
1854‘Chasing’ the diagram—the use of visualizations in algebraic reasoningReview of Symbolic Logic 10 (1): 158186. 2017.The aim of this article is to investigate the roles of commutative diagrams (CDs) in a speciﬁc mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It wi…Read more

1830Groundwork for a Fallibilist Account of MathematicsPhilosophical Quarterly 7 (4): 823844. 2021.According to the received view, genuine mathematical justification derives from proofs. In this article, I challenge this view. First, I sketch a notion of proof that cannot be reduced to deduction from the axioms but rather is tailored to human agents. Secondly, I identify a tension between the received view and mathematical practice. In some cases, cognitively diligent, wellfunctioning mathematicians go wrong. In these cases, it is plausible to think that proof sets the bar for justification …Read more

1281Forms and Roles of Diagrams in Knot TheoryErkenntnis 79 (4): 829842. 2014.The aim of this article is to explain why knot diagrams are an effective notation in topology. Their cognitive features and epistemic roles will be assessed. First, it will be argued that different interpretations of a figure give rise to different diagrams and as a consequence various levels of representation for knots will be identified. Second, it will be shown that knot diagrams are dynamic by pointing at the moves which are commonly applied to them. For this reason, experts must develop a s…Read more

1200Leopardi “Everything Is Evil”In Andrew P. Chignell (ed.), Evil: A History. pp. 351357. 2019.Giacomo Leopardi, a major Italian poet of the nineteenth century, was also an expert in evil to whom Schopenhauer referred as a “spiritual brother.” Leopardi wrote: “Everything is evil. That is to say, everything that is, is evil; that each thing exists is an evil; each thing exists only for an evil end; existence is an evil.” These and other thoughts are collected in the Zibaldone, a massive collage of heterogeneous writings published posthumously. Leopardi’s pessimism assumes a polished form i…Read more

1032What are mathematical diagrams?Synthese 200 (2): 129. 2022.Although traditionally neglected, mathematical diagrams have recently begun to attract attention from philosophers of mathematics. By now, the literature includes several case studies investigating the role of diagrams both in discovery and justification. Certain preliminary questions have, however, been mostly bypassed. What are diagrams exactly? Are there different types of diagrams? In the scholarly literature, the term “mathematical diagram” is used in diverse ways. I propose a working defin…Read more

891What is Mathematical Rigor?Aphex 25 117. 2022.Rigorous proof is supposed to guarantee that the premises invoked imply the conclusion reached, and the problem of rigor may be described as that of bringing together the perspectives of formal logic and mathematical practice on how this is to be achieved. This problem has recently raised a lot of discussion among philosophers of mathematics. We survey some possible solutions and argue that failure to understand its terms properly has led to misunderstandings in the literature.

878Envisioning Transformations – The Practice of TopologyIn Brendan Larvor (ed.), Mathematical Cultures: The London Meetings 20122014, Birkhäuser. pp. 2550. 2016.The objective of this article is twofold. First, a methodological issue is addressed. It is pointed out that even if philosophers of mathematics have been recently more and more concerned with the practice of mathematics, there is still a need for a sharp deﬁnition of what the targets of a philosophy of mathematical practice should be. Three possible objects of inquiry are put forward: (1) the collective dimension of the practice of mathematics; (2) the cognitives capacities requested to the pra…Read more

832An Inquiry into the Practice of Proving in LowDimensional TopologyIn Gabriele Lolli, Giorgio Venturi & Marco Panza (eds.), From Logic to Practice, Springer International Publishing. pp. 315336. 2015.The aim of this article is to investigate speciﬁc aspects connected with visualization in the practice of a mathematical subﬁeld: lowdimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in lowdimensional topology, justiﬁcations can be based on sequences of pictures. Three these…Read more

829Who's Afraid of Mathematical Diagrams?Philosophers' Imprint 23 (1). 2023.Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for nondiagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic…Read more

823Reconciling Rigor and IntuitionErkenntnis 86 (6): 17831802. 2020.Criteria of acceptability for mathematical proofs are fielddependent. In topology, though not in most other domains, it is sometimes acceptable to appeal to visual intuition to support inferential steps. In previous work :829–842, 2014; Lolli, Panza, Venturi From logic to practice, Springer, Berlin, 2015; Larvor Mathematical cultures, Springer, Berlin, 2016) my coauthor and I aimed at spelling out how topological proofs work on their own terms, without appealing to formal proofs which might be…Read more

336Intersubjective Propositional JustificationIn Luis R. G. Oliveira & Paul Silva Jr (eds.), Propositional and Doxastic Justification, Routledge. pp. 241262. 2022.The distinction between propositional and doxastic justification is wellknown among epistemologists. Propositional justification is often conceived as fundamental and characterized in an entirely apsychological way. In this chapter, I focus on beliefs based on deductive arguments. I argue that such an apsychological notion of propositional justification can hardly be reconciled with the idea that justification is a central component of knowledge. In order to propose an alternative notion, I st…Read more

318Conversation with John P. BurgessAphex 25. 2022.John P. Burgess is the John N. Woodhull Professor of Philosophy at Princeton University. He obtained his Ph.D. from the Logic and Methodology program at the University of California at Berkeley under the supervision of Jack H. Silver with a thesis on descriptive set theory. He is a very distinguished and influential philosopher of mathematics. He has written several books: A Subject with No Object (with G. Rosen, Oxford University Press, 1997), Computability and Logic (with G. Boolos and R. Jeff…Read more

226Recalcitrant Disagreement in Mathematics: An “Endless and Depressing Controversy” in the History of Italian Algebraic GeometryGlobal Philosophy 33 (38): 129. 2023.If there is an area of discourse in which disagreement is virtually absent, it is mathematics. After all, mathematicians justify their claims with deductive proofs: arguments that entail their conclusions. But is mathematics really exceptional in this respect? Looking at the history and practice of mathematics, we soon realize that it is not. First, deductive arguments must start somewhere. How should we choose the starting points (i.e., the axioms)? Second, mathematicians, like the rest of us, …Read more

197Objectivity and Rigor in Classical Italian Algebraic GeometryNoesis 38 195212. 2022.The classification of algebraic surfaces by the Italian School of algebraic geometry is universally recognized as a breakthrough in 20thcentury mathematics. The methods by which it was achieved do not, however, meet the modern standard of rigor and therefore appear dubious from a contemporary viewpoint. In this article, we offer a glimpse into the mathematical practice of the three leading exponents of the Italian School of algebraic geometry: Castelnuovo, Enriques, and Severi. We then bring in…Read more

139The Epistemological Subject(s) of MathematicsIn B. Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice, Springer Nature. pp. 127. 2023.Paying attention to the inner workings of mathematicians has led to a proliferation of new themes in the philosophy of mathematics. Several of these have to do with epistemology. Philosophers of mathematical practice, however, have not (yet) systematically engaged with general (analytic) epistemology. To be sure, there are some exceptions, but they are few and far between. In this chapter, I offer an explanation of why this might be the case and show how the situation could be remedied. I conten…Read more
Silvia De Toffoli
University School of Advanced Studies IUSS Pavia

University School of Advanced Studies IUSS PaviaAssistant Professor
APA Eastern Division
Areas of Specialization
Philosophy of Mathematics 
Epistemology 