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Conservapedia:Critical Thinking in Math (view source)
Revision as of 02:15, August 21, 2008
, 02:15, August 21, 2008→Draft Curriculum: Another controversy that I probably shouldn't get into.
:::::::: 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)
:::::::: 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)
I can think of at least 6 things you could say about a proof that might affect your perception of its rigor, quality, simplicity, or elegance:
#Does it use contradiction?
#Is it constructive? (only applies to theorems that say "there is a ....")
#Does it use complex numbers?
#Does it use the axiom of choice?
#Does it use mathematical induction?
#Does it involve provability or decidability within some logical framework? That is, does it relate to Gödel's incompleteness theorem?
The topic of this thread is supposed to be item 1, but it has twice gotten sidetracked, once into item 2, and once into item 6. It turns out that one can easily separate item 1 from the others. The field of "elementary" calculus has many theorems that use "pure" contradiction, unpolluted by any of the other factors. As an example, take the theorem
:Limits, when they exist, are unique.
This theorem uses the epsilon/delta formulation of limits. I won't go into the details here (I plan to put it into the limit page). But the outcome of the theorem is that
:If f(x) approaches both W and Z as a limit, with <math>W \ne Z</math>, one can set <math>\epsilon = |W-Z| / 3</math> and get a contradiction.
Does that prove that the limit is unique? Well, it proves that '''it is not possible for the limit not to be unique'''. Put another way, '''you can't find two different numbers, W and Z, both of which are limits'''.
One is free to say that such a proof is less satisfying than a really direct proof. What the proof by contradiction method is saying is that
:If you prove that it is not possible for proposition P to be false, you have proved that P is true.
Where, in this case, P means "limits are unique".
One is free to question or object to the above syllogism. At some point it comes down to the "law of the excluded middle" or intuitionistic logic. There are two things I would say about that:
*While this issue is of some interest at a deep philosophical level, all mathematicians accept proof by contradiction on a practical level, in its application to the many theorems of mathematics.
*One could pursue this question in the Critical Thinking in Math course, but it would probably take the students down a different path from the stated goals of the curriculum, and would not likely be appreciated by the audience. It might be suitable for a completely different course. Sadly, I think there would be little interest in such a course, when there are so many fascinating things in the current curriculum.
[[User:SamHB|SamHB]] 22:15, 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)