Descartes' philosophy contains an intriguing notion of the infinite, a concept labeled by the philosopher as indefinite. Even though Descartes clearly defined this term on several occasions in the correspondence with his contemporaries, as well as in his Principles of Philosophy, numerous problems about its meaning have arisen over the years. Most commentators reject the view that the indefinite could mean a real thing and, instead, identify it with an Aristotelian potential infinite. In the first part of this article, I (...) show why there is no numerical infinity in Cartesian mathematics, as such a concept would be inconsistent with the main fundamental attribute of numbers: to be comparable with each other. In the second part, I analyze the indefinite in the context of Descartes' mathematicalphysics. It is my contention that, even with no trace of infinite in his mathematics, Descartes does refer to an actual indefinite because of its application to the material world within the system of his physics. This fact underlines a discrepancy between his mathematics and physics of the infinite, but does not lead a difficulty in his mathematicalphysics. Thus, in Descartes' physics, the indefinite refers to an actual dimension of the world rather than to an Aristotelian mathematical potential infinity. In fact, Descartes establishes the reality and limitlessness of the extension of the cosmos and, by extension, the actual nature of his indefinite world. This indefinite has a physical dimension, even if it is not measurable. La filosofía de Descartes contiene una noción intrigante de lo infinito, un concepto nombrado por el filósofo como indefinido. Aunque en varias ocasiones Descartes definió claramente este término en su correspondencia con sus contemporáneos y en sus Principios de filosofía, han surgido muchos problemas acerca de su significado a lo largo de los años. La mayoría de comentaristas rechaza la idea de que indefinido podría significar una cosa real y, en cambio, la identifica con un infinito potencial aristotélico. En la primera parte de este artículo muestro por qué no hay infinito numérico en las matemáticas cartesianas, en la medida en que tal concepto sería inconsistente con el principal atributo fundamental de los números: ser comparables entre sí. En la segunda parte analizo lo indefinido en el contexto de la física matemática de Descartes. Mi argumento es que, aunque no hay rastro de infinito en sus matemáticas, Descartes se refiere a un indefinido real a causa de sus aplicaciones al mundo material dentro del sistema de su física. Este hecho subraya una discrepancia entre sus matemáticas y su física de lo infinito, pero no implica ninguna dificultad en su física matemática. Así pues, en la física de Descartes, lo indefinido se refiere a una dimensión real del mundo más que a una infinitud potencial matemática aristotélica. De hecho, Descartes establece la realidad e infinitud de la extensión del cosmos y, por extensión, la naturaleza real de su mundo indefinido. Esta indefinición tiene una dimensión física aunque no sea medible. (shrink)
This paper explores the relationship between Kant's views on the metaphysical foundations of Newtonian mathematicalphysics and his more general transcendental philosophy articulated in the Critique of Pure Reason. I argue that the relationship between the two positions is very close indeed and, in particular, that taking this relationship seriously can shed new light on the structure of the transcendental deduction of the categories as expounded in the second edition of the Critique.
In this paper I argue three things: (1) that the interactionist view underlying Benacerraf's (1973) challenge to mathematical beliefs renders inexplicable the reliability of most of our beliefs in physics; (2) that examples from mathematicalphysics suggest that we should view reliability differently; and (3) that abstract mathematical considerations are indispensable to explanations of the reliability of our beliefs.
