This paper is a small step towards the goal of constructing a coherent theory of physic and mathematics together. It is based on two ideas, the localization of mathematical systems in space or space time, and the separation of the concepts of number from number value. The separation of number from number value along with the freedom of choice of number values at different points of space or space time enables the introduction of a space or space time dependent number valuation field. The presence of a location dependent number value field affects theoretical descriptions of many physical and geometric quantities. A simple geometric example is worked out in detail, that of the length of a path. The localization of mathematical systems and the separation of number from number value or meaning both emphasize the role of observers. The separation of number from number value shows the role of observers in that value or meaning are conscious observer related concepts. Nothing, including numbers, has value or meaning to an unconscious observer. The localization of mathematical systems also shows the role of observers in that the mathematics that is potentially available to an observer is that at the same position as is the observer. This represents the mathematical knowledge that can reside in an observers brain. As an observer moves in space or space time, the mathematical knowledge potentially available to the observer is the collection of mathematical systems at the same location as the observer. It is hoped that this work, which was begun in 2010, will lead to a better understanding of the relation between the foundations of mathematics and physics, and the role that observers play in this relation.
This paper is a small step towards the goal of constructing a coherent theory of physic and mathematics together. It is based on two ideas, the localization of mathematical systems in space or space time, and the separation of the concepts of number from number value. The separation of number from number value along with the freedom of choice of number values at different points of space or space time enables the introduction of a space or space time dependent number valuation field. The presence of a location dependent number value field affects theoretical descriptions of many physical and geometric quantities. A simple geometric example is worked out in detail, that of the length of a path. The localization of mathematical systems and the separation of number from number value or meaning both emphasize the role of observers. The separation of number from number value shows the role of observers in that value or meaning are conscious observer related concepts. Nothing, including numbers, has value or meaning to an unconscious observer. The localization of mathematical systems also shows the role of observers in that the mathematics that is potentially available to an observer is that at the same position as is the observer. This represents the mathematical knowledge that can reside in an observers brain. As an observer moves in space or space time, the mathematical knowledge potentially available to the observer is the collection of mathematical systems at the same location as the observer. It is hoped that this work, which was begun in 2010, will lead to a better understanding of the relation between the foundations of mathematics and physics, and the role that observers play in this relation.
The work is based on two premises: local availability of mathematics to an observer at any space time location, and the observation that number systems, as structures satisfying axioms for the number type being considered, can be scaled by arbitrary, positive real numbers. Local availability leads to the assignment of mathematical universes, Vx, to each point, x, of space time. Vx contains all the mathematics that an observer, Ox, at x, can know. Each Vx contains many types of mathematical systems. These include the different types of numbers (natural numbers, integers, rationals, and real and complex numbers), Hilbert spaces, algebras, and many other types of systems. Space time dependent scaling of number systems is used to define representations, in Vx, of real and complex number systems in Vy. The representations are scaled by a factor ry,x relative to the systems in Vx. For y a neighbor point of x,ry,x is the exponential of the scalar product of a gauge field, A(x), and the vector from x to y. For y distant from x,ry,x is a path integral from x to y. Some consequences of the two premises will be examined. Number scaling has no effect on general comparisons of numbers obtained as computations or as experimental outputs. The effect is limited to mathematical expressions that include space or space time integrals or derivatives. The effect of A on wave packets and canonical momenta in quantum theory, and some properties of A in gauge theories, is described.
Local mathematics assumes the existence of number structures of different types, vector spaces, etc. localized at each space time point. Relations between number structures at different locations are based on two aspects: distinction between two so far conflated concepts, number and number value and the "No information at a distance" principle. This principle forbids the choice of the value of a number at one location to determine the value of the same number at another location. Value changing connections, related to a real valued field, g, move numbers between structures at different locations. The effect of the g field, or its exponential equivalent, g(y)=eα(y), on numbers extends to other mathematical structures, vector spaces, etc. The presence of α affects theoretical descriptions of quantities in physics and geometry. Two examples are described, the effect on the Dirac Lagrangian in gauge theory, and the effect on path lengths and distances in geometry. The gradient field of α, A, appears in the Lagrangian as a spin 0, real scalar field that couples to the fermion field. Any value for the mass of A is possible. The lack of direct experimental evidence for the presence of the g or α field means that the field must be essentially constant within a local region of the cosmological universe. Outside the local region there are no restrictions on the field. Possible physical candidates, (inflaton, dark matter, dark energy) for α are noted.
