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Conservapedia:Critical Thinking in Math (view source)
Revision as of 13:57, August 9, 2008
, 13:57, August 9, 2008Response to Aschlafly
::::::::: In light of [[Godel]]'s revelation that math may contain a contradiction, proofs by contradiction are particularly disfavored. One can never know logically whether the proof simply stumbled into an underlying contradiction in the math, rather than proving the proposition.--[[User:Aschlafly|Aschlafly]] 23:23, 8 August 2008 (EDT)
::::::::: In light of [[Godel]]'s revelation that math may contain a contradiction, proofs by contradiction are particularly disfavored. One can never know logically whether the proof simply stumbled into an underlying contradiction in the math, rather than proving the proposition.--[[User:Aschlafly|Aschlafly]] 23:23, 8 August 2008 (EDT)
:::::::::: Elementary proofs may certainly be superior and more satisfying -- [[Paul_Erdos|Paul Erdős']] elementary proof of the [[Prime Number Theorem]] is a very good example -- but that doesn't make proofs by contradiction insufficient or controversial. If you are referring to [[Gödel's incompleteness theorems]], his revelation was not really that math may contain contradictions but that a formal system cannot be both consistent and complete, meaning essentially that a consistent formal system will contain statements that it cannot prove true or false ''within its own system.'' The proof by contradiction that <math>\sqrt{2}</math>. is an irrational number relies on the consistency of the axioms about numbers in use and doesn't come close to worrying about the completeness of the system. <math>\sqrt{2}</math> would not be irrational only in a system with different axioms.
:::::::::: Googly's original question was simply "What's the controversy?" Is this it? Proof by contradiction, Hilbert's program and Gödel's incompleteness theorems are all already on the draft curriculum. Whether elementary proofs are better or not isn't really a controversy. Is there something else? [[User:AdrianDelmar|AdrianDelmar]] 09:57, 9 August 2008 (EDT)
*You say: "No tools beyond about 9th grade math are required, and motivated students younger than 9th grade will not have difficulty with the concepts." I think 9th-graders will struggle with Goldbach, Wiles, Hilbert... [[User:Googly|Googly]] 21:09, 6 August 2008 (EDT)
*You say: "No tools beyond about 9th grade math are required, and motivated students younger than 9th grade will not have difficulty with the concepts." I think 9th-graders will struggle with Goldbach, Wiles, Hilbert... [[User:Googly|Googly]] 21:09, 6 August 2008 (EDT)