Many researchers have defined the concept of innovation, without reaching consensus. But in any case an innovation concerns something new or the process of achieving such a thing. Since ‘new’ is a subjective qualification, the concept of innovation is weakly defined. As a consequence, the difference between an innovation and not-an-innovation (‘ordinary change’) stays unclear. This not only hinders the research of innovation and the advancement of innovation theory, but also may lead to costly mismanagement of innovation. To advance the definition of innovation, we distinguish two fundamentally different types of change: the change of the parameters of a system versus the expansion of its dimensions. The first type we identify as ordinary or first-order change and the second type as innovation or second-order change. We explain how our mathematical definition of innovation, combined with social processes of argumentation and discussion, can be operationalized methodically. Using a case of tightening the energy efficiency requirements for newly built houses, a case of business transformation, and a case of decentralization of youth care, we demonstrate how our socio-mathematical definition of innovation helps to study innovation more accurately and to understand the fundamental differences between ordinary change and innovation in their dynamics of planning, acting, and learning. Our socio-mathematical definition positions innovation management next to strategic change management, quality management and standardization management, and is easily applicable for researchers, innovation managers and policy makers.