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Mathematics and the dialogue: a symposium

Program and Abstracts

Time and Venue:
July 12th, 2021, 5pm (BST) (12.00 EDT, 09.00 PDT), Zoom channel 1, with live Q&A at 6pm (BST) (13.00 EDT, 10.00 PDT)

Speakers:

  • Benjamin Wilck (co-organizer), Doctoral Researcher in Philosophy, Humboldt-Universitaet zu Berlin, Germany: “Why Mathematicians Should Care About (Aristotelian) Dialectic“.
  • Deborah Kant (co-organizer), Doctoral Researcher in Philosophy, University of Konstanz, Germany: “Interview as Dialogue Between Mathematics and Philosophy“.
  • Yacin Hamami, Postdoctoral Researcher in Philosophy, Vrije Universiteit Brussel, Belgium: “Mathematical Rigor in Practice: Dialogical Aspects“.

Format:

The format of the symposium involves, apart from the three talks, a dialogue of the three speakers with one another (Round table), as well as with the audience (Q&A).

Abstracts of the three symposium talks:

Benjamin Wilck argues that Aristotle’s account of dialectic in the Topics, which is a codification of the Socratic dialogue, serves as a dialogical method to examine putative scientific principles. The dialectical game involves two rival parties: the questioner and the respondent. The respondent’s task is to put forward a proposition (for instance, a mathematical definition) and to defend it against the questioner’s attack. The questioner’s task is to examine the respondent’s proposition on the basis of the respondent’s concessions in dialectical debate. In particular, the questioner seeks to refute the respondent by checking whether the respondent’s proffered proposition is inconsistent with the respondent’s concessions. Aristotle expressly takes dialectic to be designed as a testing procedure for putative scientific principles such as mathematical definitions and axioms. Consequently, putative scientific principles are dialectically examined on the basis of the respondent’s concessions alone. However, why should a scientist care about dialectical testing of her proffered definitions or axioms, given that science aims at truth, whereas the respondent’s concessions may be false? The paper argues that, surprisingly, dialectical testing is effective precisely because dialectical arguments rest upon premises that need not be true, but need only be granted by the respective respondent.

Deborah Kant employs a form of methodological dialogue in her interview study with professional set theorists about their work and views pertaining to the independence phenomenon. These interviews exemplify a specific form of dialogue between a philosopher and a mathematician. The questions are formulated according to methodological criteria from the Social Sciences. These interview questions thus have a genuinely argumentative function, but they are different from the kind of philosophical question that we find in the Socratic dialogues. There may be appropriate methods other than the qualitative interview to study the set-theoretic community’s research practices and meta-views concerning set-theoretic research; for instance, one may study mathematical literature or mathematical communication between practitioners, or one could do surveys instead of interviews. However, there are two major advantages in performing a dialogue between the investigating philosopher and the mathematician who is part of the community to be investigated. Firstly, the philosopher is likely to get more information in an open, qualitative interview study and, secondly, most other methods presume that philosophers have a detailed, descriptive account of set-theoretic practices that can be tested, while the goal of an open qualitative interview study is rather to construe such an account.

Yacin Hamami investigates the specific type of dialogue that arises when the rigor or correctness of a mathematical proof is evaluated in mathematical practice. When a mathematician, or a group thereof, claims to have provided a rigorous proof of a mathematical theorem, the proof is then examined within the relevant mathematical community. In particular, members of the community can then challenge different parts of the proof by pointing out gaps in the reasoning or by exhibiting counter-examples to some of the inferences. These challenges call for responses from the author(s) who can respond by providing further details, by adjusting the formulation of the theorem, and if no response is available, by withdrawing the claim that the proof provided is rigorous. This dialogue can take place in various contexts such as discussions in workshops and conferences, during the peer-review process, or through personal communications. In this talk, Y. Hamami will propose to construe this dialogue as a game in which a defender aims to defend her claim that a certain mathematical proof P is rigorous against a challenger whose aim is to challenge this claim—he will refer to this game as the rigor game associated to P. In practice, both the defender and the challenger can be either a single mathematician or a group thereof. He will then specify the structure of rigor games, and he will argue that there is an intimate connection between correctly evaluating the rigor of a mathematical proof P in practice and possessing a winning strategy against the challenger in the rigor game associated to P.