The ontology of number

What is a number? Answering this will answer questions about its philosophical foundations - rational numbers, the complex numbers, imaginary numbers. If we are to write or talk about something, it is helpful to know whether it exists, how it exists, and why it exists, just from a common-sense point of view [Quine, 1948, p. 6]. Generally, there does not seem to be any disagreement among mathematicians, scientists, and logicians about numbers existing in some way, but currently, in the mainstream arena only definitions, descriptions of properties, and effects are presented as evidence. Enough historical description of numbers in history provides an empirical basis of number, although a case can be made that numbers do not exist by themselves empirically. Correspondingly, numbers exist as abstractions. All the while, though, these "descriptions" beg the question of what numbers are ontologically. Advocates for numbers being the ultimate reality have the problem of wrestling with the nature of reality. I start on the road to discovering the ontology of number by looking at where people have talked about numbers as already existing: history. Of course, we need to know not only what ontology is but the problems of identifying one, leading to the selection between metaphysics and provisional approaches. While we seem to be dimensionally limited, at least we can identify a more suitable bootstrapping ontology than mere definitions, leading us to the unity of opposites. The rest of the paper details how this is done and modifies Peano's Postulates

The physical cosmology of Alfred North Whitehead

Throughout the history of philosophy, cosmological theories have always deservedly enjoyed a position of special prominence. Of all recent cosmologies, or phi - losophies of Nature, perhaps the most comprehensive and satisfactory is that offered. by Alfred North Whitehead. Whitehead, always both mathematician and philosopher, enjoyed a full career as mathematician at Cambridge and London Universities before answering an invitation from Harvard University to a chair in philosophy there. His interests invariably carried him to the forefront of the advance, and his more technical mathematical works bore the imprint of a philosopher. His philosophy carried the marks of its birth in mathematics and the physical sciences.Although his Treatise on Universal Algebra (1898) won him an enviable reputation, it was his collaboration with Bertrand Russell in the first decade of the twentieth century on Principia Nathematica which proved his pioneering genius. In the middle of this decade, Whitehead offered to the Royal Society of London a memoir entitled "On Mathematical Concepts of the Material World." This memoir, which fell into oblivion, employed the symbolic technique of Principia Nathematica in solving the fundamental problem of importance to cosmological theory. Given a set of entities and a relation between those entities, Whitehead attempted to show the whole of Euclidean geometry to be an expression of the properties of the field of that relation. Certain extraneous relations served to associate the axioms with the material world of the physicists, of which Whitehead offered seven alternative concepts.The first three volumes of Princiaá Mathematica had been published, and Whitehead had begun his work on the fourth, which was to have been concerned with the application of symbolic reasoning to the foundations of geometry and the problem of space. But by this time the scientific world had been captivated by the publication of the special and general theories of relativity by Einstein. These novelties naturally attracted Whitehead, who wrote several essays on the presuppositions of relativity. Whitehead was convinced that the principle and the method introduced by Einstein constituted a revolution in physical science, but found his explanation faulty.A series of three important "Nature" volumes introduced the philosophy of "Nature" as conceived by Whitehead, using his own interpretation of the meaning of the new relativity. A powerful method of analysis, called the Method of Extensive Abstraction and having as its purpose the definition of spatial and temporal entities so as to avoid a circularity of reasoning was born at this period. The third of the volumes was devoted entirely to the development of his own theory of relativity, to which the philosophically more satisfactory interpretation of relativity could be readily applied. From his original presuppositions Whitehead offered four alternative relativity theories, one of which coincided with Einstein's, and two of which were attempts at a unified field theory. The fourth, a theory of gravitation, used a physical element, the "impetus," instead of an infinitesimal metric element, as Einstein had done. This theory proved to be empirically less satisfactory than that of Einstein. But Professor George Temple generalized this fourth theory by using a space -time of positive uniform curvature, and results more satisfactory empirically than those of Einstein followed. The philosophical advantages of Whitehead's relativity were retained. This result seems to invite a more careful consideration of Temple's generalization of ;Whitehead's relativity than has been obtained at present.But by this time Whitehead's speculations, which took as their restricted field the area of nature in which mind was irrelevant, began to concentrate on