than pressing the switch?

What if the man being pushed were a ‘villain’? Or a loved one of the observer? How would this change the ethical implications?

Can an observer make this choice without the consent of the people involved?

Research has shown most people are far more willing to press a switch than to push someone onto the tracks. This changes if the man is a ‘villain’- people are then far more willing to push him. Likewise, they are reluctant if the person being pushed is a loved one.

In Incognito: The Secret Lives of The Brain, David Eagleman writes that our brains have a distinctly different response to the idea of pushing someone and the idea of pushing a switch. When confronted with a switch, brain scans show that our rational thinking areas are activated. Changing pushing a switch to pushing a person and our emotional areas activate. Eagleman summarizes that:

People register emotionally when they have to push someone; when they only have to tip a lever, their brain behaves like Star Trek’s Mr. Spock.

The trolley problem is theoretical, but it does have real world implications. For example, the majority of people who eat meat would not be content to kill the animal themselves- they are happy to press the switch but not to push the man. Even those who do not consume meat tend to ignore the fact they are indirectly contributing to the deaths of animals due to production quotas, which mean the meat they would have eaten ends up getting wasted. They feel morally superior as they are not actively pushing anyone onto the tracks, yet are still like an observer who does not intervene in anyway. As we move towards autonomous vehicles, there may be real life instances of similar situations. Vehicles may be required to make utilitarian choices – such as swerving into a ditch and killing the driver to avoid a group of children.

Although psychology and ethics are separate fields, they often make use of the same thought experiments.

““Ford!” he said, “there’s an infinite number of monkeys outside who want to talk to us about this script for Hamlet they’ve worked out.” — Douglas Adams, The Hitchhiker's Guide to the Galaxy

The Infinite Monkey Theorem and Mathematics

The infinite monkey theorem is a mathematical thought experiment. The premise is that infinite monkeys with typewriters will, eventually, type the complete works of Shakespeare. Some versions involve infinite monkeys or a single work. Mathematicians use the monkey(s) as a representation of a device which produces letters at random.

In Fooled By Randomness, Nassim Taleb writes:

If one puts an infinite number of monkeys in front of (strongly built) typewriters, and lets them clap away, there is a certainty that one of them will come out with an exact version of the ‘Iliad.’ Upon examination, this may be less interesting a concept than it appears at first: Such probability is ridiculously low. But let us carry the reasoning one step beyond. Now that we have found that hero among monkeys, would any reader invest his life’s savings on a bet that the monkey would write the ‘Odyssey’ next?

The infinite monkey theorem is intended to illustrate the idea that any issue can be solved through enough random input, in the manner a drunk person arriving home will eventually manage to fit their key in the lock even if they do it without much finesse. It also represents the nature of probability and the idea that any scenario is workable, given enough time and resources.

To learn more about thought experiments, and other mental models, check out our book series, The Great Mental Models.