The question of an internal logic of mathematical practice is examined from a finitist point of view. The Gel"fond–Schneider theorem in transcendental number theory serves as an instance of a proof-theoretical investigation motivated and justified by constructivist foundations of logic and mathematics. Constructivist notions are emphasized by contrasting the arithmetical proof procedure of infinite descent with the principle of transfinite induction. It is argued that intuitionistic logic cannot alone provide secure foundations for constructivist mathematics and a finitist logic is briefly sketched in the framework of polynomial arithmetic.