Siteadmin, Bureaucrats, Check users, nsAm_Govt_101RO, nsAm_Govt_101RW, nsAm_Govt_101_ta, nsJudgesRO, nsJudgesRW, nsJudges_talkRO, nsJudges_talkRW, nsTeam2RO, nsTeam2RW, nsTeam2_talkRO, nsTeam2_talkRW, oversight, Administrators
125,789
edits
Changes
Jump to navigation
Jump to search
Line 81:
Line 81: + + + +
Conservapedia:Critical Thinking in Math (view source)
Revision as of 03:23, August 9, 2008
, 03:23, August 9, 2008→Draft Curriculum: reply
::::::: Proof by contradiction can be unsatisfying because often it leads to unconstructive proofs of important statements. For example, [[Euclid]] used proof by contradiction to show there are infinitely many prime numbers, but that doesn't tell us what they are, or even give an infinite set of them with some formula like <math>2^n -1</math>. Another example: there is a [[transcendental number]] in the [[reals]]. If you prove this as suggested in the transcendental number article by using [[cardinality]], you'd never actually have a transcendental number to work with. It's far more useful to actually show a number like <math>pi</math> or ''e'' is transcendental and not use contradiction to do it. -[[User:Foxtrot|Foxtrot]] 20:43, 8 August 2008 (EDT)
::::::: Proof by contradiction can be unsatisfying because often it leads to unconstructive proofs of important statements. For example, [[Euclid]] used proof by contradiction to show there are infinitely many prime numbers, but that doesn't tell us what they are, or even give an infinite set of them with some formula like <math>2^n -1</math>. Another example: there is a [[transcendental number]] in the [[reals]]. If you prove this as suggested in the transcendental number article by using [[cardinality]], you'd never actually have a transcendental number to work with. It's far more useful to actually show a number like <math>pi</math> or ''e'' is transcendental and not use contradiction to do it. -[[User:Foxtrot|Foxtrot]] 20:43, 8 August 2008 (EDT)
:::::::: Proofs are proofs, not formulas or examples. [[Euclid]]'s proof by contradiction that there are infinitely many primes is a perfectly good answer to the question "Are there infinitely many primes?" It's a terrible answer to the question "What are they?" or "What's the 10,000,000th prime?" but that's not what Euclid set out to prove. <math>pi</math> and ''e'' are very useful numbers but showing that they are transcendental doesn't tell you much about any other transcendental numbers any more than knowing that 101 is a prime number tells you whether there are an infinite number of primes -- for that you need a proof and a proof by contradiction is 100% adequate. [[User:AdrianDelmar|AdrianDelmar]] 22:49, 8 August 2008 (EDT)
:::::::: Proofs are proofs, not formulas or examples. [[Euclid]]'s proof by contradiction that there are infinitely many primes is a perfectly good answer to the question "Are there infinitely many primes?" It's a terrible answer to the question "What are they?" or "What's the 10,000,000th prime?" but that's not what Euclid set out to prove. <math>pi</math> and ''e'' are very useful numbers but showing that they are transcendental doesn't tell you much about any other transcendental numbers any more than knowing that 101 is a prime number tells you whether there are an infinite number of primes -- for that you need a proof and a proof by contradiction is 100% adequate. [[User:AdrianDelmar|AdrianDelmar]] 22:49, 8 August 2008 (EDT)
::::::::: Your argument starts with your conclusion, "proofs are proofs." In fact, esteemed [[mathematicians]] have always held some forms of proofs to be superior and preferred to others. [[Paul Erdos]], for example, felt with good reason that an [[elementary proof]] is superior.
::::::::: 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)
*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)