Visual representations of various kinds are ubiquitous in pure and applied mathematics, in the natural and social sciences, as well as in many other human activities. By investigating these visualizations, many questions arise: What are they? How do they function? What are the conditions of their correct use? Why are they, at times, such eﬀective aids to cognition? What type of knowledge can they promote?
In my research, I focus on diagrams in mathematics, not exclusively in geometry, but in diﬀerent mathematical domains as well. Despite the extreme variety of mathematical diagrams, it is possible to distill some of their characteristic properties.
Diagrams are not usually static bookkeeping devices, but displays for advancing thought in a dynamic way: experts can manipulate diagrams in order to discover and prove mathematical results. I proposed a technical deﬁnition of mathematical diagrams based on their two-dimensionality and on their operative dimension, which is constituted by the supported manipulations that correspond to mathematical operations. This deﬁnition allows me to unveil the features of diagrams that underwrite the possibility of them entering into the inferential structure of proofs.
I am also working on providing a general epistemological framework for investigating mathematics in a way that appreciates how human agents actually practice it and in particular how they engage with concrete external artifacts. I aim at showing that the normative dimension at play in mathematics is not monolithic, but multi-dimensional, with different objects of normative assessment and diﬀerent appropriate sorts of evaluation. Moreover, I am developing a notion of mathematical justification which is consistent with our fallibility and in which justification and truth can come apart.
- Silvia De Toffoli, “‘Chasing’ The Diagram – The Use of Visualizations in Algebraic Reasoning,” The Review of Symbolic Logic, Volume 10, Issue 1, pp. 158-186, 2017
- Silvia De Toffoli and Valeria Giardino, “Forms and roles of diagrams in knot theory,” Erkenntnis, Volume 79, pp. 829-842, 2014
- Silvia De Toffoli and Valeria Giardino, “Envisioning Transformations – The Practice of Topology,” in: Ed: B. Larvor, Mathematical Cultures, Birkhäuser, pp. 25-50, 2016
- Silvia De Toffoli and Valeria Giardino, “An inquiry into the practice of proving in low-dimensional topology,” in: Eds: G. Lolli, M. Panza, G. Venturi, From Logic to Practice, Boston Studies in the Philosophy and History of Science, Volume 308, pp. 315-336, 2015
- Greg Priest, Silvia De Toffoli, and Paula Findlen (Editors), Special issue of Endeavour, Tools of Reason: The Practice of Scientiﬁc Diagrammatic from Antiquity to the Present, 2018. With a co-authored introduction: “Tools of Reason: The Practice of Scientific Diagramming from Antiquity to the Present,” Endeavour, Volume 42 (2-3), pp. 49–59, 2018
- [popular mathematics book] Dario De Toffoli, Silvia De Toffoli, Dario Zaccariotto [book, pp. 274]: Numeri. Divagazioni, calcoli, giochi. (Seconda Edizione) Kangourou, 2017. (Prefazione di Angelo Lissoni, Presentazione di Furio Honsell, Introduzione di Stefano Bartezzaghi)
- Silvia De Toffoli, Degrees of Essentiality for Secants of Knots, Ph.D. Thesis, Universitätsbibliothek der Technischen Universität Berlin, 2013
- Yasuhiro Sakamoto and Silvia De Toffoli, “Einführung in Meta-Vermeer: Experimentelle Studien über originale und manipulierte Bilder,” Das Licht im Zeitalter von Rembrandt und Vermeer, Ein Handbuch der Forschumgruppe Historische Lichtgefüge, pp. 50-53, Jovis, 2012
- Silvia De Toffoli and Yasuhiro Sakamoto, “Meta-Vermeer: A Topological Reinterpretation of a Masterpiece,” Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture, pp. 499-502, 2012
WORK IN PROGRESS
- Knots and Embeddings: New Tools for Incongruent Counterpart
- The Useful Myth of Logic – Nietzsche’s View of Logic from a Fictionalist Standpoint
- The Epistemological Subject of Mathematics
- What is a Mathematical Diagram?
- Aspects of Diagrammatic Reasoning in Category Theory (with Isar Goyvaerts)
- Fruitful Over-determination: the Case of Knot Diagrams