Program details

List of keynote speakers:

Gisele Secco (Univ. Federal de Santa Maria, Brasil):

Diagrams and computers in the proof of the Four-Color Theorem: The use of diagrams and the use of computers are two significant themes within the philosophy of mathematical practice. Although case studies concerning the former are abundant – from the notorious case of Euclidean geometry to the uses of diagrams within arithmetic, analysis, topology, knot theory, and even Frege's Begriffschrift –, the latter has received less attention in the field. When it is considered, the famous case of the Four-Color Theorem (4CT) is usually mentioned. I show in my talk how the two themes – diagrams and computers – can be investigated simultaneously via an analysis of the 4CT proof. I will present the roles played by the more than 3000 diagrams and the specificities of the computational machinery mobilized in the first version of the proof (Appel & Haken 1977 and Appel, Haken & Koch 1977). By exploring the main lines of articulation between diagrams and computers in this notorious mathematical result, I will propose some criteria for discussing the identity of different versions of the 4CT proof (mainly Roberston 1997 and Gonthier 2005).

Jemma Lorenat (Pitzer College, USA):

Mathematics or moonshine: non-Euclidean geometry in The Monist at the beginning of the twentieth century: The Monist began publication in 1890 as a journal "devoted to the philosophy of science'' and  dedicated to bringing European (particularly German) texts to American readers. From the first volume, The Monist regularly featured mathematical content. Many of the regular contributors considered themselves knowledgeable amateurs, and published alongside turn-of-the-century greats such as Poincaré, Hilbert, and Veblen. The mathematical content varied from recreations to the logical foundations, but everyone had something to say about the recent changes in geometry. On one side, George Bruce Halsted ceaselessly advocated the "epoch-making" role of Lobachevsky, Bolyai, and their successors. At the other extreme, a consensus that included lawyers, reverends, philosophers and less cosmopolitan mathematicians questioned the idea that straight lines should be anything other than visibly straight. 
Nineteenth-century debates around non-Euclidean geometry are well-known within the history of continental and British mathematics. Complementing these studies, a focus on The Monist reflects the particular nationalism of the United States at a time when its academic hierarchy was still in flux and mathematical research was just beginning to be recognized abroad. Philosophical arguments navigated a delicate balance between the emerging philosophy of pragmatism and the danger of mysticism. Despite ad hominem attacks and name-calling, these exchanges document deeper debates around the relationship between the scientific method and mathematics, and the role of authority (particularly foreign authorities) in shaping the future of geometry. As one contributor inquired "how is the professional expert better fitted to see more lucidly in dealing with the elements of geometry than any other person of good geometric faculty?"

Øystein Linnebo (Univ. of Oslo, Norway):

Pluralities and sets in mathematical practice: Philosophers and logicians have recently taken a great interest in plurals, often seeking to apply the expressive resources of plurals to mathematics and its philosophy. What is the relation between pluralities and sets? This talk will pay special attention to how mathematical practice bears on this question, including (1) Cantor’s appeal to plural to explain the notion of a set and (2) a liberal view of mathematical definitions, also espoused by Cantor, which entails that every plurality defines a set. This liberal view requires us to replace the traditional logic of plurals with a more “critical” plural logic. 

Jeremy Avigad (Carnegie Mellon University, USA):

Reliability of mathematical inference: Of all the demands that mathematics imposes on its practitioners, one of the most fundamental is that proofs ought to be correct. This is also a demand that is especially hard to fulfill, given the fragility and complexity of mathematical proof. This essay considers some of ways that mathematics supports reliable assessment, which is necessary to maintain the coherence and stability of the practice.

Vincenzo De Risi (Laboratoire SPHère, CNRS-Univ. Paris 7, France):

The theory and practice of space: interactions between epistemology and expertise in early modern geometry: The talk investigates the changing views on diagrams, axioms, and space in early modern elementary geometry. Different conceptions of space, mainly provided by metaphysical investigations, seem to have gradually changed the meaning of axioms in the epistemology of mathematics, while the latter transformed the role played by diagrams in actual geometrical demonstrations. On the other hand, a system of well-established practices already regulated the use of diagrams and fixed the standards of rigor in early modern geometry. We will explore how new epistemological ideas conflicted with mathematical practices, and how they eventually changed the latter by establishing new standards and tools. This should shed some light on the relations between mathematical practice and mathematical epistemology in the course of history.