Local availability of mathematics and number scaling provide an approach to a coherent theory of physics and mathematics. Local availability of mathematics assigns separate mathematical universes, x, to each space time point, x.. The mathematics available to an observer, Ox, at x is contained in x . Number scaling is based on extending the choice freedom of vector space bases in gauge theories to choice freedom of underlying number systems. Scaling arises in the description, in x, of mathematical systems in y . If ay or ψy is a number or a quantum state in y, then the corresponding number or state in x is ry,xax or ry,xψx. Here ax and ψx are the same number and state in x as ay and ψy are in y . If y = x+ μdx is a neighbor point of x, then the scaling factor is ry,x = exp( A(x) • μdx) where A is the gradient of a scalar field. The effects of scaling and local availability of mathematics on quantum theory show that scaling has two components, external and internal. External scaling is shown above for ay and ψy. Internal scaling occurs in expressions with integrals or derivatives over space time. An example is the replacement of the position expectation value, ∫ψ*(y)yψ(y)dy, by ∫x ry,ψx* x(yx)yxψx(yx)dyx. This is an integral in x . The good agreement between quantum theory and experiment shows that scaling is negligible in a space region, L, in which experiments and calculations can be done, and results compared. L includes the solar system, but the speed of light limits the size of L to a few light years. For observers in L and events outside L, at cosmological distances, scaling is not limited by theory experiment agreement requirements.
Humans live in many different worlds, each of which has its own representation in mind space. The world of the scientist will be called the physical realms. The laws of physics are an integral part of the real world, and humans acquire a natural understanding of these laws without regard to any mathematical formulation. While the experimental physicist studies the behavior of the real world, the theoretical physicist develops and studies its representation in the mathematical world (i.e., mind space). It is not necessary to know the mathematical equations to be able to carry out a physical investigation. This can be performed through observation and recording of the data numerically, and using the processes of interpolation and extrapolation. Practicing physicists usually find that they must adopt the roles of both the theoretical as well as the experimental physicist to achieve the aims of an experiment. Observers play a vital role in physics. They become important when their state of motion changes, or when their presence affects the results of the experiment. The observer can be either animate or inanimate. Unlike inanimate observers, animate observers have access to mind space, which allows them to make decisions from collected information.
By offering mathematical frameworks and experimental techniques for characterising matter, energy, space, time, and their interactions, physics plays a crucial role in comprehending the nature of reality. The evolution of contemporary physics, from classical mechanics and electromagnetism to quantum mechanics and relativity, has profoundly altered our comprehension of the cosmos and put traditional ideas of determinism, locality, causation, and observation to the test. The contribution of modern physics theories to our comprehension of reality at the microscopic and cosmological sizes is examined in this paper. The measurement problem, entanglement, quantum information, quantum physics, and the connection between physical information and the emergence of observable reality are all given special emphasis. The study also looks at how understanding space, time, matter, and the structure of the cosmos is affected by general relativity, quantum field theory, and new theories of quantum gravity. Additionally, the potential connection between information, awareness, and physical processes is rigorously examined, separating accepted scientific findings from theoretical and philosophical interpretations. The study shows that relationships, information, and physical rules may be a more fundamental way to define reality than traditional ideas of independently existent objects by combining findings from theoretical physics, quantum information science, and philosophy of physics. The study concludes that while unanswered problems about quantum measurement, the origin of spacetime, and consciousness point to significant avenues for future multidisciplinary research, physics is still a potent framework for examining fundamental reality.
Abstract Quantum mechanics and relativity are two of the most successful theoretical frameworks in modern physics, yet their conceptual foundations remain difficult to reconcile at the deepest level. Quantum theory describes physical systems through states, observables, probabilities, measurement contexts, superposition, entanglement, and nonclassical correlations. Relativity describes physical processes through space-time structure, causal order, invariant physical law, observer-dependent decomposition of time and space, and constraints on signal propagation. This article examines the conceptual relationship between quantum measurement, relativistic causality, and space-time description. The purpose is not to present a mathematical unification but to analyze how the two frameworks organize physical explanation differently and where their interpretive tensions arise. Special attention is given to the measurement problem, the status of quantum states, the role of observers and reference frames, the meaning of locality, the distinction between correlation and causal influence, and the challenge of describing quantum processes without privileging a single global time. The article argues that many apparent conflicts between quantum theory and relativity result from importing classical intuitions into domains where they no longer apply. Quantum entanglement does not permit controllable faster-than-light communication, yet it challenges classical separability. Relativity forbids superluminal signaling, yet it does not require that all physically meaningful correlations be reducible to local classical properties. Measurement introduces another difficulty because standard quantum theory often describes state update in a way that appears instantaneous, while relativistic space-time does not provide a universal present. The discussion develops a conceptual framework in which quantum measurement is treated as a physical interaction, relativistic causality as a constraint on operational influence, and space-time description as an organizing structure for events rather than a simple background stage. This analysis is relevant for quantum field theory, quantum information, relativistic quantum foundations, and future approaches to quantum gravity.