The subject of this article is the role of inductive reasoning (in the meaning of induction by incomplete enumeration) in the methodology of mathematics. The following types of induction have been distinguished: I₁) induction which causes formulation of axioms of different mathematical theories; I₂) enumerative induction which causes formulation of theorems on the basis of a finite number of cases; I₃) induction concerning the range of application of mathematical symbols; I₄) induction generalising the properties of finite sets to the infinite case; I₅) induction pointing to analogies between problems belonging to different domains of mathematics. Inductive reasoning should not be put against the creativity of scientists, as induction is often an important, though not always conscious part of our cognition. The concept of notional metaphor can be useful for explaining the nature of inductive reasoning. With this concept, a general scheme of inductive reasoning has been proposed, which, in the opinion of the authors, is more adequate for presenting the nature of induction than traditional approaches.