scholarly journals The Idea of Limiting the Type Theory in the Philosophy of Mathematics in the Context of the Criticism of Epistemological Relativism

2019 ◽  
pp. 43-52
Author(s):  
Vsevolod A. Ladov ◽  
Keyword(s):  
Type Theory ◽  
Philosophy Of Mathematics ◽  
Epistemological Relativism

A Path to a New Logic

2020 ◽  
pp. 1-28
Author(s):  
David Corfield
Keyword(s):  
Natural Language ◽  
Type Theory ◽  
Philosophy Of Language ◽  
Philosophy Of Mathematics ◽  
Ordinary Language ◽  
Context Dependence ◽  
Ordinary Language Philosophy ◽  
Language Philosophy ◽  
Computer Scientists ◽  
Key Issues

This chapter explains how modal homotopy type theory combines ideas from two currents of thought: type theory and category theory. Despite what might appear to be rather different philosophical starting points, there has emerged an intrinsically structuralist language of great interest to computer scientists, mathematicians and physicists. This in itself should be enough to interest philosophers in the language, but further motivation is provided by addressing some of the kinds of objection raised to formalization in philosophy; in particular, those from ordinary language philosophy which emphasize the elasticity and context-dependence of natural language. We see that several of their concerns, such as that the definitional and descriptive uses of ‘is’ are conflated in logic, are addressed by the type theory. The prospect is then presented of an opportunity to use the new language to explore key issues in philosophy of mathematics, philosophy of language and metaphysics.

scholarly journals Computational Abstraction

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 213
Author(s):  
Raymond Turner
Keyword(s):  
Type Theory ◽  
Philosophy Of Mathematics ◽  
Data Abstraction ◽  
Machine Code ◽  
Physical Systems ◽  
Physical Representation ◽  
Contemporary Work ◽  
Abstract Structure ◽  
Machine Representation ◽  
High Level

Representation and abstraction are two of the fundamental concepts of computer science. Together they enable “high-level” programming: without abstraction programming would be tied to machine code; without a machine representation, it would be a pure mathematical exercise. Representation begins with an abstract structure and seeks to find a more concrete one. Abstraction does the reverse: it starts with concrete structures and abstracts away. While formal accounts of representation are easy to find, abstraction is a different matter. In this paper, we provide an analysis of data abstraction based upon some contemporary work in the philosophy of mathematics. The paper contains a mathematical account of how Frege’s approach to abstraction may be interpreted, modified, extended and imported into type theory. We argue that representation and abstraction, while mathematical siblings, are philosophically quite different. A case of special interest concerns the abstract/physical interface which houses both the physical representation of abstract structures and the abstraction of physical systems.

Type Theory and Formal Proof

2009 ◽  
Author(s):  
Rob Nederpelt ◽  
Herman Geuvers
Keyword(s):  
Type Theory ◽  
Formal Proof

Computational Models for Multilayered Composite Shells with Application to Tires

1996 ◽  
Vol 24 (1) ◽  
pp. 11-38 ◽  
Author(s):  
G. M. Kulikov
Keyword(s):  
Type Theory ◽  
Computational Models ◽  
Geometrical Nonlinearity ◽  
Higher Order ◽  
Stress Strain ◽  
Strain Fields ◽  
First Order ◽  
Timoshenko Type ◽  
Joint Influence ◽  
Discrete Layer

Abstract This paper focuses on four tire computational models based on two-dimensional shear deformation theories, namely, the first-order Timoshenko-type theory, the higher-order Timoshenko-type theory, the first-order discrete-layer theory, and the higher-order discrete-layer theory. The joint influence of anisotropy, geometrical nonlinearity, and laminated material response on the tire stress-strain fields is examined. The comparative analysis of stresses and strains of the cord-rubber tire on the basis of these four shell computational models is given. Results show that neglecting the effect of anisotropy leads to an incorrect description of the stress-strain fields even in bias-ply tires.

Mathematical Knowledge and the Interplay of Practices

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Mathematical Knowledge ◽  
A Priori ◽  
Philosophy Of Mathematics ◽  
Advanced Mathematics ◽  
New Approach ◽  
Agent Based ◽  
Epistemology Of Mathematics ◽  
Actual Experience ◽  
Elementary Math ◽  
Mathematical Tradition

This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction in the philosophy of mathematics, the book uses the crucial idea of a continuum to provide an account of the development of mathematical knowledge that reflects the actual experience of doing math and makes sense of the perceived objectivity of mathematical results. Describing a historically oriented, agent-based philosophy of mathematics, the book shows how the mathematical tradition evolved from Euclidean geometry to the real numbers and set-theoretic structures. It argues for the need to take into account a whole web of mathematical and other practices that are learned and linked by agents, and whose interplay acts as a constraint. It demonstrates how advanced mathematics, far from being a priori, is based on hypotheses, in contrast to elementary math, which has strong cognitive and practical roots and therefore enjoys certainty. Offering a wealth of philosophical and historical insights, the book challenges us to rethink some of our most basic assumptions about mathematics, its objectivity, and its relationship to culture and science.

Campus Social Climate Correlates of Environmental Type Dimensions

NASPA Journal ◽  
2004 ◽  
Vol 41 (4) ◽  
Author(s):  
Daniel W. Salter ◽  
Reynol Junco ◽  
Summer D. Irvin
Keyword(s):  
Work Environment ◽  
Personal Growth ◽  
Type Theory ◽  
General Purpose ◽  
Myers Briggs Type Indicator ◽  
Environment Scale ◽  
Type Indicator ◽  
Group Environment ◽  
System Maintenance ◽  
Environment Type

To address the ability of the Salter Environment Type Assessment (SETA) to measure different kinds of campus environments, data from three studies of the SETA with the Work Environment Scale, Group Environment Scale, and University Residence Environment Scale were reexamined (n = 534). Relationship dimension scales were very consistent with extraversion and feeling from environmental type theory. System maintenance and systems change scales were associated with judging and perception on the SETA, respectively. Results from the SETA and personal growth dimension scales were mixed. Based on this analysis, the SETA may serve as a general purpose environmental assessment for use with the Myers-Briggs Type Indicator.

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