Jump to ContentJump to Main Navigation

You are looking at 1-20 of 34 items

  • Keywords: category theory x
Clear All Modify Search

View:

CONSTRUCTIVE SET THEORIES AND THEIR CATEGORY-THEORETIC MODELS

Alex Simpson

in From Sets and Types to Topology and Analysis: Towards practicable foundations for constructive mathematics

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198566519
eISBN:
9780191713927
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566519.003.0003
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter advocates a pragmatic approach to constructive set theory, using axioms based solely on set-theoretic principles that are directly relevant to (constructive) mathematical practice. The ... More


Set Theoretic Paradoxes

Graham Priest

in In Contradiction

Published in print:
2006
Published Online:
May 2007
ISBN:
9780199263301
eISBN:
9780191718823
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199263301.003.0003
Subject:
Philosophy, Logic/Philosophy of Mathematics

In this chapter it is argued that the standard paradoxes of set theory are also dialetheias. In particular, it argues that Zermelo Fraenkel set theory does not provide a satisfactory solution to ... More


DEDUCTIVE LOGIC, REDUCTIVE LOGIC, AND PROOF-SEARCH

David J. Pym and Eike Ritter

in Reductive Logic and Proof-search: Proof Theory, Semantics, and Control

Published in print:
2004
Published Online:
September 2007
ISBN:
9780198526339
eISBN:
9780191712012
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198526339.003.0001
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter offers an explanation of the basic concepts of reductive logic and proof-search. A discussion of the mathematical prerequisites needed for this study is also included. Key basic topics ... More


CATEGORIES OF DIAGRAMS

Steve Awodey

in Category Theory

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198568612
eISBN:
9780191717567
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198568612.003.0008
Subject:
Mathematics, Algebra

This chapter presents the proof for the Yoneda Lemma, which is probably the single most used result in category theory. It is interesting how often it comes up, especially in view of the fact that it ... More


Structuralism

Geoffrey Hellman

in The Oxford Handbook of Philosophy of Mathematics and Logic

Published in print:
2005
Published Online:
July 2005
ISBN:
9780195148770
eISBN:
9780199835560
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0195148770.003.0017
Subject:
Philosophy, Logic/Philosophy of Mathematics

With developments in the 19th and early 20th centuries, structuralist ideas concerning the subject matter of mathematics have become commonplace. Yet fundamental questions concerning structures and ... More


SPACES AS COMONOIDS

A. Bucalo and G. Rosolini

in From Sets and Types to Topology and Analysis: Towards practicable foundations for constructive mathematics

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198566519
eISBN:
9780191713927
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566519.003.0012
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

The constructive approach to topology has produced a wealth of new insights about spaces, almost reaching the point of making these more elementary than sets. In particular, the point of view taken ... More


Introduction

Leila Haaparanta and Heikki J. Koskinen

in Categories of Being: Essays on Metaphysics and Logic

Published in print:
2012
Published Online:
September 2012
ISBN:
9780199890576
eISBN:
9780199980031
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199890576.003.0000
Subject:
Philosophy, Metaphysics/Epistemology, Logic/Philosophy of Mathematics

The introductory chapter gives a short survey of the various attitudes towards metaphysics in the twentieth century and describes the return of metaphysics to the analytic tradition in the second ... More


Noumena and the categories

Graham Priest

in Beyond the Limits of Thought

Published in print:
2002
Published Online:
October 2011
ISBN:
9780199254057
eISBN:
9780191698194
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199254057.003.0006
Subject:
Philosophy, Logic/Philosophy of Mathematics, Metaphysics/Epistemology

This chapter examines philosopher Immanuel Kant's Theory of Categories and how it may avoid the limits of thought. It analyses Kant's Critique of Pure Reason in order to chart the logical geography ... More


Diagrammatic Immanence: Category Theory and Philosophy

Rocco "Gangle

Published in print:
2015
Published Online:
September 2016
ISBN:
9781474404174
eISBN:
9781474418645
Item type:
book
Publisher:
Edinburgh University Press
DOI:
10.3366/edinburgh/9781474404174.001.0001
Subject:
Philosophy, Metaphysics/Epistemology

This book integrates insights from Spinoza’s metaphysics, Peirce’s semiotic theory and Deleuze’s philosophy of difference in conjunction with the formal operations of category theory. Spinoza, Peirce ... More


A Category Theory Explanation for Systematicity: Universal Constructions

Steven Phillips and William H. Wilson

in The Architecture of Cognition: Rethinking Fodor and Pylyshyn's Systematicity Challenge

Published in print:
2014
Published Online:
September 2014
ISBN:
9780262027236
eISBN:
9780262322461
Item type:
chapter
Publisher:
The MIT Press
DOI:
10.7551/mitpress/9780262027236.003.0009
Subject:
Philosophy, Philosophy of Mind

