## 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