Is there a duality in the classical acceptance of non-constructive, foundational, concepts as axiomatic?

We consider a philosophical question that is implicit in Selmer Bringsjord's paper, "The narrational case against Church's Thesis": If, as Mendelson argues, the classically accepted definitions of foundational concepts such as "partial recursive function", "function", "(Tarskian) truth", "set" etc. are vague and imprecise - hence possibly non-constructive and intuitionistically objectionable - then replacing one non-constructive concept by another may be psychologically unappealing, but it should be meta-mathematically valid and acceptable.