A long-standing debate is whether social influence improves the collective wisdom of a crowd or undermines it. This paper addresses this question based on a naive learning setting in influence systems theory: in our models individuals evolve their estimates of an unknown truth according to the weighted-average opinion dynamics. A formal mathematization is provided with rigorous theoretical analysis. We obtain various conditions for improving, optimizing and undermining the crowd accuracy, respectively. We prove that if the wisdom of finite-size group is improved, then the collective estimate converges to the truth as group size increases, provided individuals' variances are finite. We show that whether social influence improves or undermines the wisdom is determined by the social power allocation of the influence system: if the influence system allocates relatively larger social power to relatively more accurate individuals, it improves the wisdom; on the contrary, if the influence system assigns less social power to more accurate individuals, it undermines the wisdom. At a population level, individuals' susceptibilities to interpersonal influence and network centralities are both crucial. To improve the wisdom, more accurate individuals should be less susceptible and have larger network centralities. Particularly, in democratic influence networks, if relatively more accurate individuals are relatively less susceptible, the wisdom is improved; if more accurate individuals are more susceptible, the wisdom is undermined, which is consistent with the reported empirical evidence. Our investigation provides a theoretical framework for understanding the role social influence plays in the emergence of collective wisdom.