The Physical Numbers: A New Foundational Logic-Numerical Structure For Mathematics And Physics

The boundless nature of the natural numbers imposes paradoxically a high formal bound to the use of standard artificial computer programs for solving conceptually challenged problems in number theory. In the context of the new cognitive foundations for mathematics' and physics' program immersed in the setting of artificial mathematical intelligence, we proposed a refined numerical system, called the physical numbers, preserving most of the essential intuitions of the natural numbers. Even more, this new numerical structure additionally possesses the property of being a bounded object allowing us to work with quite similar axioms like the classic Peano axioms, but in a finite environment and with an enriched physical dimension. Finally, we present several enlightening examples and we conclude that the physical numbers provide a natural formal setting for approaching classic problems in number theory from a more hybrid perspective, i.e. with a potential participation in the generation of solutions, not only of working mathematicians but also of more sophisticated artificial interactive (mathematical) intelligent agents (within the context of cognitive-computational metamathematics or artificial mathematical intelligence) and more controlled by physically-inspired principles. Finally, this paper addresses a highly new, bottom-up and paradigm-shifting approach to the foundations of physics: One should refine and improve the conceptual formal setting of the language that we use for understanding physical phenomena (e.g. the mathematical grounding concepts and theories) for being able to understand better more subtle and complex physical phenomena.