What is Structure? Why Care about It?

2021 ◽  
pp. 17-51
Author(s):  
Jill North
Keyword(s):  
Special Relativity ◽  
Euclidean Plane ◽  
Reference Frames ◽  
Coordinate Systems ◽  
Newtonian Physics ◽  
Different Types ◽  
And Mathematics ◽  
Relevant Notion ◽  
Mathematics And Physics

This chapter explains the notion of structure that will be the focus of the book and illustrates it by means of examples drawn from mathematics and physics. The discussion begins with a simple example of the structure of the Euclidean plane, and goes on to explain how similar ideas apply to physical theories such as Newtonian physics and special relativity. Taken together, the examples illustrate that this notion is implicit in many aspects of our theorizing in physics and mathematics. The chapter also discusses the idea of allowable coordinate systems and reference frames; contrasts the relevant notion of structure with other related notions, including invariance, symmetry, and objectivity; and explains how to compare different types and amounts of structure.

The Equivalence Principle

2018 ◽  
Author(s):  
David M. Wittman
Keyword(s):  
General Relativity ◽  
Special Relativity ◽  
Equivalence Principle ◽  
Time Dilation ◽  
Gravitational Redshift ◽  
Coordinate Systems ◽  
Spacetime Metric

The equivalence principle is an important thinking tool to bootstrap our thinking from the inertial coordinate systems of special relativity to the more complex coordinate systems that must be used in the presence of gravity (general relativity). The equivalence principle posits that at a given event gravity accelerates everything equally, so gravity is equivalent to an accelerating coordinate system.This conjecture is well supported by precise experiments, so we explore the consequences in depth: gravity curves the trajectory of light as it does other projectiles; the effects of gravity disappear in a freely falling laboratory; and gravitymakes time runmore slowly in the basement than in the attic—a gravitational form of time dilation. We show how this is observable via gravitational redshift. Subsequent chapters will build on this to show how the spacetime metric varies with location.

Physics, Structure, and Reality

2021 ◽  
Author(s):  
Jill North
Keyword(s):  
Scientific Realism ◽  
Physical Theory ◽  
Scientific Explanation ◽  
Classical Mechanics ◽  
Careful Attention ◽  
Coordinate Systems ◽  
Spacetime Structure ◽  
The World ◽  
Background Assumption ◽  
And Mathematics

How do we figure out the nature of the world from a mathematically formulated physical theory? What do we infer about the world when a physical theory can be mathematically formulated in different ways? Physics, Structure, and Reality addresses these questions, questions that get to the heart of the project of interpreting physics—of figuring out what physics is telling us about the world. North argues that there is a certain notion of structure, implicit in physics and mathematics, that we should pay careful attention to, and that doing so sheds light on these questions concerning what physics is telling us about the nature of reality. Along the way, lessons are drawn for related topics such as the use of coordinate systems in physics, the differences among various formulations of classical mechanics, the nature of spacetime structure, the equivalence of physical theories, and the importance of scientific explanation. Although the book does not explicitly defend scientific realism, instead taking this to be a background assumption, the account provides an indirect case for realism toward our best theories of physics.

Circles on the Complex Plane

2021 ◽  
pp. 3-12
Author(s):  
A. Girsh
Keyword(s):  
Euclidean Space ◽  
Complex Plane ◽  
Euclidean Plane ◽  
Complex Space ◽  
Radical Center ◽  
Special Cases ◽  
Different Types ◽  
Different Dimensions ◽  
The Given ◽  
Made In

The Euclidean plane and Euclidean space themselves do not contain imaginary elements by definition, but are inextricably linked with them through special cases, and this leads to the need to propagate geometry into the area of imaginary values. Such propagation, that is adding a plane or space, a field of imaginary coordinates to the field of real coordinates leads to various variants of spaces of different dimensions, depending on the given axiomatics. Earlier, in a number of papers, were shown examples for solving some urgent problems of geometry using imaginary geometric images [2, 9, 11, 13, 15]. In this paper are considered constructions of orthogonal and diametrical positions of circles on a complex plane. A generalization has been made of the proposition about a circle on the complex plane orthogonally intersecting three given spheres on the proposition about a sphere in the complex space orthogonally intersecting four given spheres. Studies have shown that the diametrical position of circles on the Euclidean E-plane is an attribute of the orthogonal position of the circles’ imaginary components on the pseudo-Euclidean M-plane. Real, imaginary and degenerated to a point circles have been involved in structures and considered, have been demonstrated these circles’ forms, properties and attributes of their orthogonal position. Has been presented the construction of radical axes and a radical center for circles of the same and different types. A propagation of 2D mutual orthogonal position of circles on 3D spheres has been made. In figures, dashed lines indicate imaginary elements.

Electromagnetism and special relativity

2018 ◽  
Author(s):  
J. Pierrus
Keyword(s):  
Electromagnetic Field ◽  
Special Relativity ◽  
Point Charge ◽  
Reference Frames ◽  
Classical Electrodynamics ◽  
Covariant Form ◽  
Field Tensor ◽  
Electromagnetic Field Tensor ◽  
Preferred Reference Frame ◽  
Physical Quantities

In 1905, when Einstein published his theory of special relativity, Maxwell’s work was already about forty years old. It is therefore both remarkable and ironic (recalling the old arguments about the aether being the ‘preferred’ reference frame for describing wave propagation) that classical electrodynamics turned out to be a relativistically correct theory. In this chapter, a range of questions in electromagnetism are considered as they relate to special relativity. In Questions 12.1–12.4 the behaviour of various physical quantities under Lorentz transformation is considered. This leads to the important concept of an invariant. Several of these are encountered, and used frequently throughout this chapter. Other topics considered include the transformationof E- and B-fields between inertial reference frames, the validity of Gauss’s law for an arbitrarily moving point charge (demonstrated numerically), the electromagnetic field tensor, Maxwell’s equations in covariant form and Larmor’s formula for a relativistic charge.

scholarly journals Bhabha scattering in very special relativity at finite temperature

2020 ◽  
Vol 80 (8) ◽  
Author(s):  
Alesandro Ferreira dos Santos ◽  
Faqir C. Khanna
Keyword(s):  
Differential Cross Section ◽  
Special Relativity ◽  
Cross Section ◽  
Differential Cross ◽  
Local Form ◽  
Bhabha Scattering ◽  
Thermo Field Dynamics ◽  
Different Types ◽  
Non Local ◽  
Very Special Relativity

Abstract In this paper the differential cross section for Bhabha scattering in the very special relativity (VSR) framework is calculated. The main characteristic of the VSR is to modify the gauge invariance. This leads to different types of interactions appearing in a non-local form. In addition, using the Thermo Field Dynamics formalism, thermal corrections for the differential cross section of Bhabha scattering in VSR framework are obtained.

scholarly journals Relations Between Celestial and Selenocentric Reference Frames

1981 ◽  
Vol 63 ◽  
pp. 268-280
Author(s):  
J. Kovalevsky
Keyword(s):  
Reference Frames ◽  
Center Of Mass ◽  
Great Accuracy ◽  
Laser Ranging ◽  
Lunar Laser Ranging ◽  
Geometric Means ◽  
The Earth ◽  
Lunar Laser ◽  
Unique System ◽  
Different Types

AbstractThe very great accuracy with which the motions of the Moon can now be monitored by laser ranging, differential VLBI and occultation observations, implies that the interpretation of the measurements is conditioned by the choice and the accurate knowledge of a selenocentric, a terrestrial and a celestial frames. Two different types of selenocentric reference frames can be envisioned. The present selenographic frames are discussed but the author proposes that one should introduce a system defined by a purely geometric means. Some consequences of such a choice are discussed. One feature of the future conventional terrestrial frame is very important for Earth-Moon dynamics. Its origin should coincide with the center of mass of the Earth as determined by lunar laser ranging. As far as the quasi-inertial reference systems are concerned, the liaisons between a purely lunar dynamical system, subject to some hardly modelable effects, and purely celestial systems are analysed. The reduction of observations made with various techniques implies the use of different systems, and several problems are stated that should be solved before a unique system for Earth-Moon dynamics might be used.

scholarly journals Venues for Analytical Reasoning Problems: How Children Produce Deductive Reasoning

