SkipCaptcha, Automoderated users, edit
8,856
edits
Changes
Jump to navigation
Jump to search
Line 1:
Line 1: − + − + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Reductio Ad Absurdum (view source)
Revision as of 22:51, November 10, 2019
, 22:51, November 10, 2019remove general category at request of monitors, leave solely in specific subcategory
'''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. An example of this is [[Euclid]]'s proof of the infinitude of the [[Prime number|primes]].
'''''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.
[[Category:Mathematics]]
==In Mathematics==
''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
#Follow a series of axiomatically valid steps
#Reach a contradiction
#Therefore, the initial assumption is incorrect
===Proof of Proposition ''P'' by Contradiction===
#Suppose ~P.
#...
#Therefore, Q.
#...
#Therefore, ~Q
#Hence, Q and ~Q, a contradiction
#Thus, P
===Example===
An example of this is [[Euclid]]'s proof of the [[infinitude]] of the [[Prime number|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
#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
#Therefore, our initial assumption ("there are finitely many primes") is invalid
==See also==
*[[Stolen concept]], a fallacy that can be exposed using a ''reductio ad absurdum''
==References==
<references/>
[[Category:Logic]]