OLOFOS - On the Ontological Interest of Adaptive Mathematical Theories


19 décembre 2018



Salle Jean Ladrière - Collège Mercier - Place cardinal Mercier, 14

Rencontre avec Diderik Batens (UGent) organisée dans le cadre des séminaires OLOFOS "Formal methods towards scientific ontology


A position on the ontological import of mathematical theories will be presented and defended. It will serve as a framework for assessing the ontological import of mathematical theories that are inconsistent as well as adaptive. Two examples of such theories will be considered.

Adaptive Peano Arithmetic (APA) [1] provides insights into theoretical and practical effects that would materialize if Peano Arithmetic would turn out inconsistent - in view of Gödel's second incompleteness theorem, the latter possibility apparently cannot be eliminated by absolute means.

I shall consider Adaptive Fregean Set Theories (AFS) as recently developed by me [2], informed by earlier work, especially by Peter Verdée [3, 4]. As Fregean set theory is known to be inconsistent, the situation of AFS theories is very different from that for APA. The point at issue is whether AFS should be developed. The answer turns out positive: the AFS enable one to describe consistent sets that are beyond the grasp of any of the known set theories that are presumed consistent.

The interest of the results is amply sufficient to grant the status of existing mathematical entities to APA and its inconsistent predicates (or functions) and, with even more reason, to the AFS, their sets and their membership relations.



[1] Diderik Batens. The consistency of Peano Arithmetic. A defeasible perspective. In Patrick Allo and Bart Van Kerkhove, editors, Modestly Radical or Radically Modest. Festschrift for Jean Paul van Bendegem on the Occasion of His 60th Birthday, volume 24 (sometimes 22) of Tributes, pages 11-59. College Publications, London, 2014.

[2] Diderik Batens. Adaptive Fregean set theory. Being refereed.

[3] Peter Verdée. Non-monotonic set theory as a pragmatic foundation of mathematics. Foundations of Science, 18:655{680, 2013.

[4] Peter Verdée. Strong, universal and provably non-trivial set theory by means of adaptive logic. Logic Journal of the IGPL, 21(1):108{125, 2013.