Fundamental notions Husserl introduced in Ideen I, such as epochè, reality, and empty X as substrate, might be useful for elucidating how mathematicalphysics concepts are produced. However, this is obscured in the context of Husserl’s phenomenology itself. For this possibility, the author modifies Husserl’s fundamental notions introduced for pure phenomenology, which found all sciences on the absolute Ego. Subsequently, the author displaces Husserl's phenomenological notions toward the notions operating inside scientific activities themselves and shows this using a (...) case study of the construction of noncommutative geometry. The perspective in Ideen I about geometry and mathematicalphysics includes points that are inappropriate to modern geometry and to modern physics, especially to noncommutative geometry and to quantum physics. The first point relates to the intuitive character of geometrical objects in Husserl. The second is linked to the notion of locality related to the notion of extension, by which Husserl characterizes the essence of physical things. The points show that the notion of empty X as a substrate, developed in “Phenomenology of Reason” in Ideen I, is helpful for considering the notions of physical reality and of geometrical space, especially reality in quantum physics and space in noncommutative geometry. The salient conclusions include the proposition that aphilosophical study of the relationship between the physical object X, which imparts a unity to what is given to sensibility, and the geometrical space X, which imparts a unity of sense to various mathematical operations, opens a reinterpretation of Husserl’s interpretation, supporting an epistemology of mathematicalphysics. (shrink)
The point of departure for this article is Werner Heisenberg’s remark, made in 1929: “It is not surprising that our language [or conceptuality] should be incapable of describing processes occurring within atoms, for … it was invented to describe the experiences of daily life, and these consist only of processes involving exceedingly large numbers of atoms. … Fortunately, mathematics is not subject to this limitation, and it has been possible to invent a mathematical scheme—the quantum theory [quantum mechanics]—which seems (...) entirely adequate for the treatment of atomic processes.” The cost of this discovery, at least in Heisenberg’s and related interpretations of quantum mechanics (such as that of Niels Bohr), is that, in contrast to classical mechanics, the mathematical scheme in question no longer offers a description, even an idealized one, of quantum objects and processes. This scheme only enables predictions, in general, probabilistic in character, of the outcomes of quantum experiments. As a result, a new type of the relationships between mathematics and physics is established, which, in the language of Eugene Wigner adopted in my title, indeed makes the effectiveness of mathematics unreasonable in quantum but, as I shall explain, not in classical physics. The article discusses these new relationships between mathematics and physics in quantum theory and their implications for theoretical physics—past, present, and future. (shrink)
Probability plays a crucial role regarding the understanding of the relationship which exists between mathematics and physics. It will be the point of departure of this brief reflection concerning this subject, as well as about the placement of Poincaré’s thought in the scenario offered by some contemporary perspectives.
It is shown by means of general principles and specific examples that, contrary to a long-standing misconception, the modern mathematicalphysics of compressible fluid dynamics provides a generally consistent and efficient language for describing many seemingly fundamental physical phenomena. It is shown to be appropriate for describing electric and gravitational force fields, the quantized structure of charged elementary particles, the speed of light propagation, relativistic phenomena, the inertia of matter, the expansion of the universe, and the physical nature (...) of time. New avenues and opportunities for fundamental theoretical research are thereby illuminated. (shrink)
This is a comprehensive discussion of complexity as it arises in physical, chemical, and biological systems, as well as in mathematical models of nature. Common features of these apparently unrelated fields are emphasised and incorporated into a uniform mathematical description, with the support of a large number of detailed examples and illustrations. The quantitative study of complexity is a rapidly developing subject with special impact in the fields of physics, mathematics, information science, and biology. Because of the (...) variety of the approaches, no comprehensive discussion has previously been attempted. This book will be of interest to graduate students and researchers in physics (nonlinear dynamics, fluid dynamics, solid-state, cellular automata, stochastic processes, statistical mechanics and thermodynamics), mathematics (dynamical systems, ergodic and probability theory), information and computer science (coding, information theory and algorithmic complexity), electrical engineering and theoretical biology. (shrink)
To find exact traveling wave solutions to nonlinear evolution equations, we propose a method combining symmetry properties with trial polynomial solution to nonlinear ordinary differential equations. By the method, we obtain some exact traveling wave solutions to the Burgers-KdV equations and a kind of reaction-diffusion equations with high order nonlinear terms. As a result, we prove that the Burgers-KdV equation does not have the real solution in the form a 0+a 1tan ξ+a 2tan 2 ξ, which indicates that some types (...) of the solutions to the Burgers-KdV equation are very limited, that is, there exists no new solution to the Burgers-KdV equation if the degree of the corresponding polynomial increases. For the second equation, we obtain some new solutions. In particular, some interesting structures in those solutions maybe imply some physical meanings. Finally, we discuss some classifications of the reaction-diffusion equations which can be solved by trial equation method. (shrink)
A mathematical model of the natural origin of our universe is presented. The model is based only on well-established physics. No claim is made that this model uniquely represents exactly how the universe came about. But the viability of a single model serves to refute any assertions that the universe cannot have come about by natural means.