2020 ◽  
Vol 10 (6) ◽  
pp. 169
Author(s):  
Susana Carreira ◽  
Nélia Amado ◽  
Hélia Jacinto
Keyword(s):  
Content Analysis ◽  
Deductive Reasoning ◽  
Qualitative Content Analysis ◽  
Web Based ◽  
School Competition ◽  
Analytical Reasoning ◽  
Reasoning Problem ◽  
Different Types ◽  
Per Se ◽  
And Mathematics

The research on deductive reasoning in mathematics education has been predominantly associated with the study of proof; consequently, there is a lack of studies on logical reasoning per se, especially with young children. Analytical reasoning problems are adequate tasks to engage the solver in deductive reasoning, as they require rule checking and option elimination, for which chains of inferences based on premises and rules are accomplished. Focusing on the solutions of children aged 10–12 to an analytical reasoning problem proposed in two separate settings—a web-based problem-solving competition and mathematics classes—this study aims to find out what forms of deductive reasoning they undertake and how they express that reasoning. This was done through a qualitative content analysis encompassing 384 solutions by children participating in a beyond-school competition and 102 solutions given by students in their mathematics classes. The results showed that four different types of deductive reasoning models were produced in the two venues. Moreover, several representational resources were found in the children’s solutions. Overall, it may be concluded that moderately complex analytical reasoning tasks can be taken into regular mathematics classes to support and nurture young children’s diverse deductive reasoning models.

Discourse cohesion in text and tutorial dialogue

2007 ◽  
Vol 15 (3) ◽  
pp. 199-213 ◽  
Author(s):  
Arthur C. Graesser ◽  
Moongee Jeon ◽  
Yan Yan ◽  
Zhiqiang Cai
Keyword(s):  
College Students ◽  
Natural Language ◽  
Semantic Similarity ◽  
Software Tool ◽  
Pedagogical Agent ◽  
Newtonian Physics ◽  
Tutorial Dialogue ◽  
Logical Operators ◽  
Different Types ◽  
Human Tutoring

Discourse cohesion is presumably an important facilitator of comprehension when individuals read texts and hold conversations. This study investigated components of cohesion and language in different types of discourse about Newtonian physics: A textbook, textoids written by experimental psychologists, naturalistic tutorial dialoguebetween expert human tutors and college students, andAutoTutor tutorial dialogue between a computer tutor and students (AutoTutor is an animated pedagogical agent that helps students learn about physics by holding conversations in natural language). We analyzed the four types of discourse with Coh-Metrix, a software tool that measures discourse on different components of cohesion, language, and readability. The cohesion indices included co-reference, syntactic and semantic similarity, causal cohesion, incidence of cohesion signals (e.g., connectives, logical operators), and many other measures. Cohesion data were quite similar for the two forms of discourse in expository monologue (textbooks and textoids) and for the two types of tutorial dialogue (i.e., students interacting with human tutors and AutoTutor), but very different between the discourse of expository monologue and tutorial dialogue. Coh-Metrix was also able to detect subtle differences in the language and discourse of AutoTutor versus human tutoring.

Late Scholastic Analyses of Inductive Reasoning

2020 ◽  
Vol 17 (1) ◽  
pp. 35-66
Author(s):  
Miroslav Hanke ◽  
Keyword(s):  
Modern Philosophy ◽  
Early Modern ◽  
Inductive Reasoning ◽  
The Other ◽  
Macro Level ◽  
Newtonian Physics ◽  
Academic Philosophy ◽  
Micro Level ◽  
Scholastic Philosophy ◽  
Mathematics And Physics

The late scholastic era was, among others, contemporary to the “emergence of probability”, the German academic philosophy from Leibniz to Kant, and the introduction of Newtonian physics. Within this era, two branches of the late-scholastic analysis of induction can be identified, one which can be thought of as a continual development of earlier scholastic approaches, while the other one absorbed influences of early modern philosophy, mathematics, and physics. Both branches of scholastic philosophy share the terminology of modalities, probability, and forms of (inductive) arguments. Furthermore, induction was commonly considered valid as a result of being a covert syllogism. Last but not least, there appears to be a difference in emphasis between the two traditions’ analyses of induction: while Tolomei discussed the theological presuppositions of induction, Amort’s “leges contingentium” exemplify the principles of induction by aleatory phenomena and Boscovich’s rules for inductive arguments are predominately concerned with the generalisation of macro-level observations to the micro-level.

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