scholarly journals Formalism and Hilbert’s understanding of consistency problems

2021 ◽  
Author(s):  
Michael Detlefsen
Keyword(s):  
Philosophy Of Mathematics ◽  
Model Construction ◽  
Traditional Model ◽  
Consistency Proof ◽  
Salient Points ◽  
Consistency Problem ◽  
Late Nineteenth ◽  
Similarity And Difference ◽  
Traditional Understanding

AbstractFormalism in the philosophy of mathematics has taken a variety of forms and has been advocated for widely divergent reasons. In Sects. 1 and 2, I briefly introduce the major formalist doctrines of the late nineteenth and early twentieth centuries. These are what I call empirico-semantic formalism (advocated by Heine), game formalism (advocated by Thomae) and instrumental formalism (advocated by Hilbert). After describing these views, I note some basic points of similarity and difference between them. In the remainder of the paper, I turn my attention to Hilbert’s instrumental formalism. My primary aim there will be to develop its formalist elements more fully. These are, in the main, (i) its rejection of the axiom-centric focus of traditional model-construction approaches to consistency problems, (ii) its departure from the traditional understanding of the basic nature of proof and (iii) its distinctively descriptive or observational orientation with regard to the consistency problem for arithmetic. More specifically, I will highlight what I see as the salient points of connection between Hilbert’s formalist attitude and his finitist standard for the consistency proof for arithmetic. I will also note what I see as a significant tension between Hilbert’s observational approach to the consistency problem for arithmetic and his expressed hope that his solution of that problem would dispense with certain epistemological concerns regarding arithmetic once and for all.

Robustness Consideration in Modelling a Prototype by Quality Engineering

1994 ◽  
Author(s):  
Chang-Chung Li ◽  
Tsong-Shin Sheu ◽  
Yu-Cheng Lee
Keyword(s):  
Dimensional Analysis ◽  
Robust Design ◽  
Model Theory ◽  
Model Construction ◽  
Traditional Model ◽  
Quality Engineering ◽  
Prediction Equations ◽  
High Quality

Abstract Traditional model theory uses dimensional analysis and similarity to obtain design conditions for model construction and then uses prediction equations to predict the performance of a prototype. No consideration of quality is involved. In this paper, through model theory, we illustrate with examples that a high quality prototype can be obtained by quality engineering using robust design.

The equivalence of NF-style set theories with “tangled” type theories; the construction of ω-models of predicative NF (and more)

1995 ◽  
Vol 60 (1) ◽  
pp. 178-190 ◽  
Author(s):  
M. Randall Holmes
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
The Other ◽  
Natural Numbers ◽  
Consistency Proof ◽  
Natural Behaviour ◽  
New Foundations ◽  
Consistency Problem

AbstractAn ω-model (a model in which all natural numbers are standard) of the predicative fragment of Quine's set theory “New Foundations” (NF) is constructed. Marcel Crabbé has shown that a theory NFI extending predicative NF is consistent, and the model constructed is actually a model of NFI as well. The construction follows the construction of ω-models of NFU (NF with urelements) by R. B. Jensen, and, like the construction of Jensen for NFU, it can be used to construct α-models for any ordinal α. The construction proceeds via a model of a type theory of a peculiar kind; we first discuss such “tangled type theories” in general, exhibiting a “tangled type theory” (and also an extension of Zermelo set theory with Δ0 comprehension) which is equiconsistent with NF (for which the consistency problem seems no easier than the corresponding problem for NF (still open)), and pointing out that “tangled type theory with urelements” has a quite natural interpretation, which seems to provide an explanation for the more natural behaviour of NFU relative to the other set theories of this kind, and can be seen anachronistically as underlying Jensen's consistency proof for NFU.

