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'''Reductio ad absurdum''', also called '''proof by contradiction''', is a method of [[mathematical proof]].  It involves assuming the opposite of what one is trying to [[prove]], and showing that this would lead to a [[contradiction]]. It works by the [[law of the excluded middle]]. The proof typically follows this structure:

'''''Reducto ad absurdum''''' is [[Latin]] for "reduction to the absurd."  It is a form of argument that seeks to disprove a proposition by showing that that proposition inevitably leads to an absurdity, or of proving a proposition by assuming its negation and then showing that that assumption inevitably leads to an absurdity.<ref>[http://www.thefreedictionary.com/reductio+ad+absurdum Reductio ad absurdum]</ref>  One use is in [[mathematics]], as will be described in greater detail below.  ''Reductio ad absurdum'' should not be confused with the [[straw man fallacy]] or the [[slippery slope]] fallacy.

 

''Reductio ad absurdum,'' also called "proof by contradiction" or "proof by assuming the opposite," is a method of [[mathematical proof]].  It involves assuming the opposite of what one is trying to [[prove]], and showing that this would lead to a [[contradiction]]. It works by the [[law of the excluded middle]]. The proof typically follows this structure:

#Create an initial assumption

#Create an initial assumption

#Follow a series of axiomatically valid steps

#Follow a series of axiomatically valid steps

#Reach a contradiction

#Reach a contradiction

#Therefore the initial assumption is incorrect

#Therefore, the initial assumption is incorrect

An example of this is [[Euclid]]'s proof of the [[infinitude]] of the [[Prime number|primes]]:

An example of this is [[Euclid]]'s proof of the [[infinitude]] of the [[Prime number|primes]]:

#Assume there are finitely many primes

#Assume there are finitely many primes

#Take the product of all primes and call it ''N''. Since ''N+1'' is not in our finite set of primes, it must be composite

#Take the product of all primes and call it ''N''. Since ''N''+1 is not in our finite set of primes, it must be composite

#By the [[fundamental theorem of arithmetic]], ''N+1'' has a prime factorization. But ''N+1'' is not divisible by any of the previous primes

#By the [[fundamental theorem of arithmetic]], ''N''+1 has a prime factorization. But ''N+1'' is not divisible by any of the previous primes

#Since ''N+1'' is composite, there must be a prime missing from our set of primes. But this set contains all primes

#Since ''N''+1 is composite, there must be a prime missing from our set of primes. But this set contains all primes

#Therefore, our initial assumption ("there are finitely many primes") is invalid

#Therefore, our initial assumption ("there are finitely many primes") is invalid

There is an academic controversy over whether proofs by contradiction (see also [[Excluded Middle]]) should be accepted qua proofs; either due to concerns regarding computability, or [[Intuitionist Mathmematics]]. These are not necessarily held to be distinct concerns. Most mathematicians disregard any such controversy and accept proofs by contradiction.

==See also==

*[[Stolen concept]], a fallacy that can be exposed using a ''reductio ad absurdum''

 

[[Category:Mathematics]]

[[Category:Logic]]