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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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    $\begingroup$ The first are "constructivists", the second are "mathematicians". $\endgroup$ – Lord Shark the Unknown Mar 26 at 6:11
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    $\begingroup$ The first ones are the idiots and the second the cool guys. ;-) $\endgroup$ – amsmath Mar 26 at 6:12
  • $\begingroup$ Useful reading is Eric Schechter's 2001 expository paper Constructivism is difficult. $\endgroup$ – Dave L. Renfro Mar 26 at 6:34
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Both descriptions can include multiple perspectives. The first view could describe either constructivists or people who don't accept the low 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.

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