Methodology in Late Nineteenth- and Early Twentieth-Century Analytic Philosophy

2014 ◽  
Author(s):  
Scott Soames
Keyword(s):  
Twentieth Century ◽  
Driving Force ◽  
Philosophy Of Mathematics ◽  
Analytic Philosophy ◽  
Linguistic Analysis ◽  
Philosophical Analysis ◽  
Logical Syntax ◽  
Logical Atomism ◽  
Late Nineteenth ◽  
Philosophical Questions

This chapter discusses the methodology that guided logico-linguistic analysis from Gottlob Frege’s 1879 Begriffsschrift to Rudolf Carnap’s 1934 The Logical Syntax of Language. In the first four decades of this period, culminating with Bertrand Russell’s 1918–19 lectures on The Philosophy of Logical Atomism, analysis was viewed as an increasingly powerful tool employed in the service of solving traditional philosophical problems. The logicist reduction of arithmetic to what was taken to be logic was the driving force, providing the exemplar of philosophical analysis and the model for extending it beyond the philosophy of mathematics. The methodology is indicated by the role played by A2 in answering Frege’s guiding philosophical questions Q1 and Q2.

scholarly journals LOGICISM, INTERPRETABILITY, AND KNOWLEDGE OF ARITHMETIC

2014 ◽  
Vol 7 (1) ◽  
pp. 84-119 ◽  
Author(s):  
SEAN WALSH
Keyword(s):  
Mathematical Logic ◽  
Philosophy Of Mathematics ◽  
The Other ◽  
Problem Structure ◽  
Begging The Question ◽  
Arithmetical Knowledge ◽  
Consistency Problem ◽  
Closure Conditions ◽  
A Charge ◽  
Crucial Part

AbstractA crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here, an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions of representation from contemporary mathematical logic. It is argued that the theory-based versions of such logicism are either too liberal (the plethora problem) or are committed to intuitively incorrect closure conditions (the consistency problem). Structure-based versions must on the other hand respond to a charge of begging the question (the circularity problem) or explain how one may have a knowledge of structure in advance of a knowledge of axioms (the signature problem). This discussion is significant because it gives us a better idea of what a notion of representation must look like if it is to aid in realizing some of the traditional epistemic aims of logicism in the philosophy of mathematics.

Contents of Western Sociology of Religion as a Science and Educational Discipline

2002 ◽  
pp. 106-110
Author(s):  
Liudmyla O. Fylypovych
Keyword(s):  
Nineteenth Century ◽  
International Society ◽  
Sociology Of Religion ◽  
Scientific Study ◽  
Late Nineteenth Century ◽  
The West ◽  
Late Nineteenth

Sociology of religion in the West is a field of knowledge with at least 100 years of history. As a science and as a discipline, the sociology of religion has been developing in most Western universities since the late nineteenth century, having established traditions, forming well-known schools, areas related to the names of famous scholars. The total number of researchers of religion abroad has never been counted, but there are more than a thousand different centers, universities, colleges where religion is taught and studied. If we assume that each of them has an average of 10 religious scholars, theologians, then the army of scholars of religion is amazing. Most of them are united in representative associations of researchers of religion, which have a clear sociological color. Among them are the most famous International Society for the Sociology of Religion (ISSR) and the Society for Scientific Study of Religion (SSSR).

The Introduction of Plantations into East Sumatra and its Socio-economic Impacts, 1863-1929

2020 ◽  
Vol 6 (5) ◽  
pp. 563-576
Keyword(s):  
Twentieth Century ◽  
Social Transformation ◽  
Rapid Expansion ◽  
Local Resistance ◽  
Short Period ◽  
Late Nineteenth ◽  
Fierce Competition ◽  
Plantation Crops ◽  
Colonial Government ◽  
Economic Regions

The goal of this article is to examine the introduction of plantations into East Sumatra (Indonesia) in the late nineteenth and early twentieth century. Attention is given to the five most important plantation crops, namely tobacco, rubber, oil palm, tea, and fiber. The article analyzes the economic and social transformation of the region as a consequence of the rapid expansion of plantations. Within a short period of time, East Sumatra emerged to become one of the most dynamic economic regions of Southeast Asia. The development of the region and the needs of a source of protection for Dutch planters in face of fierce competition from other Western companies and local resistance encouraged the Dutch colonial government to establish effective authority in East Sumatra. Received 4th June 2020; Revised 15th September 2020; Accepted 26th September 2020

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