Concepts are the means by which we structure our understanding of the world and consequently the primary means by which we encounter it. It is commonly assumed that one of the essential characteristics of concepts – regardless of referent – is their stability, tending toward stasis;and, indeed, it can be hard to see how concepts can otherwise be systematic and productive, inthe way they are conventionally taken to be. Even the question has been raised whether conceptscan change; on some prominent accounts, they cannot. The Unified Conceptual Space Theory(UCST) – an extension of Conceptual Spaces Theory – makes the controversial claim that conceptsnot only are subject to change over an iterative lifecycle but that, at an underlying level, they are in a state of continuous motion; indeed, they must be to function as they do. Mere openness to change is not enough. Even the most seemingly fixed of concepts – mathematical concepts are the paradigm example – can be seen to evolve and continually be evolving as our understanding of mathematics evolves. UCST suggests that concepts possess an intrinsic tension that appears topresent a contradiction: to be able to apply in more or less the same way across unboundedly many contexts (systematicity) and to be able to combine coherently with other concepts (productivity),they must be relatively stable; and yet, since each new application context is, in some nontrivial way, different from every previous context in ways that do not fit within neat conceptual boundaries,they must adapt each time to fit. In a physical world we have reason to view as ultimately one of fluidity, of processes and motion rather than stable entities, concepts should probably have a similar nature.