Fundamental Concepts of Mathematics and Quantum Formalism

Frederico Rodrigues Pfrimer

What is the nature of reality? How should be an answer to this question? At this level, we are so deep that all our concepts are obscure. Quantum theory (QT) is at this level. The quest for interpreting it fails because the clarity of our actual fundamental concepts in ordinary language cannot match the precision of the theory's mathematical formalism. The interpretation of quantum mechanics should define and bring clarity to fundamental concepts; not simply rely over them. Quantum theory requires a new worldview, the one that will make it understandable. We discuss on the formulation and interpretation of closed, and final theories, in other words, completely axiomatic theories that define their own conceptual framework, and closed theories that cannot be made simpler. We claim that everything we can properly say about a theory follows from its interpretation, and that interpreting is saying the meaning of each of its mathematical statements. We argue that quantum mechanics is a closed theory; and, therefore, that part of it is part of the final or fundamental theory of physics (FTP). We propose a new entirely axiomatic formulation of part of quantum theory (including all its mathematical framework) in which new powerful and useful elements are added to the existing formalism. Our interpretation is straightforward: it allows us reading or saying the meaning of each of the theory's axiom, theorem or element. We state that the proposed theory is the FTP. Its interpretation may dissolve most of the existing paradoxes around QT. The principle of excluded middle is not valid in general. Both epistemic and ontic elements are present in the theory since it is a theory of knowledge and truth. This and other fundamental concepts are defined by the theory, which can represent both reality and our knowledge about it. Aesthetics is one of its key features.

Doreen Fraser

An important way in which quantum field theory (QFT) differs from non-relativistic quantum mechanics is that in QFT systems generically possess an infinite number of degrees of freedom. This chapter examines the proposals for non-fundamental roles that the particle concept plays in particle physics. The fields which feature in the QFT formalism are mathematical expressions; work needs to be done to determine the ontological properties of the entities that are represented by the mathematical field formalism. One difficulty for the quanta interpretation of QFT is that the analysis in terms of quanta offered in the preceding paragraphs is restricted to free systems. The no-go results for interacting systems inspire a different approach to regarding particles as non-fundamental entities. The fields which feature in the QFT formalism are mathematical expressions; work needs to be done to determine the ontological properties of the entities that are represented by the mathematical field formalism.

Stephen J. Gustafson, Israel Michael Sigal

Stephen J. Gustafson, Israel Michael Sigal

Vasil Dinev Penchev

A concept of formal transcendentalism is utilized. The fundamental and definitive property of the totality suggests for “the totality to be all”, thus, its externality (unlike any other entity) is contained within it. This generates a fundamental (or philosophical) “doubling” of anything being referred to the totality, i.e. considered philosophically. Thus, that doubling as well as transcendentalism underlying it can be interpreted formally as an elementary choice such as a bit of information and a quantity corresponding to the number of elementary choices to be defined. This is the quantity of information defined both transcendentally and formally and thus, philosophically and mathematically. If one defines information specifically, as an elementary choice between finiteness (or mathematically, as any natural number of Peano arithmetic) and infinity (i.e. an actually infinite set in the meaning of set theory), the quantity of quantum information is defined. One can demonstrate that the so-defined quantum information and quantum information standardly defined by quantum mechanics are equivalent to each other. The equivalence of the axiom of choice and the well-ordering “theorem” is involved. It can be justified transcendentally as well, in virtue of transcendental equivalence implied by the totality. Thus, all can be considered as temporal as far anything possesses such a temporal counterpart necessarily. Formally defined, the frontier of time is the current choice now, a bit of information, furthermore interpretable as a qubit of quantum information.