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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,

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^{[2]}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.

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^{[4]}It is a way of reasoning that has been used throughout the history of mathematics and philosophy from classical antiquity onwards.

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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.

## History

The phrase can be traced back to the Greek *η εις άτοπον απαγωγή* (*hê eis átopon apagogê*). This phrase means "reduction to the impossible".^{[3]} It was often used by Aristotle.^{[5]} The method is used a number of times in *Euclid's Elements*.

## Method

*Reduction ad absurdum* can be a tool of discovery.^{[6]}

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,

## Related pages

## References

- ↑ "The Definitive Glossary of Higher Mathematical Jargon: Proof by Contraction" (in en-US). 2019-08-01. https://mathvault.ca/math-glossary/#contradiction.
- ↑
^{2.0}^{2.1}Weston, Anthony. (2009).*A Rulebook for Arguments,*pp. 43-44. - ↑
^{3.0}^{3.1}^{3.2}"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.