Reductio ad absurdum

You can prove a thing is true. First, pretend the opposite is true.

If that leads to a silly idea, you win! This is called a proof by contradiction.

It is a very old way to think. People in Ancient Greece used it. They used it to solve hard puzzles.

Math people love this tool too. One man called it a fine weapon. It helps them find the truth.

Imagine the Earth is flat. People would fall off the edge! Since that is silly, the Earth is not flat.

This helps us understand our world.

How do you prove something is true? Sometimes, you do it by showing the opposite is impossible. This is called reductio ad absurdum. It is a Latin name. It means "reduction to absurdity." You pretend the opposite of your idea is true. Then, you follow that idea to see where it goes. If it leads to a silly or impossible result, you win! You have shown the original idea must be true.

This way of thinking is very old. People in Ancient Greece used it. A thinker named Socrates used it in debates. He would lead people to a contradiction. A contradiction is when two things cannot both be true. For example, a thing cannot be both true and false at once.

Math experts love this tool. A man named G. H. Hardy called it a fine weapon. He said it is even better than a move in chess. Mathematicians use it to solve big puzzles. One example is about the Earth. If the Earth were flat, people would fall off the edge. Since people do not fall off, the Earth cannot be flat. This logic helps us find the truth.

Imagine you want to prove that a certain rule is always true. Instead of proving it directly, you start by pretending the rule is false. You follow that false idea to see where it leads you. If that path leads to a result that is silly, impossible, or breaks the rules of logic, you have found a contradiction. This means your starting guess must have been wrong. This clever way of thinking is called reductio ad absurdum. It is a Latin phrase that means "reduction to absurdity."

To use this method, you must follow a very specific set of steps. First, you pick the statement you want to prove. Next, you assume that the statement is actually false. You then use logic to see what happens if that false idea were real. If you reach a point where two things are both true and false at the same time, you have hit a wall. This is called the law of non-contradiction. Because the false idea led to an impossible wall, you can conclude the original statement is true.

This way of arguing is very old and comes from Ancient Greece. A poet named Xenophanes of Colophon used it around 570 BCE. He argued that if gods had human bodies, then animals would draw gods that look like horses or oxen. Since gods cannot look like both, his point was made. Later, the philosopher Socrates used a similar method in his debates. He would lead people through a train of reasoning until they had to admit their idea was impossible. This helped him reach a state called aporia, where a person realizes they do not know as much as they thought.

Many famous thinkers have used this tool in math and philosophy. The mathematician G. H. Hardy called it one of a mathematician's finest weapons. He even said it was better than any move in a game of chess. Great mathematicians like Euclid of Alexandria and Archimedes of Syracuse used it too. In Buddhist philosophy, a thinker named Nāgārjuna used these arguments to show that ideas about things having a permanent essence were unsustainable. He used them to explain that things are subject to change.

You can see this logic in many parts of life. For example, you can use it to show the Earth is not flat. If the Earth were flat and finite, people would eventually fall off the edge. Since we do not see people falling off, the flat Earth idea must be wrong. You can also use it with numbers. If you assume there is a smallest positive rational number, you can show that half of that number would be even smaller. This creates a contradiction because you assumed you already had the smallest one.

Reductio ad absurdum is a powerful form of logical reasoning used to establish a claim. The name is Latin for "reduction to absurdity." It is also known as an apagogical argument or a proof by contradiction. This method works by attempting to show that a contrary proposition leads to an impossible result. If following a specific logic leads to a contradiction, that logic must be flawed. In mathematics, this technique is frequently used to provide a proof by contradiction. This is often called an indirect proof because it does not prove a statement directly. Instead, it proves a statement by showing that its opposite cannot possibly be true.

To perform a mathematical proof by contradiction, a thinker follows a strict sequence of steps. First, identify the proposition you wish to prove, which we can call P. Next, assume that P is actually false. This is written as the negation of P, or ¬P. You then use logical reasoning to see what happens if ¬P is true. The goal is to derive two mutually contradictory assertions, such as Q and ¬Q. This process relies on the law of non-contradiction, which states that a proposition cannot be both true and false at once. Once you reach this contradiction, you have proven that your initial assumption was wrong. Therefore, the original proposition P must be true.

There are different ways this technique can manifest in reasoning. One common version is the existence proof by contradiction. In this case, you want to show that an object with a specific property exists. To do this, you assume that no such object exists. You then show that this assumption leads to a logical contradiction. Another version is refutation by contradiction, also called a proof of negation. In a refutation, you are trying to prove that a statement is false. You assume the statement is true and then derive a falsehood to conclude it is actually false. While these terms are often used interchangeably in math, they are formally distinct.

This method has a deep history reaching back to Ancient Greek philosophy. A satirical poem attributed to Xenophanes of Colophon around 570 to 475 BCE provides an early example. He argued that if gods had human forms, then animals would draw gods with horse or ox bodies. Since the gods cannot have both forms, the idea was a contradiction. Later, the philosopher Socrates used a formal dialectical method known as the Socratic method. He would lead an opponent through a step-by-step train of reasoning. He would use background assumptions to force the opponent to admit their assertion led to an absurd conclusion. This often left the opponent in a state of aporia, or a realization of doubt.

Great mathematicians have relied on this tool for centuries. Euclid of Alexandria and Archimedes of Syracuse used it to prove fundamental mathematical propositions. The mathematician G. H. Hardy famously described proof by contradiction as one of a mathematician's finest weapons. He claimed it was a far finer gambit than any move in a game of chess. Hardy noted that while a chess player might sacrifice a single piece, a mathematician offers the entire game. In modern logic, the principle is expressed through propositional formulas. It can be justified by the law of the excluded middle, which states that a proposition either holds or it does not.

Beyond the West, reductio ad absurdum is central to Buddhist philosophy. The Madhyamaka school uses these arguments, known as prasaṅga in Sanskrit, to challenge essentialist ideas. The philosopher Nāgārjuna used this in his work, the Mūlamadhyamakakārikā. He aimed to show that theories of permanent essence were unsustainable. For example, he argued that if a "young man" had a permanent essence, he could never grow old. Because change is possible, the idea of a fixed, unchanging essence must be false. This helped demonstrate that phenomena are empty of essential existence.

In modern times, philosophers still use these appeals to highlight irrationality. Lewis White Beck used them to criticize mechanism philosophy in 1974. Robert L. Holmes used the technique in 1989 to critique deterrence theory. He argued that relying on the threat of war to prevent war was irrational. Interestingly, the technique is not universally accepted in all logic systems. In intuitionistic logic, proof by contradiction is not generally considered valid. This is because intuitionistic logic requires a constructive method to prove a statement. Without a direct way to build the result, the contradiction alone is not always enough.