The Relationship Between Nontrivial Automorphisms and Order of Fields

1968 ◽  
Vol 61 (3) ◽  
pp. 246-250
Author(s):  
Victor Keiser
Keyword(s):  
Mathematics Teacher ◽  
Definite Article ◽  
Real Numbers ◽  
The Real ◽  
Nontrivial Automorphism ◽  
Ordered Fields ◽  
The Relationship

In the article “A Nontrivial Automorphism of the Field of Real Numbers” (THE MATHEMATICS TEACHER, December 1966), Robert F. Lawler defines operations * and Δ on a certain set F which he refers to as the field of real numbers. Before going further, let us point out that the use of the definite article in the phrase “the field of real numbers” is justified by the well-known theorem stating that any two complete ordered fields are isomorphic; it does not arise from the existence of some particular distinguished set of objects which we call the real numbers.

scholarly journals A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007 ◽  
Vol 72 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Ehud Hrushovski ◽  
Ya'acov Peterzil
Keyword(s):  
Real Numbers ◽  
The Real ◽  
First Order ◽  
Real Closed Fields ◽  
Minimal Structure ◽  
Closed Fields ◽  
New Construction ◽  
The Relationship

AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

Banach games

1984 ◽  
Vol 49 (2) ◽  
pp. 343-375 ◽  
Author(s):  
Chris Freiling
Keyword(s):  
Real Number ◽  
Perfect Information ◽  
Real Numbers ◽  
The Real ◽  
Infinite Game ◽  
Image Position ◽  
The Relationship

Abstract.Banach introduced the following two-person, perfect information, infinite game on the real numbers and asked the question: For which sets A ⊆ R is the game determined?Rules: The two players alternate moves starting with player I. Each move an is legal iff it is a real number and 0 < an, and for n > 1, an < an−1. The first player to make an illegal move loses. Otherwise all moves are legal and I wins iff exists and .We will look at this game and some variations of it, called Banach games. In each case we attempt to find the relationship between Banach determinacy and the determinacy of other well-known and much-studied games.

Nonstandard Student Conceptions About Infinitesimals

2010 ◽  
Vol 41 (2) ◽  
pp. 117-146 ◽  
Author(s):  
Robert Ely
Keyword(s):  
Real Number ◽  
Nonstandard Analysis ◽  
Number Line ◽  
Real Numbers ◽  
Student Conceptions ◽  
The Real ◽  
New Perspective ◽  
Real Number Line ◽  
The Relationship

This is a case study of an undergraduate calculus student's nonstandard conceptions of the real number line. Interviews with the student reveal robust conceptions of the real number line that include infinitesimal and infinite quantities and distances. Similarities between these conceptions and those of G. W. Leibniz are discussed and illuminated by the formalization of infinitesimals in A. Robinson's nonstandard analysis. These similarities suggest that these student conceptions are not mere misconceptions, but are nonstandard conceptions, pieces of knowledge that could be built into a system of real numbers proven to be as mathematically consistent and powerful as the standard system. This provides a new perspective on students' “struggles” with the real numbers, and adds to the discussion about the relationship between student conceptions and historical conceptions by focusing on mechanisms for maintaining cognitive and mathematical consistency.

Frege on the Real Numbers

2019 ◽  
pp. 343-383
Author(s):  
Eric Snyder ◽  
Stewart Shapiro
Keyword(s):  
Light Intensity ◽  
Detailed Examination ◽  
Real Numbers ◽  
Detailed Review ◽  
The Real ◽  
Cardinal Numbers ◽  
Main Complaint ◽  
The Relationship

This paper is concerned with Gottlob Frege’s theory of the real numbers as sketched in the second volume of his masterpiece Grundgesetze der Arithmetik. It is perhaps unsurprising that Frege’s theory of the real numbers is intimately intertwined with and largely motivated by his metaphysics. The account raises interesting, and surprisingly underexplored, questions about Frege’s metaphysics: Can this metaphysics even accommodate mass quantities like water, gold, light intensity, or charge? Frege’s main complaint with his contemporaries Cantor and Dedekind is that their theories of the real numbers do not build the applicability of the real numbers directly into the construction. In taking Cantor and Dedekind’s Arithmetic theories to be insufficient, clearly Frege takes it to be a desideratum on a theory of the real numbers that their applicability be essential to their construction. We begin with a detailed review of Frege’s theory, one that mirrors Frege’s exposition in structure. This is followed by a critique, outlining Frege’s linguistic motivation for ontologically distinguishing the cardinal numbers from the real numbers. We briefly consider how Frege’s metaphysics might need to be developed, or amended, to accommodate some of the problems. Finally, we offer a detailed examination of Frege’s Application Constraint – that the reals ought to have their applicability built directly into their characterization. It bears on deeper questions concerning the relationship between sophisticated mathematical theories and their applications.

scholarly journals Collaborative calculation and application of interreal and real interval relations to protect privacy