the enlarged field of cosmological theory in its points of contact with metaphysics. The most important discovery he believed he had made was that in this enlarged area, all the more special physical and extensive properties of nature were dependent for their existence upon process.Now in his sixties, Whitehead accepted Harvard's invitation to a chair in philosophy. Within a very few years he returned to the United Kingdom to deliver the Gifford Lectures at the University of Edinburgh, in which the implications of adopting process as the central principle in the universe were systematically presented.One outstanding; feature of these lectures has been unfortunately ignored; it is a major and original suggestion of this thesis that the categoreal scheme of Process and Reality is really the axiomatic scheme of "On Mathematical Concepts of the Material World" generalized on the metaphysical level. An attempt at the application of the symbolic method to the axioms (categories of explanation and obligation) is made here. Thus the generalized problem in Process and Reality becomes, "Given a set of onto - logical existents and the operation of creativity, what axioms regarding the operation of creativity will have as their result that the more specialized discoveries of the humanities and the sciences follow from the properties of those entities forming the field of creativity?"These lectures, although they offered a comprehensive metaphysical system justifying the operation of physical field theories, suffered under the misfa' tune that they were given at just the time when the quantum mechanics revolution was precipitated in the physical sciences. From the point of view of quantum mechanics, therefore, the philosophy of organism does not supply a satisfactory cosmology within which it can operate. This is especially unfortunate in view of his possibly superior physical theory of relativity; possible points of expansion to allow for quantum mechanics are indicated, although they do violence to the base of the philosophy of organism.As the chief exemplification of the metaphysical principles, Whitehead postulated a brilliantly conceived metaphysical God who was important in physical cosmology. It is suggested that this metaphysical God is, nevertheless, inadequate to satisfy the demands of the religious conscience.Despite the originality of most of the elements introduced by Whitehead, a full understanding of his meaning and an appreciation of his novelties is possible only by referring his writings to their proper settings. Thus, the philosophy of organism is explained against the background of the process philosophies of Bergson, Alexander, and Horgan. Because of its many similarities in respect to the setting of the cosmological problem and the essentials of the solution to the Timaeus, a special chapter is devoted to the correspondence between the two. Whitehead's relativity and philosophy of Nature requires an understanding of the development of the theory of relativity, the world- models of the relativistic cosmologies, and the attempts at a unified field theory. Similarly, the memoir of 1905 is described in a more general back ground setting forth a broad picture of the state of geometry, physical science, and philosophy at the turn of the century.As a final reflection, certain presuppositions at the base of Whitehead's philosophy of organism are investigated and evaluated. The points believed by the present writer to be especially vulnerable in the philosophy of organism are exposed. An experiment in suggesting the prospectus of an alternative system which might avoid the difficulties, and incorporate the advantages of, the philosophy of organism, is made with the warning that it is no more than a suggestion.Throughout the thesis, certain dominant strains of "Ihitehead's thinking can be detected: the importance in his mind of the axiomatic -deductive method in the sciences; the realization that prevalent habits of thinking need to be altered by new discoveries, but are resisted; the conviction that the sciences must be ontologically centered; the faith in field theories; and the conviction that cosmology must be the search for the forms in the facts; to designate the more outstanding convictions

Mathematical metaphors and philosophical structures

The purpose of this study was to examine relationships between mathematics and philosophy. The first part of the study examined the history and basic doctrines of idealism, realism, pragmatism, and existentialism. This was a basic overview which would familiarize the reader with the teachings of each philosophical system. Mathematical topics and structure were then used to model and evaluate each of the philosophies. By using mathematical metaphors to evaluate each philosophical structure, the reader could decide which beliefs would have worth to his or her life. The second part of the study addressed the problem of choice. The belief that humans have few choices and that only one of those choices would bring success was evaluated using the binomial distribution to mathematically model the Greek dialectic. The belief that humans have an infinite number of choices was evaluated using Georg Cantor's mathematical argument that there are infinitely many decimal fractions on the finite line segment between zero and one

An attitude of complexity: thirteen essays on the nature and construction of reality under the challenge of Zeno's Paradox

