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Reductio ad absurdum
Reductio ad absurdum is a Latin phrase which means "reduction to the absurd". The phrase describes a kind of indirect proof. It is a proof by contradiction, and is a common form of argument. It shows that a statement is true because its denial leads to a contradiction, or a false or absurd result. It is a way of reasoning that has been used throughout the history of mathematics and philosophy from classical antiquity onwards.
The ridiculous or "absurdum" conclusion of a reductio ad absurdum argument can have many forms. For example,
- Rocks have weight, otherwise we would see them floating in the air.
- Society must have laws, otherwise there would be chaos.
- There is no smallest positive rational number, because if there were, it could be divided by two to get a smaller one.
The phrase can be traced back to the Greek η εις άτοπον απαγωγή (hê eis átopon apagogê). This phrase means "reduction to the impossible". It was often used by Aristotle. The method is used a number of times in Euclid's Elements.
Reduction ad absurdum can be a tool of discovery.
The method of proving something works by first assuming something about it. Then other things are deduced from that. If there is a contradiction, it shows that the first something cannot be correct. For example,
- "The Definitive Glossary of Higher Mathematical Jargon: Proof by Contraction" (in en-US). 2019-08-01. https://mathvault.ca/math-glossary/#contradiction.
- Weston, Anthony. (2009). A Rulebook for Arguments, pp. 43-44.
- "Reductio ad Absurdum | Internet Encyclopedia of Philosophy" (in en-US). https://iep.utm.edu/reductio/.
- "Reductio ad absurdum | logic" (in en). https://www.britannica.com/topic/reductio-ad-absurdum.
- Heath, Thomas Little 1908. The Thirteen Books of Euclid's Elements, Vol. 1, p. 136.
- Polya, Goerge. (2008). How to Solve It: A New Aspect of Mathematical Method, p. 169.