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{{merge from|Reductio ad absurdum}}

'''''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''', 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:

 

''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

==Proof of Proposition ''P'' by Contradiction==

===Proof of Proposition ''P'' by Contradiction===

#Suppose ~P.

#Suppose ~P.

#...

#...

#Thus, P

#Thus, P

==Example==

===Example===

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 Mathematics]]. These are not necessarily held to be distinct concerns. Most mathematicians disregard any such controversy and accept proofs by contradiction.

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 Mathematics]]. These are not necessarily held to be distinct concerns. Most mathematicians disregard any such controversy and accept proofs by contradiction.

 

[[Category:Mathematics]]

[[Category:Mathematics]]