Some relations between physics and finitary and infinitary mathematics are explored in the context of a many-minds interpretation of quantum theory. The analogy between mathematical ``existence'' and physical ``existence'' is considered from the point of view of philosophical idealism. Some of the ways in which infinitary mathematics arises in modern mathematical physics are discussed. Empirical science has led to the mathematics of quantum theory. This in turn can be taken to suggest a picture of reality involving possible minds and the physical laws which determine their probabilities. In this picture, finitary and infinitary mathematics play separate roles. It is argued that mind, language, and finitary mathematics have similar prerequisites, in that each depends on the possibility of possibilities. The infinite, on the other hand, can be described but never experienced, and yet it seems that sets of possibilities and the physical laws which define their probabilities can be described most simply in terms of infinitary mathematics.
Some relations between physics and finitary and infinitary mathematics are explored in the context of a many-minds interpretation of quantum theory. The analogy between mathematical existence'' and physical existence'' is considered from the point of view of philosophical idealism. Some of the ways in which infinitary mathematics arises in modern mathematical physics are discussed. Empirical science has led to the mathematics of quantum theory. This in turn can be taken to suggest a picture of reality involving possible minds and the physical laws which determine their probabilities. In this picture, finitary and infinitary mathematics play separate roles. It is argued that mind, language, and finitary mathematics have similar prerequisites, in that each depends on the possibility of possibilities. The infinite, on the other hand, can be described but never experienced, and yet it seems that sets of possibilities and the physical laws which define their probabilities can be described most simply in terms of infinitary mathematics.
The article discusses approaches to understanding the nature of probability in mathematics and physics with the consequences of these approaches, as well as the difficulties that arise. The ontological status of probability is discussed. It is shown that probability theory, contrary to R. Mises, should be classified as a mathematical science, using the definition formulated by A.N. Kolmogorov. The application of this definition in practice is provided within the framework of the classical approach, provided that the connection with physical reality is defined through the concept of equal opportunity. To do this, the concepts of simple and complex events are introduced, and the method of decomposing complex events into simple ones is explained. The point of view is substantiated, according to which a mathematical probability determined and related to physical reality in this way can be considered an objective probability, that is, existing independently of any tests. Various interpretations of quantum mechanics and approaches to understanding probability are also revealed. It is shown that in physics today the existing approaches to understanding the nature of probability contradict each other. In the Bohm interpretation, probability is an analogue of the classical approach – statistical objective probability, which has a mathematical formalization, and in quantum Bayesianism, probability quantifies the degree of reliability and depends on the subject conducting the experiment. The Copenhagen interpretation of quantum mechanics tries to eliminate the difficulties associated with combining the wave and corpuscular properties of light and electron beams into a single picture by giving the ontological status of probability, which leads to the problem of understanding physical reality. The article argues for the need for a new connection between the mathematical description of probability and physical reality. The empirical (frequency) approach of R. Mises is analyzed, which causes a lot of criticism, as it does not have a clear consistent formalization. The article argues that this approach is inconsistent with quantum mechanics. The probability given by M. Born in the Copenhagen interpretation of quantum mechanics is also considered, which formulates a new definition of probability, which does not exclude the difficulties associated with the physical interpretation of probability. Born’s approach is based on the classical approach and is reduced to a formal axiomatic definition, which assumes the interpretation of probability as a mathematical construct. The connection of probability with physical reality and the consistency of the statistical method of finding probability values remain unresolved problems, which creates a dualism of understanding probability: in mathematics – one thing, in physics – another. If in the mathematical approach, the solution of these problems is carried out through the connection of probability with physical properties, then in quantum mechanics, in the Copenhagen interpretation, problems persist, which is caused by giving the wave function an ontological status.
The argument of this chapter is governed by the double meaning of its title: (a) the relationships between the concepts of discreteness and continuity in modern mathematics and physics; and (b) the relationships, both continuous and discontinuous, between mathematics and physics, as, from Galileo on, a mathematical-experimental science, with mathematics coming first in this conjunction. The project of modern physics was, thus, defined by an essential continuity between mathematics and physics, as a mathematical representation of nature, principally by means of continuous functions and then calculus, until the rise of quantum theory, specifically as quantum mechanics (QM) and quantum field theory (QFT). With QM, at least in certain interpretations, such as the one adopted in this chapter, quantum theory disconnected the ultimate constitution of nature from any mathematical representation, thus, making physics discontinuous with mathematics. As this chapter will argue, however, this new epistemological situation did not disconnect quantum physics from mathematics, but, on the contrary, led to relating abstract mathematics, such as that of Hilbert spaces over ℂ and operator algebra there, to discrete physical phenomena in terms of probabilities, thus reestablishing the connection, which was no longer representational, between mathematics and physics. This chapter will also argue that the development of the concepts of continuity and discontinuity, and the relationships between them, has acquired a great richness and complexity through the nineteenth and twentieth centuries in mathematics itself, richness and complexity that found their way into physics, in particular relativity and quantum theory. One of the intriguing aspects of this development is the idea, advanced by, among others, A. Grothendieck, following B. Riemann's comment on the subject, that the continuous may serve as an approximation of the discontinuous, rather than seeing, as is more common, the discontinuous as a mode of technical approach to the continuous. This chapter will discuss this idea, and the relationships between continuity and discontinuity in, and between, mathematics and physics in terms of two new concepts: reality without realism (RWR), applicable in both mathematics and physics, and the relationships between them, and ideality without idealism (IWI), the version of the concept of RWR, applicable specifically to mathematics.
