Quantum mechanics provides fundamental laws of the physical world that have successfully described wide-ranging phenomena from electronic devices to the formation of gravitational structure in the universe, but it forces us to abandon conventional views of the physical world. To reconstruct our understanding of the world there are two big questions: an epistemological question concerning the process of cognition and an ontological question concerning the nature of reality. Although these questions have been subject to intense investigation, they have, for the most part, been studied separately. The goal of this project is to construct a mathematical framework for coherent explorations of those two questions with logical, algebraic, and operational methods for field and particle physics in order to mathematically reconstruct Heisenberg's uncertainty principle as a cognitive doctrine in distinction to Bohr's complementarity principle which is an ontological doctrine. This approach aims to make both such doctrines testable and universally valid.