Several theories have been proposed to describe the transition from process to object in mathematical thinking. Yet, what is the nature of this ''object'' produced by the ''encapsulation'' of a process? Here, we outline the development of some of the theories (including Piaget, Dienes, Davis, Greeno, Dubinsky, Sfard, Gray, and Tall) and consider the nature of the mental objects (apparently) produced through encapsulation and their role in the wider development of mathematical thinking. Does the same developmental route occur in geometry as in arithmetic and algebra? Is the same development used in axiomatic mathematics? What is the role played by imagery?\ud \u
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