Constructive mathematics and non-constructive mathematics are both valuable. Rejecting the latter is crackpottery.
As Andrej Bauer likes to point out—see, for example, Section 3.1 on p. 486 of http://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-... —non-constructive mathematics is a special case of mathematics (in essentially the same way that untyped = unityped programming is a special case of typed programming).