This book is about the construction of reality. The central aim of this study is to understand how gravity works and how it may be focused and manipulated. While I do not have an answer to this question, the discoveries along the way have been worth collecting into a single volume for future reference

An attitude of complexity: thirteen essays on the nature and construction of reality under the challenge of Zeno's Paradox

This book is about the construction of reality. The central aim of this study is to understand how gravity works and how it may be focused and manipulated. While I do not have an answer to this question, the discoveries along the way have been worth collecting into a single volume for future reference

Classical and quantum aspects of topological solitons: (using numerical methods)

In Introduction, we review integrable and topological solitons. In Numerical Methods, we describe how to minimize functionals, time-integrate configurations and solve eigenvalue problems. We also present the Simulated Annealing scheme for minimisation in solitonic systems. In Classical Aspects, we analyse the effect of the potential term on the structure of minimal- energy solutions for any topological charge n. The simplest holomorphic baby Skyrme model has no known stable minimal-energy solution for n > 1. The one-vacuum baby Skyrme model possesses non-radially symmetric multi-skyrmions that look like 'skyrmion lattices' formed by skyrmions with n = 2. The two-vacua baby Skyrme model has radially symmetric multi- skyrmions. We implement Simulated Annealing and it works well for higher order terms. We find that the spatial part of the six-derivative term is zero. In Quantum Aspects, we find the first order quantum mass correction for the Ф(^4) kink using the semi-classical expansion. We derive a trace formula which gives the mass correction by using the eigenmodes and values of the soliton and vacuum perturbations. We show that the zero mode is the most important contribution. We compute the mass correction of Ф(^4) kink and Sine-Gordon numerically by solving the eigenvalue equations and substituting into the trace formula

The Poetry of Logical Ideas: Towards a Mathematical Genealogy of Media Art

In this dissertation I chart a mathematical genealogy of media art, demonstrating that mathematical thought has had a significant influence on contemporary experimental moving image production. Rather than looking for direct cause and effect relationships between mathematics and the arts, I will instead examine how mathematical developments have acted as a cultural zeitgeist, an indirect, but significant, influence on the humanities and the arts. In particular, I will be narrowing the focus of this study to the influence mathematical thought has had on cinema (and by extension media art), given that mathematics lies comfortably between the humanities and sciences, and that cinema is the object par excellence of such a study, since cinema and media studies arrived at a time when the humanities and sciences were held by many to be mutually exclusive disciplines. It is also shown that many media scholars have been implicitly engaging with mathematical concepts without necessarily recognizing them as such. To demonstrate this, I examine many concepts from media studies that demonstrate or derive from mathematical concepts. For instance, Claude Shannon's mathematical model of communication is used to expand on Stuart Hall's cultural model, and the mathematical concept of the fractal is used to expand on Rosalind Krauss' argument that video is a medium that lends itself to narcissism. Given that the influence of mathematics on the humanities and the arts often occurs through a misuse or misinterpretation of mathematics, I mobilize the concept of a productive misinterpretation and argue that this type of misreading has the potential to lead to novel innovations within the humanities and the arts. In this dissertation, it is also established that there are many mathematical concepts that can be utilized by media scholars to better analyze experimental moving images. In particular, I explore the mathematical concepts of symmetry, infinity, fractals, permutations, the Axiom of Choice, and the algorithmic to moving images works by Hollis Frampton, Barbara Lattanzi, Dana Plays, T. Marie, and Isiah Medina, among others. It is my desire that this study appeal to scientists with an interest in cinema and media art, and to media theorists with an interest in experimental cinema and other contemporary moving image practices

A Unifying Field in Logics: Neutrosophic Logic.

The author makes an introduction to non-standard analysis, then extends the dialectics to “neutrosophy” – which became a new branch of philosophy. This new concept helps in generalizing the intuitionistic, paraconsistent, dialetheism, fuzzy logic to “neutrosophic logic” – which is the first logic that comprises paradoxes and distinguishes between relative and absolute truth. Similarly, the fuzzy set is generalized to “neutrosophic set”. Also, the classical and imprecise probabilities are generalized to “neutrosophic probability”