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1994, Cadernos de História e Filosofia da Ciência (UNICAMP)
Metascientific criteria used for explaining or constraining physical space dimensionality and their historical relationship to prevailing causal systems are discussed. The important contributions by Aristotle, Kant and Ehrenfest to the dimensionality of space problem are considered and shown to be grounded on different causal explanations: {\it causa materialis} for Aristotle, {\it causa efficiens} for young Kant and an ingenious combination of {\it causa efficiens} and {\it causa formalis} for Ehrenfest. The prominent and growing rôle played by {\it causa formalis} in modern physical approaches to this problem is emphasized.
2015 •
In this article it is shown that a careful analysis of Kant’s Gedanken von der wahren Schätzung der lebendigen Kräfte und Beurtheilung der Beweise leads to a conclusion that does not match the usually accepted interpretation of Kant’s reasoning in 1747, according to which the young Kant supposedly establishes a relationship between the tridimensionality of space and Newton’s law of universal gravitation. Indeed, it is argued that this text does not yield a satisfactory explanation of space dimensionality, and actually restricts itself to justifying the tridimensionality of extension.
This paper recovers an important, century-old debate regarding the methodological and metaphysical foundations of dimensional analysis. Consideration of Richard Tolman's failed attempt to install the principle of similitude-the relativity of size-as the founding principle of dimensional analysis both clarifies the method of dimensional analysis and articulates two metaphysical positions regarding quantity dimensions. Tolman's position is quantity dimension fundamentalism. This is a commitment to dimensional realism and a set of fundamental dimensions which ground all further dimensions. The opposing position, developed primarily by Bridgman, is quantity dimension conventionalism. Conventionalism is an anti-realism regarding dimensional structure, holding our non-representational dimensional systems have basic quantity dimensions fixed only by convention. This metaphysical dispute was left somewhat unsettled. It is shown here that both of these positions face serious problems: fundamentalists are committed to surplus dimensional structure; conventionalists cannot account for empirical constraints on our dimensional systems nor the empirical success of dimensional analysis. It is shown that an alternative position is available which saves what is right in both: quantity dimension functionalism.
2012 •
The Historical Background of Dimensionality as far as I know, the first modern mathematician conscious about dimensionality or space dimension is Bernhard Riemann. In his third doctoral paper(, Friedrich Gauss reviewed it and gave him the oral examination for it, then, he was so fascinated by the theory that he fell down a gutter on the way home after the interview.) he established so called in today Riemann Geometry. This is also known as mathematics Albert Einstein applied it when he established his theory of general relativity. In his thesis, he offered a tricky but profound idea of dimensionality by applying Pythagorean theorem; we know it is defined by a^2+b^2=c^2 in 2-dimensional space, and it is actually a^2+b^2+c^2=d^2 in 3-dimensional space. So he inductively generalized it as a^2+b^2+c^2+d^2+⋯=z^2 in higher dimensional space. Such a concept of higher dimensional space is succeeded to Einstein inspired by Theodor Kaluza and Oscar Klein in the process of establishing his unified theory including general relativity in his later life. It is also known as the Kaluza-Klein theory in today's theoretical physics. The K-K theory insists on that; there is higher dimensional space more than our 3-dimesnional space like 4, 5, 6, 7 dimensional one, but we can't detect it because it is compactified very very small. These days, physicists consider that there is wider space but we can't move to there just like an ant stuck to the wall(2-dimensional plane) don't know 3-dimensional space; in other words, 2-dimensional space is subset of 3-dimensional space. However, these ideas are really true? In a Dawn of October 5th in 1991 One night of October 4th in 1991, I was thinking of a strangeness why and how photon or electron have a duality of particle and wave in my bed; certainly quantum field theory answers the question by introducing various convenient fields, but I wanted to think of the duality without such fields or so called "mode". I couldn't fell asleep yet in the midnight because I was still pondering of its strangeness. Afterward, the time dawn breaks was coming, but still imagining a picture that the circle was growing without medium. At the time, God whispered it to me(; suddenly I got an inspiration.) That was a new idea of coordinate geometry based on quite a different concept from which Rene Descartes got about 400 years ago!
Fundamenta Scientiae
On the Physical Problem of Spatial Dimensions: An Alternative Procedure to Stability ArgumentsWhy is space 3{dimensional? The rst answer to this question, entirely based on Physics, was given by Ehrenfest, in 1917, who showed that the stability requirement for n-dimensional two-body planetary system very strongly constrains space dimensionality, favouring 3-d. This kind of approach will be generically called "stability postulate" throughout this paper and was shown by Tangherlini, in 1963, to be still valid in the framework of general relativity as well as for quantum mechanical hydrogen atom, giving the same constraint for space-dimensionality. In the present work, before criticizing this methodology, a brief discussion has been introduced, aimed at stressing and clarifying some general physical aspects of the problem of how to determine the number of space dimensions. Then, the epistemological consequences of Ehrenfest's methodology are critically reviewed. An alternative procedure to get at the proper number of dimensions, in which the stability postulate (and the implicit singularities in three-dimensional physics) are not an essential part of the argument, is proposed. In this way, the main epistemological problems contained in Ehrenfest's original idea are avoided. The alternative methodology proposed in this paper is realized by obtaining and discussing the n-dimensional quantum theory as expressed in Planck's law, de Broglie relation and the Heisenberg uncertainty relation. As a consequence, it is possible to propose an experiment, based on thermal nêutron diffraction by crystals, to directly measure space dimensionality. Finally the distinguished role of Maxwell's electromagnetic theory in the determination of space dimensionality is stressed.
