Difference between revisions of "Reductio Ad Absurdum"
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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: | '''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: | ||
Revision as of 16:19, August 5, 2011
It has been suggested that Reductio ad absurdum be merged with this article or section. (Discuss)
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:
- 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 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
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.
