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What we are proposing in the following pages is an analysis of the structure of mathematics such as we have represented it in diagram (D). An essential organizational factor of the diagram is the opposition Discrete (on the left) — Continuous (on the right), which constitutes the horizontal axis X'X. The descendant vertical axis Y'Y describes, on the contrary, the nature of the generativity which is active in all of the mentioned theories. The form of a structure — i.e. an ensemble of operations capable of acting in a given space — is by definition tied to the algebraic properties of the structure. What essentially intervenes is the nature of the generativity of the operations, that is to say, the structure's capacity to extend itself through the use of permitted operations, and in particular, through concatenation, alias the composition g o f of the two operations f and g (f being followed by g), when these are permitted.

DOI: 10.1007/978-94-011-5690-5_6

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