Quantum mechanics presents a reality that defies classical intuitions, with the universal wavefunction offering one of the most profound perspectives in modern physics. This paper delves into the mathematical structure of the universal wavefunction, its role in the Many-Worlds Interpretation, and the metaphysical implications for determinism, reality, and consciousness. By blending rigorous mathematical formalism with philosophical exploration, we investigate the conceptual and mathematical underpinnings of quantum reality and its relevance to the nature of existence. Recognizing the speculative nature of certain topics, we clearly distinguish established science from conjectural ideas. Additionally, we provide a critical analysis of various interpretations of quantum mechanics, compare their strengths and weaknesses, and discuss potential experimental tests and observable consequences of the universal wavefunction theory. This work also addresses challenges, such as the derivation of the Born rule and the interpretation of probability in the multiverse.
Beginning in 1900 with 1-iax Planck's hypothesis of quanta, physics has been embarked on a course which has raised serious questions about the true nature of physical reality, about the foundational assumptions and structures of science, and about the relationships between physical reality and physical theory. In particular, attempts to give a physical interpretation to quantum theory seem to result in "pictures" of nature which are paradoxical. A subatomic entity is described by the theory as exhibiting the properties and behavior of both particle-like phenomena and wavelike phenomena. Since it seems unreasonable that a single entity could be both a point-like mass and a wave spread out through a region of space, quantum theory would seem to be less than satisfactory or less than complete at the interpretational level. The purpose of this dissertation is to investigate the underlying causes of this paradox. The dissertation proceeds by reviewing the development of quantum theory and the emergence of the paradox. Next, the problem of physical interpretation is examined and several of the more significant existing interpretations are discussed and analyzed. Finally, the foundations of science and the general structure of scientific theory are examined in a search for the origins of the paradox. As a result of these investigations and analyses, it is concluded that the paradoxical nature of the physical interpretation of quantum theory is the consequence of certain paradigms and preconceptualizations about physical reality which have been incorporated into the theory at the foundational level. These fundamental assU14ptions produce, within quantum theory, a forced marriage of the plenum and atomic views of physical reality. They also produce an ambiguity with respect to the nature of causality which is charaeteristic of the theory. These problems are reflected in the mathematical formalisms and physical concepts of the theory as well as in its metaphysical foundations. Lastly, methods of resoling the paradox are discussed. It is argued that a successful solution is possible, but that it would involve the creation of a new metaphor-an extension of the present language--which would lead science to a deeper, subquantum level in terms of which the present paradox would vanish.
The argument of this article is threefold. First, the article argues that from its rise in the sixteenth century to our own time, the advancement of modern physics as mathematical-experimental science has been defined by the invention of new mathematical structures. Second, the article argues that quantum theory, especially following quantum mechanics, gives this thesis a radically new meaning by virtue of the following two features: on the one hand, quantum phenomena are defined as essentially different from those found in all previous physics by purely physical features; and on the other, quantum mechanics and quantum field theory are defined by purely mathematical postulates, which connect them to quantum phenomena strictly in terms of probabilities, without, as in all previous physics, representing or otherwise relating to how these phenomena physically come about. While these two features may appear discordant, if not inconsistent, I argue that they are in accord with each other, at least in certain interpretations (including the one adopted here), designated as “reality without realism”, RWR, interpretations. This argument also allows this article to offer a new perspective on a thorny problem of the relationships between continuity and discontinuity in quantum physics. In particular, rather than being concerned only with the discreteness and continuity of quantum objects or phenomena, quantum mechanics and quantum field theory relate their continuous mathematics to the irreducibly discrete quantum phenomena in terms of probabilistic predictions while, at least in RWR interpretations, precluding a representation or even conception of how these phenomena come about. This subject is rarely, if ever, discussed apart from previous work by the present author. It is, however, given a new dimension in this article which introduces, as one of its main contributions, a new principle: the mathematical complexity principle.
The argument of this article is threefold. First, the article argues that, from its rise in the sixteenth century to our own time, the advancement of modern physics, as mathematical-experimental science, has been defined by the invention of new mathematical structures. Secondly, the article argues that quantum theory, specifically with quantum mechanics, gave this thesis a radically new meaning by virtue of the following two features: one the one hand, quantum phenomena themselves are defined by purely physical features, as essentially different from all previous physics; and on the other, quantum mechanics and quantum field theory are defined by purely mathematical postulates, which connect them to quantum phenomena strictly in terms of probabilities, without representing or otherwise relating to how these phenomena physically come about. While these two features may appear discordant, if not inconsistent, I argue that they are in accord with each other, at least in certain interpretations (including the one adopted here), designated as “reality without realism,” RWR, interpretations. This argument allows this article to offer a new perspective on a thorny problem of the relationships between continuity and discontinuity in quantum physics. In particular, rather than being concern only with the discreteness and continuity of quantum objects or phenomena, quantum mechanics and quantum field theory relate their continuous mathematics to the irreducibly discrete quantum phenomena, in terms of probabilistic predictions, while, at least in RWR interpretations, precluding a representation or even conception of how these phenomena come about. This subject is rarely, if ever discussed, apart from previous work by the present author. It will, however, be given a new dimension in this article which introduces, as one of its main contributions, a new principle: the mathematical complexity principle.