A theory of cognitive architecture should explain why systematicity is a necessary, not just possible consequence of the theory's core principles and base assumptions, without relying on arbitrary ... More


A Path to a New Logic

David Corfield

in Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy

Published in print:
2020
Published Online:
March 2020
ISBN:
9780198853404
eISBN:
9780191888069
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198853404.003.0001
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

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 ... More


Diagrams of Difference: Adjunctions and Topoi

Rocco Gangle

in Diagrammatic Immanence: Category Theory and Philosophy

Published in print:
2015
Published Online:
September 2016
ISBN:
9781474404174
eISBN:
9781474418645
Item type:
chapter
Publisher:
Edinburgh University Press
DOI:
10.3366/edinburgh/9781474404174.003.0007
Subject:
Philosophy, Metaphysics/Epistemology

This chapter examines two advanced constructions in category theory: adjoint functors and topoi. Pairs of adjoint functors, or adjunctions, are formally defined and then illustrated with several ... More


Diagrams of Variation: Functor Categories and Presheaves

Rocco Gangle

in Diagrammatic Immanence: Category Theory and Philosophy

Published in print:
2015
Published Online:
September 2016
ISBN:
9781474404174
eISBN:
9781474418645
Item type:
chapter
Publisher:
Edinburgh University Press
DOI:
10.3366/edinburgh/9781474404174.003.0005
Subject:
Philosophy, Metaphysics/Epistemology

This chapter begins by posing the question of linguistic meaning within the context of the mathematics of category theory. It then explains the notions of functor categories, natural transformations ... More


Structure and Foundations

John P. Burgess

in Rigor and Structure

Published in print:
2015
Published Online:
May 2015
ISBN:
9780198722229
eISBN:
9780191789076
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198722229.003.0004
Subject:
Philosophy, Logic/Philosophy of Mathematics

If contemporary mathematics treats structures “in abstraction from”—which is to say, ignoring—the nature of their elements and considering only relations among them, category theory operates at an ... More


Diagrams of Structure: Categories and Functors

Rocco Gangle

in Diagrammatic Immanence: Category Theory and Philosophy

Published in print:
2015
Published Online:
September 2016
ISBN:
9781474404174
eISBN:
9781474418645
Item type:
chapter
Publisher:
Edinburgh University Press
DOI:
10.3366/edinburgh/9781474404174.003.0003
Subject:
Philosophy, Metaphysics/Epistemology

This chapter introduces the elementary concepts and operations of category theory. It introduces the notion of category by generalizing from partial orders and systems of functions and then ... More


Mathematics Based on Fuzzy Logic

Radim Bělohlávek, Joseph W. Dauben, and George J. Klir

in Fuzzy Logic and Mathematics: A Historical Perspective

Published in print:
2017
Published Online:
June 2017
ISBN:
9780190200015
eISBN:
9780190200039
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780190200015.003.0005
Subject:
Philosophy, Logic/Philosophy of Mathematics, General

Mathematical reasoning is governed by the laws of classical logic, based on the principle of bivalence. With the acceptance of intermediate truth degrees, the situation changed substantially. This ... More


Scalars, Monads, and Categories

Dion Coumans and Bart Jacobs

in Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse

Published in print:
2013
Published Online:
May 2013
ISBN:
9780199646296
eISBN:
9780191747847
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199646296.003.0007
Subject:
Mathematics, Applied Mathematics

This chapter describes interrelations between: algebraic structure on sets of scalars, properties of monads associated with such sets of scalars, and structure in categories (especially Lawvere ... More


Theory and Meta-theory; Logic and Meta-logic I: Philosophical and Foundational Studies

Stewart Shapiro

in Varieties of Logic

Published in print:
2014
Published Online:
September 2014
ISBN:
9780199696529
eISBN:
9780191784866
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199696529.003.0006
Subject:
Philosophy, Logic/Philosophy of Mathematics

Logic is ubiquitous. Whenever there is deductive reasoning, there is logic. So there are issues similar to those for a global relativism. What is the proper logic for perspectives from which one ... More


Canonical Maps

Jean-Pierre Marquis

in Categories for the Working Philosopher

Published in print:
2017
Published Online:
January 2018
ISBN:
9780198748991
eISBN:
9780191811593
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198748991.003.0006
Subject:
Philosophy, History of Philosophy

Canonical maps play a central role in contemporary mathematics. However, their exact definition eludes present-day mathematics. We claim in this chapter that category theory allows us to identify ... More


Unfolding FOLDS: A Foundational Framework for Abstract Mathematical Concepts

Jean-Pierre Marquis

in Categories for the Working Philosopher

Published in print:
2017
Published Online:
January 2018
ISBN:
9780198748991
eISBN:
9780191811593
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198748991.003.0008
Subject:
Philosophy, History of Philosophy

FOLDS, first-order logic with dependent sorts, has been introduced by the logician Michael Makkai as a foundational framework to capture the abstract nature of contemporary mathematical concepts. In ... More


View: