5e Journée Mathématiques en action

Michael Friedman

Starting from the middle of the 19th century the construction of physical models in mathematics and chemistry – especially in Germany – was an integral part of the research. In chemistry, the cardboard models of Van ‘t Hoff were an essential part of the visual culture of (organic) chemistry. In mathematics, plaster and string models were used to visualize curves and surfaces. In both domains, but especially in mathematics, these models were regarded not only as means of visualization of an abstract entity, but also as an epistemological object: i.e. as what may prompt new discoveries and research questions. Models which were made of folded paper or cardboard also played a small but essential role in this tradition.

The talk aims to highlight two failed attempts of folded models to prompt these discoveries; This failure might be seen either with the abandonment of the visual-material techniques (as with van ‘t Hoff models) or with a recognition of its insufficiency to visualize the mathematical formulas, as seen with the crisis of intuition at the end of the 19th century. Hence, one may claim that these attempts point towards the epistemic boundaries of visualization practices at the end of the 19th century.

Colin J. Rittberg

The aim of this talk is to present and investigate mathematical paper folding as a material reasoning practice. We show that the patterns of mathematical activity observed in mathematical paper folding are, at least since the end 19th century, sufficiently stable to be considered as a practice. Moreover, we will argue that this practice is material. It is controlled by the physical realities of paper-like material, whilst claims to generality of some reasoning operations are supported by arguments from other mathematical idioms. The controlling structure provided by this material side of the practice is tight enough to allow for non-textual rigorous argument and wide enough to provide sufficiently many problems for a practice to be formed. The upshot is that mathematical paper folding is a non-propositional and non-diagrammatic reasoning practice, which demands the attention of any philosophical account of the epistemic force of mathematical proof.