scholarly journals Carnap’s Structuralist Thesis

2020 ◽  
pp. 383-420
Author(s):  
Georg Schiemer
Keyword(s):  
Present Article ◽  
Structural Properties ◽  
Philosophy Of Mathematics ◽  
Mathematical Structure ◽  
Mathematical Structuralism ◽  
Definition Of ◽  
Philosophical Debates ◽  
Implicit Definitions

This chapter investigates Carnap’s structuralism in his philosophy of mathematics of the 1920s and early 1930s. His approach to mathematics is based on a genuinely structuralist thesis, namely that axiomatic theories describe abstract structures or the structural properties of their objects. The aim in the present article is twofold: first, to show that Carnap, in his contributions to mathematics from the time, proposed three different (but interrelated) ways to characterize the notion of mathematical structure, namely in terms of (i) implicit definitions, (ii) logical constructions, and (iii) definitions by abstraction. The second aim is to re-evaluate Carnap’s early contributions to the philosophy of mathematics in light of contemporary mathematical structuralism. Specifically, the chapter discusses two connections between his structuralist thesis and current philosophical debates on structural abstraction and the on the definition of structural properties.

scholarly journals Transfer Principles, Klein’s Erlangen Program, and Methodological Structuralism

2020 ◽  
pp. 106-141
Author(s):  
Georg Schiemer
Keyword(s):  
Present Article ◽  
Transformation Groups ◽  
Theoretic Approach ◽  
Structural Identity ◽  
Erlangen Program ◽  
Structural Equivalence ◽  
Mathematical Structuralism ◽  
Principle Of Duality ◽  
Philosophical Debates ◽  
Transfer Principles

The present article investigates Felix Klein’s mathematical structuralism underlying his Erlangen program. The aim here is twofold. The first aim is to survey the geometrical background of his 1872 article, in particular, work on the principle of duality and so-called transfer principles in projective geometry. The second aim is more philosophical in character and concerns Klein’s structuralist account of geometrical knowledge. The chapter will argue that his group-theoretic approach is best characterized as a kind of “methodological structuralism” regarding geometry. Moreover, one can identify at least two aspects of the Erlangen program that connect his approach with present philosophical debates, namely (i) the idea to specify structural properties and structural identity conditions in terms of transformation groups and (ii) an account of the structural equivalence of geometries in terms of transfer principles.

scholarly journals What are Implicit Definitions?

Erkenntnis ◽  
2019 ◽  
Author(s):  
Eduardo N. Giovannini ◽  
Georg Schiemer
Keyword(s):  
Alternative Model ◽  
Axiomatic Theory ◽  
Mathematical Concepts ◽  
Systematic Assessment ◽  
General Terms ◽  
History Of ◽  
Definition Of ◽  
Philosophical Debates ◽  
Implicit Definitions ◽  
Structural Definition

Abstract The paper surveys different notions of implicit definition. In particular, we offer an examination of a kind of definition commonly used in formal axiomatics, which in general terms is understood as providing a definition of the primitive terminology of an axiomatic theory. We argue that such “structural definitions” can be semantically understood in two different ways, namely (1) as specifications of the meaning of the primitive terms of a theory and (2) as definitions of higher-order mathematical concepts or structures. We analyze these two conceptions of structural definition both in the history of modern axiomatics and in contemporary philosophical debates. Based on that, we give a systematic assessment of the underlying semantics of these two ways of understanding the definiens of such definitions, by considering alternative model-theoretic and inferential accounts of meaning.

scholarly journals Composing Arugat ha-Bosem: How Piyyut Commentary Became Associated with Ḥasidei Ashkenaz

2021 ◽  
Author(s):  
Elisabeth Hollender
Keyword(s):  
Present Article ◽  
Thirteenth Century ◽  
Open Book ◽  
Recent Scholarship ◽  
Definition Of ◽  
New Research ◽  
Hasidei Ashkenaz

AbstractBased on Ivan Marcus’s concept of “open book” and considerations on medieval Ashkenazic concepts of authorship, the present article inquires into the circumstances surrounding the production of SeferArugat ha-Bosem, a collection of piyyut commentaries written or compiled by the thirteenth-century scholar Abraham b. Azriel. Unlike all other piyyut commentators, Abraham ben Azriel inscribed his name into his commentary and claims to supersede previous commentaries, asserting authorship and authority. Based on the two different versions preserved in MS Vatican 301 and MS Merzbacher 95 (Frankfurt fol. 16), already in 1939 Ephraim E. Urbach suggested that Abraham b. Azriel might have written more than one edition of his piyyut commentaries. The present reevaluation considers recent scholarship on concepts of authorship and “open genre” as well as new research into piyyut commentary. To facilitate a comparison with Marcus’s definition of “open book,” this article also explores the arrangement and rearrangement of small blocks of texts within a work.

