scholarly journals On mathematical concepts of the material world

1906 ◽  
Vol 77 (517) ◽  
pp. 290-291 ◽  
Keyword(s):  
Euclidean Space ◽  
Euclidean Geometry ◽  
Ordinary Language ◽  
Abstract Form ◽  
Mathematical Concepts ◽  
Material World ◽  
Mathematical Investigation ◽  
Definition Of ◽  
Set Of Points

The object of this memoir is to initiate the mathematical investigation of various possible ways of conceiving the nature of the Material World. In so far as its results are worked out in precise mathematical detail, the memoir is concerned with the possible relations to space of the ultimate entities which (in ordinary language) constitute the “stuff” in space. An abstract logical statement of this limited problem, in the form in which it is here conceived, is as follows:—Given a set of entities which form the field of a certain polyadic ( i. e ., many-termed) relation R . What “axioms” satisfied by R have as their consequence that the theorems of Euclidean Geometry are the expression of certain properties of the field of R ? If the set of entities are themselves to be the set of points of the Euclidean Space, the problem, thus set, narrows itself down to the problem of the axioms of Euclidean Geometry. The solution of this narrower problem of the axioms of geometry is assumed ( cf . Part II, Concept I) without proof in the form most convenient for this wider investigation. Poincaré has used language which might imply the belief that, with the proper definitions, Euclidean Geometry can be applied to express properties of the field of any polyadic relation whatever. His context, however, suggests that his thesis is, that in a certain sense (obvious to mathematicians) the Euclidean and certain other geometries are interchangeable, so that, if one can be applied, then each of the others can also be applied. Be that as it may, the problem here discussed is to find various formulations of axioms concerning R , from which, with appropriate definitions, the Euclidean Geometry issues as expressing properties of the field of R . In view of the existence of change in the Material World, the investigation has to be so conducted as to introduce, in its abstract form, the idea of time, and to provide for the definition of velocity and acceleration.

scholarly journals XIV. On mathematical concepts of the material world

1906 ◽  
Vol 205 (387-401) ◽  
pp. 465-525 ◽  
Keyword(s):  
Physical Science ◽  
Euclidean Geometry ◽  
Ordinary Language ◽  
Abstract Form ◽  
Mathematical Concepts ◽  
Material World ◽  
Mathematical Investigation ◽  
Definition Of ◽  
Physical Laws ◽  
Set Of Points

The object of this memoir is to initiate the mathematical investigation of various possible ways of conceiving the nature of the material world. In so far as its results are worked out in precise mathematical detail, the memoir is concerned with the possible relations to space of the ultimate entities which (in ordinary language) constitute the “stuff” in space. An abstract logical statement of this limited problem, in the form in which it is here conceived, is as follows: Given a set of entities which form the field of a certain polyadic ( i. e ., many-termed) relation R, what “axioms” satisfied by R have as their consequence, that the theorems of Euclidean geometry are the expression of certain properties of the field of R ? If the set of entities are themselves to be the set of points of the Euclidean space, the problem, thus set, narrows itself down to the problem of the axioms of Euclidean geometry. The solution of this narrower problem of the axioms of geometry is, assumed ( cf . Part II., Concept I.) without proof in the form most convenient for this wider investigation. But in Concepts III., IV., and V., the entities forming the field of R are the “stuff,” or part of the “stuff,” constituting the moving material world. Poincaré has used language which might imply the belief that, with the proper definitions, Euclidean geometry can be applied to express properties of the field of any polyadic relation whatever. His context, however, suggests that his thesis is, that in a certain sense (obvious to mathematicians) the Euclidean and certain other geometries are interchangeable, so that, if one can be applied, then each of the others can also be applied. Be that as it may, the problem, here discussed, is to find various formulations of axioms concerning R, from which, with appropriate definitions, the Euclidean geometry issues as expressing properties of the field of R. In view of the existence of change in the material world, the investigation has to be so conducted as to introduce, in its abstract form, the idea of time, and to provide for the definition of velocity and acceleration. The general problem is here discussed purely for the sake of its logical ( i. e ., mathematical) interest. It has an indirect bearing on philosophy by disentangling the essentials of the idea of a material world from the accidents of one particular concept. The problem might, in the future, have a direct bearing upon physical science if a concept widely different from the prevailing concept could be elaborated, which allowed of a simpler enunciation of physical laws. But in physical research so much depends upon a trained imaginative intuition, that it seems most unlikely that existing physicists would, in general, gain any advantage from deserting familiar habits of thought.

On One Property of a Circle on the Coordinate Plane

2017 ◽  
Vol 5 (2) ◽  
pp. 13-24
Author(s):  
Графский ◽  
O. Grafskiy ◽  
Пономарчук ◽  
Yu. Ponomarchuk
Keyword(s):  
Euclidean Geometry ◽  
Coordinate Plane ◽  
Non Euclidean Geometry ◽  
Geometrical Form ◽  
Constant Distance ◽  
Intersection Points ◽  
Definition Of ◽  
Set Of Points ◽  
First Time

Descartes’ and Fermat's method allowed to define many geometrical forms, including circles, on the coordinate plane by means of the arithmetic equations and to make necessary analytical operations in order to solve many problems of theoretical and applied research in various scientific areas, for example. However, the equations of a circle and other conics in the majority of research topics are used in the subsequent analysis of applied problems, or for analytical confirmation of constructive solutions in geometrical research, according to Russian geometrician G. Monge and others, including. It is natural to consider a circle as a locus of points, equidistant from a given point — a center of the circle, with a constant distance R. There is another definition of a circle: a set of points from which a given segment is visible under constant directed angle. Besides, a circle is accepted to model the Euclid plane in the known scheme of non-Euclidean geometry of Cayley-Klein, it is the absolute which was given by A. Cayley for the first time in his memoirs. It is possible to list various applications of this geometrical form, especially for harmonism definition of the corresponding points, where the diametral opposite points of a circle are accepted as basic, and also for construction of involutive compliances. The construction of tangents to a circle can be considered as a classical example. Their constructive definition is simple, but also constructions on the basis of known projective geometry postulates are possible (a hexagon when modeling a series of the second order, Pascal's lines). These postulates can be applied to construction of tangents to a circle (to an ellipse and hyperboles to determination of imaginary points of intersection of a circle and a line. This paper considers the construction of tangents to a circle without the use of arches of auxiliary circles, which was applied in order to determine the imaginary points of intersection of a circle and a line (an axis of coordinates). Besides, various dependences of parameter p2, which is equal to the product of the values of the intersection points’ coordinates of a circle and coordinate axes, are analytically determined.

Pascal's Prism

2012 ◽  
Vol 96 (536) ◽  
pp. 213-220
Author(s):  
Harlan J. Brothers
Keyword(s):  
Probability Theory ◽  
Fractal Geometry ◽  
Euclidean Geometry ◽  
Fibonacci Series ◽  
Pascal’S Triangle ◽  
Number Sequences ◽  
Definition Of ◽  
Image Position ◽  
Pascal's Triangle

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

Mutual Determination, Concrescence and Transition. Whitehead’s Speculative Conception of Temporal Subjectivity Interpreted from a Merleau-Pontyan Standpoint

2020 ◽  
Vol 22 ◽  
pp. 101-117
Author(s):  
Luca Vanzago ◽  
Keyword(s):  
Mathematical Concepts ◽  
Material World ◽  
Natural Knowledge ◽  
Constant Change ◽  
Interpretive Approach ◽  
Dual Notion ◽  
Mathematics And Physics ◽  
Processual Approach

The interpretive approach adopted in this paper is influenced by Merleau-Ponty’s philosophy and in particular by his understanding of Nature, which in turn takes into consideration Whitehead’s work. Whitehead’s philosophy of organism is seen by its author as the metaphysical generalization of problems found in his investigation of natural knowledge. Whitehead admits that a speculative approach is necessitated by the very questions arising from the mathematical concepts of the material world and the revolutions undergone in logic, mathematics and physics at the turn of the century.Whitehead’s understanding of nature is framed from the beginning in terms of a processual approach. However, this notion of process is not fully worked out in the epistemological works and requires a metaphysical deepening. This is due to the fact that the notion of duration adopted in the epistemological works is not sufficient to convey the notion of process. This lack of adequacy is coupled by Whitehead with the need to interpret process in terms of experience. In turn, this notion of experience is wider than the usual one, for it implies that there is experience from the lowest levels onwards. Matter itself experiences. Seen in this perspective, reality is thus conceived in terms of a whole in constant change, whose parts are in mutual connection. This conception derives from Whitehead’s criticism of Aristotle’s substantialism and from his preference for a relationist ontology. The outcome of this approach is a speculative conception of reality in terms of a twofold notion of process: concrescence and transition, which Whitehead sees as the two faces of the creative advance of nature. This dual notion of process is interpreted in this essay in a merleau-pontyan perspective.

scholarly journals A Helly-type theorem on a sphere

1967 ◽  
Vol 7 (3) ◽  
pp. 323-326 ◽  
Author(s):  
M. J. C. Baker
Keyword(s):  
Euclidean Space ◽  
Convex Sets ◽  
Type Theorem ◽  
Dimensional Euclidean Space ◽  
Dimensional Sphere ◽  
Strongly Convex ◽  
Set Of Points

The purpose of this paper is to prove that if n+3, or more, strongly convex sets on an n dimensional sphere are such that each intersection of n+2 of them is empty, then the intersection of some n+1 of them is empty. (The n dimensional sphere is understood to be the set of points in n+1 dimensional Euclidean space satisfying x21+x22+ …+x2n+1 = 1.)

scholarly journals KONSISTENSI AKSIOMA-AKSIOMA TERHADAP ISTILAH-ISTILAH TAKTERDEFINISI GEOMETRI HIPERBOLIK PADA MODEL PIRINGAN POINCARE

EDUPEDIA ◽  
2018 ◽  
Vol 2 (2) ◽  
pp. 161
Author(s):  
Febriyana Putra Pratama ◽  
Julan Hernadi
Keyword(s):  
Half Plane ◽  
Hyperbolic Geometry ◽  
Euclidean Geometry ◽  
Cross Ratio ◽  
Literature Study ◽  
Disk Model ◽  
Non Euclidean Geometry ◽  
The Cross ◽  
Definition Of ◽  
Poincaré Disk

This research aims to know the interpretation the undefined terms on Hyperbolic geometry and it’s consistence with respect to own axioms of Poincare disk model. This research is a literature study that discusses about Hyperbolic geometry. This study refers to books of Foundation of Geometry second edition by Gerard A. Venema (2012), Euclidean and Non Euclidean Geometry (Development and History)  by Greenberg (1994), Geometry : Euclid and Beyond by Hartshorne (2000) and Euclidean Geometry: A First Course by M. Solomonovich (2010). The steps taken in the study are: (1) reviewing the various references on the topic of Hyperbolic geometry. (2) representing the definitions and theorems on which the Hyperbolic geometry is based. (3) prepare all materials that have been collected in coherence to facilitate the reader in understanding it. This research succeeded in interpret the undefined terms of Hyperbolic geometry on Poincare disk model. The point is coincide point in the Euclid on circle . Then the point onl γ is not an Euclid point. That point interprets the point on infinity. Lines are categoried in two types. The first type is any open diameters of   . The second type is any open arcs of circle. Half-plane in Poincare disk model is formed by Poincare line which divides Poincare field into two parts. The angle in this model is interpreted the same as the angle in Euclid geometry. The distance is interpreted in Poincare disk model defined by the cross-ratio as follows. The definition of distance from  to  is , where  is cross-ratio defined by  . Finally the study also is able to show that axioms of Hyperbolic geometry on the Poincare disk model consistent with respect to associated undefined terms.

Material efficiency in a multi-material world

2013 ◽  
Vol 371 (1986) ◽  
pp. 20120002 ◽  
Author(s):  
Reid Lifset ◽  
Matthew Eckelman
Keyword(s):  
Economic Effects ◽  
Material World ◽  
Multiple Resources ◽  
Material Efficiency ◽  
Material Choice ◽  
Multiple Materials ◽  
Materials Used ◽  
Environmental Goals ◽  
Definition Of ◽  
The Impact

Material efficiency—using less of a material to make a product or supply a service—is gaining attention as a means for accomplishing important environmental goals. The ultimate goal of material efficiency is not to use less physical material but to reduce the impacts associated with its use. This article examines the concept and definition of material efficiency and argues that for it to be an effective strategy it must confront the challenges of operating in a multi-material world, providing guidance when materials are used together and when they compete. A series of conceptions of material efficiency are described, starting with mass-based formulations and expanding to consider multiple resources in the supply chain of a single material, and then to multiple resources in the supply chains of multiple materials used together, and further to multiple environmental impacts. The conception of material efficiency is further broadened by considering material choice, exploring the technical and economic effects both of using less material and of materials competition. Finally, this entire materials-based techno-economic system is considered with respect to the impact of complex policies and political forces. The overall goal here is to show how the concept of material efficiency when faced with more expansive—and yet directly relevant—definitional boundaries is forced to confront analytical challenges that are both familiar and difficult in life cycle assessment and product-based approaches.

Interactive Web-Based Tools for Learning Mathematics

2012 ◽  
pp. 274-306
Author(s):  
Barry Cherkas ◽  
Rachael M. Welder
Keyword(s):  
Mathematics Teachers ◽  
Dynamic Capabilities ◽  
Preservice Teacher Education ◽  
Mathematical Concepts ◽  
Web Based ◽  
Working Definition ◽  
Learning Mathematics ◽  
Collegiate Mathematics ◽  
Definition Of ◽  
Interactive Resources

There is an abundance of Web-based resources designed for mathematics teachers and learners at every level. Some of these are static, while others are interactive or dynamic, giving mathematics learners opportunities to develop visualization skills, explore mathematical concepts, and obtain solutions to self-selected problems. Research into the efficacy of online mathematics demonstrations and interactive resources is lacking, but it is clear that not all online resources are equal from a pedagogical viewpoint. In this chapter, a number of popular and relevant websites for collegiate mathematics and collegiate preservice teacher education are examined. They are reviewed and investigated in terms of their interactivity, dynamic capabilities, pedagogical strengths and weaknesses, the practices they employ, and their potential to enhance mathematical learning both inside and outside of the collegiate classroom. Culled from these reviews is a working definition of “best practices”: condensing difficult mathematical concepts into representations and models that clarify ideas with minimal words, thereby enabling a typical student to grasp, quickly and easily, the underlying mathematics.

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