### Mathematical reasoning and external symbolic systems

The aim of this paper is to discuss the exact status of external symbolic systems with respect to mathematical reasoning and mathematical practice. The standpoint adopted is a combination of philosophical analysis with focus on empirical studies on numerical cognition (ranging from cognitive science to developmental psychology and anthropology) and on the history of notations. Indeed, the investigation takes into account three different levels: the synchronic level of a mathematician doing mathematics at a given point; the diachronic, developmental level of how a given individual learns mathematics; and the diachronic, historical level of the development of mathematics as a discipline throughout the centuries. It will be argued that the use of external symbolic systems is constitutive of mathematical reasoning and mathematical practice in a fairly strong sense of ‘constitutive’, but not in the sense that manipulating notations is the *only* route to mathematical insight. Indeed, two case studies will illustrate the qualification: a man with acquired savant syndrome and a blind mathematician.

In what sense is a sketch of curves approximated by a series of tangent lines equivalent to a "hand drawn fractal"?

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