This paper approaches the question of space dimensionality by discussing a neglected argument proposed by Hermann Weyl in 1955. In Why is the World Four-Dimensional? (1955), Weyl offered a different argument from the one generally attributed to him and presented in Raum-Zeit-Materie. In the first sections of the paper, this new argument and its features are spelled-out, and in the last section, I shall develop some useful remarks on the concept of topology of causation that can still inform our reflection on the dimensionality of the world.
The Nature of Time: Geometry, Physics and Perception
Geometry of time and Dimensionality of Space2003 •
A general sketch on how the problem of space dimensionality depends on anthropic arguments is presented. Several examples of how life has been used to constraint space dimensionality (and vice-versa) are reviewed. In particular, the influences of three-dimensionality in the solar system stability and the origin of life on Earth are discussed. New constraints on space dimensionality and on its invariance in very large spatial and temporal scales are also stressed. Preamble More and more researchers devote themselves to understand or at least shed some light onto two apparently uncorrelated scientific issues: The origin of life [1-6] and the threefold nature of physical space [7-9] The first is per se a fascinating subject and probably one of the most difficult problems to be solved; the second is related to the dream of a Grand Unification in Physics. The issue of the origin of life has a continuously renewed interest due to improvements in the accuracy of measuring instruments. Another contribution comes from new data originated from space telescopes and satellites, as COBE, as will be stressed in this paper. The discovery of planets beyond our Solar system using data from the new ground and space-based telescopes, such as Kepler, is an additional source of interest. Furthermore, the interest on the quest of space dimensionality is now being renewed as the search for experimental evidences of extra dimensions is part of the contemporary tendency to investigate physics " beyond the Standard Model " in Collider Experiments [10]. A modern and comprehensive survey of dimensionality can be found elsewhere [11]. From a modern perspective, the idea these two open problems can be somehow entangled can be traced back to Schrödinger's seminal ideas collected in his 1944 famous book " What is life? " [12]. There, it is claimed that life should be understood in terms of the new Quantum Physics, relating the stability of genes to the discontinuous (quantum) transitions of physical states. Therefore, it is expected a significant contribution of Physics for the understanding of life. In particular, the influence of the topology of physical space (like its dimensionality) on the kind of life we know should not be neglected. In spite of this expectancy, there has been an enormous amount of speculation about the origin of life, with little heed to constraints that might be imposed by the physical settings [3]. This is particularly true concerning the influence of topological properties of space, like its dimensionality. The main scope of this brief report is to summarize how physical and philosophical approaches to the problem of space dimensionality are related to life through a version of the Anthropic Principle. The paper also aims to discuss the temporal scale of each kind of restriction imposed onto space dimensionality. Finally, it attempts to argue that recent analysis of the microwave background radiation spectrum gives rise to the first constraint on dimensionality for a given temporal scale larger than that usually required for the existence of life. The Historical Roots That space is three dimensional seems to be so obvious to laymen and even to scientists that one can easily disregard it as a scientific problem. Indeed, in almost all physical and chemical theories developed along centuries, dimensionality-an essential topological feature of space-is merely assumed as a given truth, as an unquestionable matter of fact supported by visual, tactile and kinesthetic perception of space. However, in this review it will be shown that things can go in a different way, and that we can improve our comprehension about the possibility of having physical space dimensionality fluctuating or not in a large spatial and temporal scale. How this could affect the origin of life, and, vice-versa, how the conditions for the existence of life are used to constrain the dimensionality are issues that will also be addressed. The way Aristotle and Kant caught a glimpse of the possibility of relating space dimensionality with the perceived World and life should be recognized and emphasized as two keystones. It is well known that Aristotle did not develop a theory of space. He instead weaved a theory of place (topos), which is mainly discussed in his Physics. In a modern language, the Aristotelian topos has always been a bi-dimensional surface (the inner limit of a containing body) and space has been a kind of collection of all possible places. However, it is not in this book that he treated the question of dimensionality, which is considered, instead, in his cosmological and biological texts, namely On the Heavens (De Caelo) and Movement of Animals. In fact, in the Book I of De Caelo he says that " we cannot pass beyond body to a further kind, as we passed from length to surface, and to surface to body'' [13], since he admitted body alone to be determined by the three dimensions. However, it was only in the Movement of Animals that the Stagirite tried to develop a theory of dimensions based on the study of movement. Since he had a hierarchical conception of space
2004 •
As has been shown and extensively dealt with in early and more recent scholarship, Neo-Assyrian palatial wall reliefs went through many thematic changes throughout their two hundred and fifty years of existence. One of their conspicuous traits was a gradual abandoning of magical-religious subject matters, represented by protective supernatural beings, in favour of larger and more detailed historical compositions — mostly of a belligerent nature — revealing, for the first time in antiquity, a truer sense of narrative display. As the narrative-historical themes were rightly considered to be an innovative and prominent contribution of Assyrian imagery to the history of art, extensive efforts have been devoted to the study of these compositions within the context of Assyrian palaces.In the present contribution I intend, however, to concentrate on the “losing” side of Assyrian palatial decoration, namely to focus on the visibility of apotropaic fantastic creatures rendered on wall relief...
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