Saturday 29 March 2025
A new approach has been proposed that seeks to bridge the gap between two seemingly disparate theories in mathematics: Prawitz’s theory of grounds and Girard’s Ludics. For decades, these two frameworks have been studied separately, each attempting to explain the nature of evidence and proof. However, researchers have long recognized that there may be a deeper connection between them.
Prawitz’s theory of grounds posits that evidence is derived from primitive operations that yield normal or canonical objects. In other words, our understanding of the world is built upon fundamental building blocks that are themselves self-evident. Girard’s Ludics, on the other hand, is a framework that views proof as an interactive game between two parties: the prover and the verifier. Here, evidence is seen as the result of a constructive act that corresponds to reduction or cut-elimination over non-primitive steps.
The connection between these two theories lies in their shared concern with the nature of evidence and proof. Both frameworks recognize that evidence is not simply a matter of arbitrary assertion, but rather must be derived from some deeper foundation. However, they differ significantly in their approach to achieving this foundation.
Prawitz’s theory of grounds relies on a notion of primitive operations that yield normal or canonical objects. These operations are seen as the fundamental building blocks of our understanding of the world. In contrast, Girard’s Ludics views proof as an interactive game between two parties. Here, evidence is seen as the result of a constructive act that corresponds to reduction or cut-elimination over non-primitive steps.
The new approach proposed seeks to reconcile these two frameworks by viewing Prawitz’s primitive operations as the fundamental building blocks of Girard’s Ludics. In this way, the theory of grounds can be seen as providing a deeper foundation for the interactive game of proof that is at the heart of Ludics.
This connection has significant implications for our understanding of evidence and proof. It suggests that our understanding of the world is built upon fundamental building blocks that are themselves self-evident. This in turn implies that our knowledge claims must be grounded in some deeper foundation, rather than simply being arbitrary assertions.
The proposed approach also raises interesting questions about the nature of interaction in proof.
Cite this article: “Harmonizing Grounds and Ludics: A New Approach to Evidence and Proof”, The Science Archive, 2025.
Mathematics, Theory, Grounds, Ludics, Evidence, Proof, Primitive Operations, Normal Objects, Interactive Game, Foundations
Reference: Davide Catta, Antonio Piccolomini d’Aragona, “Game of grounds” (2025).