Erkundungen Von Gegenwelten: Zur Orientierungsleistung «mythischer» Reisen Am Beispiel Zweier Mesopotamischer Texte Erkundungen Von Gegenwelten: Zur Orientierungsleistung «mythischer» Reisen Am Beispiel Zweier Mesopotamischer Texte

Numen ◽  
2005 ◽  
Vol 52 (2) ◽  
pp. 226-254
Author(s):  
Daria Pezzoli-Olgiati
Keyword(s):  
Present Article ◽  
Religious Orientation ◽  
Human Life ◽  
Religious Symbol ◽  
Symbol Systems ◽  
Literary Sources ◽  
Similarities And Differences ◽  
Gilgamesh Epic ◽  
Definition Of

AbstractThe present article focuses on the function of mythic journeys with regard to the problem of death and the transience of human life in two selected Mesopotamian literary sources: the Gilgamesh-Epic IX–XI and the Descent of Ishtar to the Underworld. The selected texts are analysed and compared from the perspective of a functionalist definition of religious symbol systems, with particular attention to the transformation involved in travelling through different cosmic regions. The structure of the journey, the characterisation of the different regions visited by the protagonist, and the changes provoked by the mythic travel evince similarities and differences in the strategies employed to produce a religious orientation dealing with the ineluctable limits of life.

scholarly journals Lagal aspects of water users organizations´ business

2016 ◽  
Vol 4 (3) ◽  
pp. 0-0
Author(s):  
Умиджон Улугов ◽  
Umidjon Ulugov
Keyword(s):  
Present Article ◽  
Economic Activity ◽  
Public Authorities ◽  
Water Users ◽  
The Earth ◽  
Operational Management ◽  
Rational Management ◽  
The Republic ◽  
Definition Of ◽  
Enterprise Activity

Modern Tajikistan experiences considerable difficulties in a solution of the problem of rational management of objects of water infrastructure which don´t belong to maintaining public authorities, and are step by step transferred to hands of water users. In the present article the actual problems of economic (enterprise) activity of associations of water users in the Republic of Tajikistan urged to solve problems of maintaining, operational management of an internal irrigational network of the Republic of Tajikistan are considered. In article the separate data which have become a basis for reforming of sector of agriculture, to reforming of the former collective farms and state farms in modern Dehkan (farmer) farms and also on lack of essential measures on definition of the property right to constructions, the equipment, the earth of internal irrigational networks of Tajikistan are entered. Way out creation of a new civil form – associations of water users is considered – to whom functions on the maintenance of an internal irrigational network due to economic activity are assigned. Arguments concerning equivalence of economic activity commercial and as result – a contradiction to standards of the civil legislation of the Republic of Tajikistan are given in article.

The anatytic conception of truth and the foundations of arithmetic

2000 ◽  
Vol 65 (1) ◽  
pp. 33-102 ◽  
Author(s):  
Peter Apostoli
Keyword(s):  
Philosophy Of Mathematics ◽  
Logical Truth ◽  
Numerical Identity ◽  
Hume’S Principle ◽  
Contextual Definition ◽  
Definition Of ◽  
Axiomatic Set Theory ◽  
In Virtue Of ◽  
Second Order Logic ◽  
Foundations Of Arithmetic

Until very recently, it was thought that there couldn't be any current interest in logicism as a philosophy of mathematics. Indeed, there is an old argument one often finds that logicism is a simple nonstarter just in virtue of the fact that if it were a logical truth that there are infinitely many natural numbers, then this would be in conflict with the existence of finite models. It is certainly true that from the perspective of model theory, arithmetic cannot be a part of logic. However, it is equally true that model theory's reliance on a background of axiomatic set theory renders it unable to match Frege's Theorem, the derivation within second order logic of the infinity of the number series from the contextual “definition” of the cardinality operator. Called “Hume's Principle” by Boolos, the contextual definition of the cardinality operator is presented in Section 63 of Grundlagen, as the statement that, for any concepts F and G,the number of F s = the number of G sif, and only if,F is equinumerous with G.The philosophical interest in Frege's Theorem derives from the thesis, defended for example by Crispin Wright, that Hume's principle expresses our pre-analytic conception of assertions of numerical identity. However, Boolos cites the very fact that Hume's principle has only infinite models as grounds for denying that it is logically true: For Boolos, Hume's principle is simply a disguised axiom of infinity.

scholarly journals Observation of a Most Phenomenal Computed Calligraphy in Quran

2016 ◽  
Vol 9 (3) ◽  
pp. 42
Author(s):  
Baback Khodadoost
Keyword(s):  
Present Article ◽  
Recent Article ◽  
Mathematical Structure ◽  
Font Size ◽  
Computational Structure ◽  
The Given ◽  
Word Frequencies ◽  
Mathematical Design

