Mathematical reasoning and external symbolic systems

(Cross-posted at NewAPPS)

A few weeks ago I wrote a post on blind mathematicians, discussing the case of Bernard Morin and the eversion of the sphere in particular. I had been thinking about blind mathematicians then because I was working on a paper on the role of external symbolic systems (written systems such as notations in particular) for mathematical reasoning and mathematical practice. I have now completed a first, preliminary draft of the paper, and uploaded it on my academia website (it's on top of the list under 'Papers'). Should anyone be interested in taking a look, comments would be most welcome! I discuss the case of Bernard Morin all the way at the end of the paper, as well as the case of Jason Padgett, the man with acquired savant syndrome who sees shapes as fractals and can hand-draw fractals of pretty much any image you can think of. Here is the abstract:

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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.