On Mathematical Proving

What is it about?

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.

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Why is it important?

Keywords: mathematical proof, proof-event, problem solving, agents, calculus of events, Kolmogorov’s calculus of problems, Polymath project, T. Gowers, J. Goguen, R. Kowalski, A.N. Kolmogorov.