about two differrent opinions in mathematics

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My question is: what is the name of mathematicians who ignore the proofs by contradiction and say all of the proofs should be constructive, and what is the name of opposite opinion?

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Both descriptions can include multiple perspectives. The first view could describe either constructivists or people who don't accept the law of non-contradiction; this latter group is called paraconsistent logicians. The second view could describe either classical logicians or those who are neither classical nor paraconsistent, such as the intuitionists.