Duality in contemporary mathematics —
philosophical aspects
Workshop at the Bergische Universität Wuppertal,
September 3-5, 2015
organized by David Corfield and Ralf Krömer

The Princeton Companion of Mathematics tells us that "Duality is an important general theme which has manifestations in almost every area of mathematics...Despite the importance of duality in mathematics, there is no single definition which covers all instances of the phenomenon." ([GBGL08], III. 19 Duality, p. 187). This claim notwithstanding, over the past few decades attempts have been made to give precise mathematical definitions of the concept of duality in general, raising the question of how the various manifestations became seen to share common properties. At the same time, the discovery and exploration of dualities have often coincided with deep reflection by mathematicians on their broader significance. [See for example [A]]

We think that the philosophy of mathematics should therefore attack the theme of duality in mathematics. To begin such an investigation, we propose the following questions:

  • Is there a general theory of duality? How can dualities be classified? What is the role played by ideas of completion?
  • Why are dualities so frequently encountered in contemporary mathematics, and why do we hear so little about triality?
  • Is it the case that category theory is best adapted to make sense of duality? And if so, what does it tell us? Which roles are played by category-theoretic ways of representation or identification of mathematical objects?
  • Many dualities mediate between entities of a seemingly different nature, e.g., algebra and geometry; does this tell us something about the nature of these entities?
  • The use of some dualities is motivated by the claim that one side of the duality is simpler. Is such a simplification generated by transfer a merely psychological effect?
  • Is it the case that dualities occur at more than chance level at the interface of mathematics and physics, and what would this tell us about the relationship between mathematics and physics?

The aim of the workshop is to bring together specialists working with dualities in various fields of mathematical and other research, in order to initiate a collective reflection on these questions; we want to learn more about how these dualities are used.

Our long-term aim is to produce a collective volume accounting for large parts of the history and the philosophy of duality in mathematics; we see this workshop as the start of the writing process on the philosophical part. Manifestations of duality include

  • classical dualities in mathematics, from projective geometry and Boolean algebra via vector space theory and algebraic topology to Pontrjagin duality, duality in Fourier analysis, Hodge duality and matroids;
  • dualities in theoretical physics, from electromagnetic to string-theoretic dualities and mirror symmetry and their connection to Langlands duality;
  • duality in logic between theories and their models, from Lindenbaum and Tarski up to the category-theoretic treatment by Awodey and Forssell [AF].
  • the dualities of stress and strain or of controllability and observability in mechanical engineering;
  • dualities in computer science, optimization theory...

The workshop is supported by the ANR-DFG project "Mathematics: Objectivity by representation" (MathObRe) led by G. Heinzmann (Nancy) and H. Leitgeb (Munich) and falls under the subtask "Arrows, diagrams, dualization: the role of representation in category theory". We are particularly interested in the role played by representations in the constitution of mathematical objects and in mathematical reasoning, and hope the workshop will shed some light on this role in the case of dualities.

  • [A] Sir Michael Atiyah. Duality in Mathematics and Physics, lecture delivered in 2007 at the Institut de Matemàtica de la Universitat de Barcelona (IMUB). Available online via http://www.fme.upc.edu/arxius/butlleti-digital/riemann/071218_conferencia_atiyah-d_article.pdf.
  • [AF] - Steve Awodey and J. Henrik Forssell. First-order logical duality. Arxiv
  • [GBGL08] - Timothy Gowers, June Barrow-Green, and Imre Leader, editors. The Princeton Companion to Mathematics. Princeton University Press, 2008.