From the beginning of chaos research until today, the unpredictability of chaos has been a central theme. It is widely believed and claimed by philosophers, mathematicians and physicists alike that chaos has a new implication for unpredictability, meaning that chaotic systems are unpredictable in a way that other deterministic systems are not. Hence, one might expect that the question ‘What are the new implications of chaos for unpredictability?’ has already been answered in a satisfactory way. However, this is not the (...) case. I will critically evaluate the existing answers and argue that they do not fit the bill. Then I will approach this question by showing that chaos can be defined via mixing, which has never before been explicitly argued for. Based on this insight, I will propose that the sought-after new implication of chaos for unpredictability is the following: for predicting any event, all sufficiently past events are approximately probabilistically irrelevant. (shrink)

The contemporary use of the term ‘complexity’ frequently indicates that it is considered a unified concept. This may lead to a neglect of the range of different theories that deal with the implications related to the notion of complexity. This paper, integrating both the English and the Latin traditions of research associated with this notion, suggests a more nuanced use of the term, thereby avoiding simplification of the concept to some of its dominant expressions only. The paper further explores the (...) etymology of ‘complexity’ and offers a chronological presentation of three generations of theories that have shaped its uses; the epistemic and socio‐cultural roots of these theories are also introduced. From an epistemological point of view, this reflection sheds light on the competing interpretations underlying the definition of what is considered as complex. Also, from an anthropological perspective it considers both the emancipatory as well as the alienating dimensions of complexity. Based on the highlighted ambiguities, the paper suggests in conclusion that contributions grounded in contemporary theories related to complexity, as well as critical appraisals of their epistemological and ethical legitimacy, need to follow the recursive feedback loops and dynamics that they constitute. In doing so, researchers and practitioners in education should consider their own practice as a learning process that does not require the reduction of the antagonisms and the complementarities that shape its own complexity. (shrink)

The general theme of this article is the actual practice of how definitions are justified and formulated in mathematics. The theoretical insights of this article are based on a case study of topological definitions of chaos. After introducing this case study, I identify the three kinds of justification which are important for topological definitions of chaos: natural-world-justification, condition-justification and redundancy-justification. To my knowledge, the latter two have not been identified before. I argue that these three kinds of justification are widespread (...) in mathematics. After that, I first discuss the state of the art in the literature about the justification of definitions in the light of actual mathematical practice. I then go on to criticize Lakatos’s account of proof-generated definitions—the main account in the literature on this issue—as being limited and also misguided: as for topological definitions of chaos, in nearly all mathematical fields various kinds of justification are found and are also reasonable. (shrink)