On Mathematical Proving

Abstract

This paper outlines a logical representation of certain aspects of the process of mathematical proving that are important from the point of view of Artificial Intelligence. Our starting-point is the concept of proof-event or proving, introduced by Goguen, instead of the traditional concept of mathematical proof. The reason behind this choice is that in contrast to the traditional static concept of mathematical proof, proof-events are understood as processes, which enables their use in Artificial Intelligence in such contexts, in which problem-solving procedures and strategies are studied. We represent proof-events as problem-centered spatio-temporal processes by means of the language of the calculus of events, which captures adequately certain temporal aspects of proof-events (i.e. that they have history and form sequences of proof-events evolving in time). Further, we suggest a "loose" semantics for the proof-events, by means of Kolmogorov's calculus of problems. Finally, we expose the intented interpretations for our logical model from the fields of automated theoremproving and Web-based collective proving.

Search for mathematical proof in the Web environment as a process of creative collective discovery

“…This concept incorporates not only the concluding outcome of the process of mathematical proving, but also the following aspects the process itself [5, 113- [8,9]. Using the language of the calculus of events, we can talk about proof events and their sequences, evolving in time [10]. The semantics of proof events follow a logic that can be expressed in terms of Kolmogorov's calculus of problems [11], initially used for explication of intuitionistic logic [12,10].…”

“…Using the language of the calculus of events, we can talk about proof events and their sequences, evolving in time [10]. The semantics of proof events follow a logic that can be expressed in terms of Kolmogorov's calculus of problems [11], initially used for explication of intuitionistic logic [12,10].…”

Embracing Inference as Action: A Step Towards Human-Level Reasoning

“…treating action inferences as a relation between possible worlds). For example, although our concept of action-as-inference was primarily motivated by the later work of Jean Piaget [6], new promising work based on the concept of proof-events is emerging [11], which considers proving as a process that unfolds through time. It is outside of the scope of this paper to fully compare all alternatives.…”