PLoS ONE ◽  
2021 ◽  
Vol 16 (12) ◽  
pp. e0261213
Author(s):  
Shaofeng Lu ◽  
Yuefeng Lu ◽  
Ying Sun
Keyword(s):  
Real Number ◽  
Data Privacy ◽  
Basic Problem ◽  
Homomorphic Encryption ◽  
Real Interval ◽  
Real Numbers ◽  
Sign Function ◽  
The Real ◽  
The Relationship

The determination of the relation between a number and a numerical interval is one of the core problems in the scientific calculation of privacy protection. The calculation of the relationship between two numbers and a numerical interval to protect privacy is also the basic problem of collaborative computing. It is widely used in data queries, location search and other fields. At present, most of the solutions are still fundamentally limited to the integer level, and there are few solutions at the real number level. To solve these problems, this paper first uses Bernoulli inequality generalization and a monotonic function property to extend the solution to the real number level and designs two new protocols based on the homomorphic encryption scheme, which can not only protect the data privacy of both parties involved in the calculation, but also extend the number domain to real numbers. In addition, this paper designs a solution to the confidential cooperative determination problem between real numbers by using the sign function and homomorphism multiplication. Theoretical analysis shows that the proposed solution is safe and efficient. Finally, some extension applications based on this protocol are given.

Joshua Kates, Essential History: Jacques Derrida and the Development of Deconstruction (Evanston, Illinois: Northwestern University Press, 2005), 352pp, $29.95 (USD), ISBN 10: 0810123274, ISBN-13: 978-0810123274

Derrida Today ◽  
2008 ◽  
Vol 1 (1) ◽  
pp. 131-133
Author(s):  
Gary Banham
Keyword(s):  
Jacques Derrida ◽  
The Real ◽  
Northwestern University ◽  
Relationship Of ◽  
The Relationship

This book promises a ‘radical reappraisal’ (Kates 2005, xv) of Derrida, concentrating particularly on the relationship of Derrida to philosophy, one of the most vexed questions in the reception of his work. The aim of the book is to provide the grounds for this reappraisal through a reinterpretation in particular of two of the major works Derrida published in 1967: Speech and Phenomena and Of Grammatology. However the study of the development of Derrida's work is the real achievement of the book as Kates discusses major works dating from the 1954 study of genesis in Husserl's phenomenology through to the essays on Levinas and Foucault in the early 1960's as part of his story of how Derrida arrived at the writing of the two major works from 1967.

scholarly journals Adaptive Model for Biofeedback Data Flows Management in the Design of Interactive Immersive Environments

2021 ◽  
Vol 11 (11) ◽  
pp. 5067
Author(s):  
Paulo Veloso Gomes ◽  
António Marques ◽  
João Donga ◽  
Catarina Sá ◽  
António Correia ◽  
...  
Keyword(s):  
Real Time ◽  
Emotional State ◽  
Adaptive Model ◽  
Multimodal Data ◽  
The Real ◽  
Time Interaction ◽  
Data Flows ◽  
Immersive Systems ◽  
Immersive Environment ◽  
The Relationship

The interactivity of an immersive environment comes up from the relationship that is established between the user and the system. This relationship results in a set of data exchanges between human and technological actors. The real-time biofeedback devices allow to collect in real time the biodata generated by the user during the exhibition. The analysis, processing and conversion of these biodata into multimodal data allows to relate the stimuli with the emotions they trigger. This work describes an adaptive model for biofeedback data flows management used in the design of interactive immersive systems. The use of an affective algorithm allows to identify the types of emotions felt by the user and the respective intensities. The mapping between stimuli and emotions creates a set of biodata that can be used as elements of interaction that will readjust the stimuli generated by the system. The real-time interaction generated by the evolution of the user’s emotional state and the stimuli generated by the system allows him to adapt attitudes and behaviors to the situations he faces.

Knowing by Doing: The Role of Geometrical Practice in Aristotle’s Theory of Knowledge

Elenchos ◽  
2015 ◽  
Vol 36 (1) ◽  
pp. 45-88 ◽  
Author(s):  
Monica Ugaglia
Keyword(s):  
Scientific Knowledge ◽  
Theory Of Knowledge ◽  
States Of Mind ◽  
General Process ◽  
The Real ◽  
Analogical Model ◽  
Primary Function ◽  
The Relationship ◽  
The Way

Abstract Aristotle’s way of conceiving the relationship between mathematics and other branches of scientific knowledge is completely different from the way a contemporary scientist conceives it. This is one of the causes of the fact that we look at the mathematical passages we find in Aristotle’s works with the wrong expectation. We expect to find more or less stringent proofs, while for the most part Aristotle employs mere analogies. Indeed, this is the primary function of mathematics when employed in a philosophical context: not a demonstrative tool, but a purely analogical model. In the case of the geometrical examples discussed in this paper, the diagrams are not conceived as part of a formalized proof, but as a work in progress. Aristotle is not interested in the final diagram but in the construction viewed in its process of development; namely in the figure a geometer draws, and gradually modifies, when he tries to solve a problem. The way in which the geometer makes use of the elements of his diagram, and the relation between these elements and his inner state of knowledge is the real feature which interests Aristotle. His goal is to use analogy in order to give the reader an idea of the states of mind involved in a more general process of knowing.

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