This paper aims to shed light on an overlooked but essential aspect of informal reasoning and its radical implication to mathematics education research: Decentralising mathematics. We start to problematise that previous studies on informal reasoning implicitly overfocus on what students infer. Based on Walton's distinction between reasoning and argument, and Ernest's concept of intrapersonal dialogue, we propose two theoretical perspectives for understanding the roles of informal reasoning in argumentation: the semi-formal, and the negotiation perspectives. From the latter perspective, we can say that informal reasoning involves creating alternatives, eschewing the relatively unpromising ones, and choosing the most promising one. To illustrate the advantage of the negotiation perspective over the semi-formal perspective, we present two examples of students' statistical written reports from a previous study. These examples illustrate that spontaneous concepts influenced the students' creation of multiple alternatives, and choice of the most promising one, in informal reasoning. Therefore, to better understand the development of mathematical concepts, we need to recognise the role of spontaneous concepts through decentralising mathematics. Finally, we introduce inferentialism as an additional theoretical perspective for investigating both the mathematical development of spontaneous concepts, and the spontaneous development of mathematical concepts. The inferentialist idea of the game of giving and asking for reasons indicates how to empirically investigate the mutual development of spontaneous and mathematical concepts.