<span style="font-size: 10pt; font-family: 'Times New Roman','serif'; color: black; mso-bidi-font-size: 9.0pt; mso-fareast-font-family: 宋体; mso-themecolor: text1; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA;" lang="EN-US">Observation of a multifaceted mathematical-computational structure of Quran through analysis of its letter and word frequencies and important implications of such observations have been extensively explained and discussed in a recent article: </span><span style="font-size: 10pt; font-family: 'Times New Roman','serif'; color: black; mso-bidi-font-size: 8.0pt; mso-fareast-font-family: 宋体; mso-themecolor: text1; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA;" lang="EN-US">“Khodadoost B. (2015) The Computed Scripture: Exponentially Based Fourier Regulated Construct of Quran and its fundamentally important Consequences"</span><span style="font-size: 10pt; font-family: 'Times New Roman','serif'; color: black; mso-bidi-font-size: 9.0pt; mso-fareast-font-family: 宋体; mso-themecolor: text1; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA;" lang="EN-US">. In the present article we report observation of yet another facet of this mathematical structure of Quran which is a phenomenal "parametric name-printing”. This observation has been made through a systematic compute-plot algorithm which uses the given name and chapter frequencies of letters in Quran as its input and shows in the output, calligraphic printing in Arabic of the same name. Several names of God, Major Prophets, and even some physicists are shown to clearly manifest these calligraphic effects. Sensitivities of these observations to changes in letter frequencies in Quran are so high that increase or decrease of even one letter and only in one chapter of Quran can completely demolish the calligraphic effects. These astonishing observations not only are extremely important and interesting in their own right, but also point to an immensely complicated and intricate super-intelligent mathematical design of Quran and reinforce "Mathematically Fully constrained Writing" or MFCW identity of this scripture and its consequences, as have been explained in the above article.</span>

A Unified Taxonomy Framework of Trust

2011 ◽  
pp. 29-50 ◽  
Author(s):  
Weiliang Zhao ◽  
Vijay Varadharajan ◽  
George Bryan
Keyword(s):  
Trust Management ◽  
Lower Layer ◽  
Mathematical Structure ◽  
Formal Definition ◽  
Trust Relationship ◽  
Base Level ◽  
Trust Relationships ◽  
Analysis Design ◽  
Definition Of

In this chapter, we provide a formal definition of trust relationship with a strict mathematical structure that can reflect many of the commonly used notions of trust. Based on this formal definition, we propose a unified taxonomy framework of trust. Under the taxonomy framework, we discuss classification of trust. In particular, we address the base level authentication trust at the lower layer and a hierarchy of trust relationships at a higher level. We provide a set of definitions, propositions, and operations based on the relations of trust relationships. Then we define and discuss properties of trust direction and trust symmetry. We define the trust scope label in order to describe the scope and diversity of trust relationship. All the definitions about the properties of trust become elements of the unified taxonomy framework of trust. Some example scenarios are provided to illustrate the concepts in the taxonomy framework. The taxonomy framework of trust will provide accurate terms and useful tools for enabling the analysis, design, and implementation of trust. The taxonomy framework of trust is first part of research for the overall methodology of trust relationships and trust management in distributed systems.

Dic cur hic?

2006 ◽  
Vol 11 ◽  
pp. 103-158
Author(s):  
Sascha Salatowsky
Keyword(s):  
Present Article ◽  
Good Life ◽  
Basic Problem ◽  
The Other ◽  
Human Being ◽  
Ethical Life ◽  
Theological Virtues ◽  
Aristotle's Ethics ◽  
The One ◽  
Definition Of

In order to attain a deeper understanding of Aristotelian philosophy in the Renaissance, it is necessary to consider the theological implications of given facts. This article discusses a basic problem centring on the reception of Aristotle’s Ethics. The Nicomachean Ethics was widely regarded as the basis for a virtuous ethical life, yet how could a pagan philosophy, with its concepts of happiness, virtue, justice, etc., be the basis of a Christian society? The aim of the present article is to show how Lutheran scholars solved this problem in confrontation with Catholic and Calvinist scholars of the time. The first part deals with the two basic components of Aristotle’s Ethics, namely the doctrines of happiness (Eudaimonologia) and virtue (Aretologia), and attempts to show that Aristotle’s Ethics should not be understood as a system of rules, but rather as a handbook for the cultivation of practical habits in the free human being who strives to live a good life. The second part examines two key ideological confrontations in relation to Aristotle’s philosophy: between Lutherans and Calvinists in respect of definition of theology and philosophical and theological virtues on the one hand, and between Lutherans and ›the Enthusiasts‹ in respect of the concept of virtues on the other.

What the Word “Function” Means to Algebra Teachers

1943 ◽  
Vol 36 (5) ◽  
pp. 212-218
Author(s):  
Lee J. Cronbach
Keyword(s):  
High School ◽  
Present Article ◽  
College Mathematics ◽  
High School Algebra ◽  
Definition Of ◽  
Word Function ◽  
Technical Terms

Teachers are well aware of the fact that the pupil who can repeat the definition of a word may not really understand what that word means. Since understanding of technical terms in a subject like mathematics is essential, it is important for the teacher to determine whether the pupil has really mastered basic words. In one of a series of studies, the writer sought to construct a test which would determine how well pupils understand the word function, a term generally considered basic in work in advanced high school algebra and college mathematics. While attempting to build the test, it was found that teachers often did not agree as to whether a given expression should be called a function; this suggested that it might be important to determine just what is being taught as the meaning of function, since agreement as to what is being tested is necessary before a test can be constructed. The present article reports an attempt to determine what typical teachers of algebra mean when they speak of a function.

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