Many arguments found in the physics literature involve concepts that are not well-defined by the usual standards of mathematics. I argue that physicists are entitled to employ such concepts without rigorously defining them so long as they restrict the sorts of mathematical arguments in which these concepts are involved. Restrictions of this sort allow the physicist to ignore calculations involving these concepts that might lead to contradictory results. I argue that such restrictions need not be ad hoc, but (...) can sometimes be justified by considering some of the metaphysical issues surrounding the question of the applicability of mathematics to physical reality. 1 Introduction 2 Rejecting inferential permissiveness 3 The agreement problem 4 Independent objections to the liberal view. (shrink)
Causal claims in physics may have two familiar kinds of support: theoretical and experimental. This paper claims that a rigorous mathematical derivation in a realistic model is necessary, though not sufficient, for full theoretical support. The support is not provided by the derivation itself; but rather it comes from a detailed back-tracing through the derivation, matching the mathematical dependencies, point by point, with details of the causal story. This back-tracing is not enough to pick out the correct (...) causal story, however; a good deal of background causal knowledge is required as well. These claims are illustrated by a detailed example of what causes the Lamb dip in gas lasers. (shrink)
I outline an intrinsic (coordinate-free) formulation of classical particle mechanics, making no use of set theory or second-order logic. Physical quantities are accepted as real, but are constrained only by elementary axioms. This contrasts with the formulations of Field and Burgess, in which space-time regions are accepted as real and are assumed to satisfy second-order comprehension axioms. The present formulation is both logically simpler and physically more realistic. The theory is finitely axiomatizable, elementary, and even quantifier-free, but is provably empirically (...) equivalent to the standard coordinate formulations. (shrink)
The problem of how mathematics and physics are related at a foundational level is of interest. The approach taken here is to work towards a coherent theory of physics and mathematics together by examining the theory experiment connection. The role of an implied theory hierarchy and use of computers in comparing theory and experiment is described. The main idea of the paper is to tighten the theory experiment connection by bringing physical theories, as mathematical structures over C, (...) the complex numbers, closer to what is actually done in experimental measurements and computations. The method replaces C by C n which is the set of pairs, R n ,I n , of n figure rational numbers in some basis. The properties of these numbers are based on those of numerical measurement outcomes for continuous variables. A model of space and time based on R n is discussed. The model is scale invariant with regions of constant step size interrupted by exponential jumps. A method of taking the limit n→∞ to obtain locally flat continuum-based space and time is outlined. Also R n based space is invariant under scale transformations. These correspond to expansion and contraction of space relative to a flat background. The location of the origin, which is a space and time singularity, does not change under these transformations. Some properties of quantum mechanics, based on C n and on R n space are briefly investigated. (shrink)
As an approach to a Theory of Everything a framework for developing a coherent theory of mathematics and physics together is described. The main characteristic of such a theory is discussed: the theory must be valid and and sufficiently strong, and it must maximally describe its own validity and sufficient strength. The mathematical logical definition of validity is used, and sufficient strength is seen to be a necessary and useful concept. The requirement of maximal description of its own (...) validity and sufficient strength may be useful to reject candidate coherent theories for which the description is less than maximal. Other aspects of a coherent theory discussed include universal applicability, the relation to the anthropic principle, and possible uniqueness. It is suggested that the basic properties of the physical and mathematical universes are entwined with and emerge with a coherent theory. Support for this includes the indirect reality status of properties of very small or very large far away systems compared to moderate sized nearby systems. Discussion of the necessary physical nature of language includes physical models of language and a proof that the meaning content of expressions of any axiomatizable theory seems to be independent of the algorithmic complexity of the theory. Gödel maps seem to be less useful for a coherent theory than for purely mathematical theories because all symbols and words of any language must have representations as states of physical systems already in the domain of a coherent theory. (shrink)
If physicalism is true, everything is physical. In other words, everything supervenes on, or is necessitated by, the physical. Accordingly, if there are logical/mathematical facts, they must be necessitated by the physical facts of the world. In this paper, I will sketch the first steps of a physicalist philosophy of mathematics; that is, how physicalism can account for logical and mathematical facts. We will proceed as follows. First we will clarify what logical/mathematical facts actually are. Then, we (...) will discuss how these facts can be accommodated in the physicalist ontology. This might sound like immanent realism (as in Mill, Armstrong, Kitcher, or Maddy), according to which the mathematical concepts and propositions reflect some fundamental features of the physical world. Although, in my final conclusion I will claim that mathematical and logical truths do have contingent content in a sophisticated sense, and they are about some peculiar part of the physical world, I reject the idea, as this thesis is usually understood, that mathematics is about the physical world in general. In fact, I reject the idea that mathematics is about anything. In contrast, the view I am proposing here will be based on the strongest formalist approach to mathematics. (shrink)
This volume is about searching for fundamental theory in physics which has become somewhat elusive in recent decades. Like a group of blind men investigating an elephant, one physicist postulates the trunk as a hose, another a leg as a tree, the body a wall or barrier, the tail a rope and the ears as a fan. The organizers of the Vigier series symposia strongly believe cross polination by exploring many avenues of seemingly disparate research is key to breakthrough (...) discovery and solicited papers on all areas of physics deemed pertinent in Astrophysics, Cosmology, nuclear physics, quantum theory, electromagnetism, thermodynamics, vacuum field theory and topology. (shrink)
The present paper examines the role of exact results in the theory of many‐body physics, and specifically the example of the Mermin‐Wagner theorem, a rigorous result concerning the absence of phase transitions in low‐dimensional systems. While the theorem has been shown to hold for a wide range of many‐body models, it is frequently ‘violated’ by results derived from the same models using numerical techniques. This raises the question of how scientists regulate their theoretical commitments in such cases, given that (...) the models, too, are often described as approximations to the underlying ‘full’ many‐body problem. (shrink)
Sir Arthur Eddington, the celebrated astrophysicist, made great strides towards his own 'theory of everything'in his last two books published in 1936 and 1946. Unlike his earlier lucid and authoritative works, these are strangely tentative and obscure - as if he were nervous of the significant advances that he might be making. This volume examines both how Eddington came to write these uncharacteristic books - in the context of the physics and history of the day - and what value (...) they have to modern physics. The result is an illuminating description of the development of theoretical physics, in the first half of the twentieth century, from a unique point of view: how it affected Eddington's thought. This will provide fascinating reading for scholars in the philosophy of science, theoretical physics, applied mathematics and the history of science. (shrink)
The notion of order as a universal and fundamental conceptual category is discussed as being based on sets of similar differences and different similarities. A discussion of relationships between order and disorder is followed by a proposal for a mathematical theory based on non-ordinality which could also have relevance for indistinguishables in physics.
Pre-existing mathematical formulations are generally used for the treatment of new scientific problems. In this note we show that the construction of mathematical structures from open physical, chemical, and biological problems leads to new intriguing mathematics of increasing complexity called iso-, geno-, and hyper-mathematics for the treatment of matter in reversible, irreversible, and multi-valued conditions, respectively, plus anti-isomorphic images called isodual mathematics for the treatment of antimatter. These novel mathematics are based on the lifting of the multiplicative unit (...) of ordinary fields (with characteristic zero) from its traditional value +1 into: (1) invertible, Hermitean, and single-valued units for isomathematics; (2) invertible, non-Hermitean, and single-valued units for genomathematics; and (3) invertible, non-Hermitean, and multi-valued units for hypermathematics; with corresponding liftings of the conventional associative product and consequential lifting of all branches of mathematics admitting a (left and right) multiplicative unit. An anti-Hermitean conjugation applied to the totality of quantities and their operation of the preceding mathematics characterizes the isodual mathematics. Intriguingly, the emerging formulations preserve the abstract axioms of conventional mathematics (that based on the unit +1). As such, the new formulations result to be new realizations of existing abstract mathematical axioms. We then show that the above mathematical advances permit corresponding liftings of conventional classical and quantum theories with a resolution of basic open problems in physics, chemistry, and biology, numerous experimental verifications, as well as new industrial applications. (shrink)
This paper considers the role of mathematics in the process of acquiring new knowledge in physics and astronomy. The defining of the notions of continuum and discreteness in mathematics and the natural sciences is examined. The basic forms of representing the heuristic function of mathematics at theoretical and empirical levels of knowledge are studied: deducing consequences from the axiomatic system of theory, the method of generating mathematical hypotheses, “pure” proofs for the existence of objects and processes, mathematical (...) modelling, the formation of mathematics on the basis of internal mathematical principles and the mathematical theory of experiment. (shrink)
This article is a brief formulation of a radical thesis. We start with the formalist doctrine that mathematical objects have no meanings; we have marks and rules governing how these marks can be combined. That's all. Then I go further by arguing that the signs of a formal system of mathematics should be considered as physical objects, and the formal operations as physical processes. The rules of the formal operations are or can be expressed in terms of the laws (...) of physics governing these processes. In accordance with the physicalist understanding of mind, this is true even if the operations in question are executed in the head. A truth obtained through (mathematical) reasoning is, therefore, an observed outcome of a neuro-physiological (or other physical) experiment. Consequently, deduction is nothing but a particular case of induction. (shrink)