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LeBrad
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I recently saw something that said some mathematicians won't acknowledge proof by contradiction. What is the reason for that? Could somebody elaborate on this for me.
I recently saw something that said some mathematicians won't acknowledge proof by contradiction. What is the reason for that? Could somebody elaborate on this for me.
Yeah, I don't know what one would do without it. To me, some of the neatest proofs by contradiction are those where you can hypothesize a "least counterexample", and then proceed to construct a smaller one. Someone recently gave a proof like that here for Sylvester's line problem.1+1=1 said:Proof by contradiction? This useful technique assisted me in all of my proofs classes while in college. To me, using a proof by contradiction is great. You set the proof up for contradiction and soon the proof comes tumbling down...
A proof of [itex]\neg A[/itex] is a procedure that transforms any hypothetical proof of [itex]A[/itex] into a proof of a contradiction."
"Intuitionists" sounds familiar, I'm pretty sure that's what it was talking about.robert Ihnot said:This is about the Intuitionists and their rejection of the logic of the excluded middle, that is, the acceptance of the "Either A or not A" case.
This is about the Intuitionists and their rejection of the logic of the excluded middle, that is, the acceptance of the "Either A or not A" case.
Obviously they do not accept the Axiom of Choice, since, obviously no one can constructively make these choices.
It's funny that Brouwer was a constructivist, since the proof of his fixed point theorem is usually given as a proof by contradiction.
that is "recall" in the mathematical sense of "see for the first time".
"Ain't No Proof By Contradiction" is a phrase used in the scientific community to express the concept that a theory or hypothesis cannot be proven true by simply disproving its opposite or contradictory statement.
"Ain't No Proof By Contradiction" is an important concept in the scientific method as it highlights the need for evidence and data to support a theory, rather than just relying on the absence of evidence for a contradictory theory.
Yes, "Ain't No Proof By Contradiction" is a fundamental principle in all scientific fields, as it emphasizes the need for empirical evidence and logical reasoning to support theories and hypotheses.
"Ain't No Proof By Contradiction" and falsifiability are related concepts, but they are not the same. Falsifiability refers to the ability to test a hypothesis and potentially prove it false, while "Ain't No Proof By Contradiction" focuses on the need for positive evidence to support a theory.
While "Ain't No Proof By Contradiction" is a widely accepted principle in science, there may be rare cases where contradictory evidence is strong enough to disprove a theory. However, this is not the norm and should not be used to dismiss the need for empirical evidence in scientific research.