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Conservapedia:Critical Thinking in Math (view source)
Revision as of 23:14, August 20, 2008
, 23:14, August 20, 2008Contradiction or Counterexample?
::::::::::::: You're not addressing my basic point, which I've repeated twice now, so I probably won't pursue this discussion further at this time. Godspeed to you.--[[User:Aschlafly|Aschlafly]] 18:34, 9 August 2008 (EDT)
::::::::::::: You're not addressing my basic point, which I've repeated twice now, so I probably won't pursue this discussion further at this time. Godspeed to you.--[[User:Aschlafly|Aschlafly]] 18:34, 9 August 2008 (EDT)
:::::::: I wonder if you aren't confusing contradiction and counterexample. Today I was reading ''Poincaré's Prize'' by George Szpiro and came across this passage about Poul Heegard finding a counterexample to Poincaré's proof of the duality theorem (p. 85):
:::::::: <blockquote>Let us recall that according to the theorem, the ''k''-th Betti numbers must be equal to the (''n-k'')-th Betti number. Heegard constructed an example of a three-dimensional manifold -- an intersection of a certain cone with a cylinder -- whose Betti numbers are (1,1,2,1). This contradicts the duality theorem. Finding a counterexample to a theorem can mean either that the counterexample is wrong, or that the theorem's proof is wrong, or that everything is based on a misunderstanding. In this case, it was the third alternative...</blockquote>
:::::::: A counterexample is much more "unconstructive" that a proof by contradiction, in the sense that a counterexample simply shows that the proof as stated isn't right, but doesn't say that it couldn't reformulated as Poincaré and other mathematicians went on to do for the duality theorem. -[[User:AdrianDelmar|AdrianDelmar]] 19:14, 20 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)