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## Tracking Reason: Proof, Consequence, and Truth

*Jody Azzouni*

- Published in print:
- 2006
- Published Online:
- October 2011
- ISBN:
- 9780195187137
- eISBN:
- 9780199850570
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195187137.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

When ordinary people—mathematicians among them—take something to follow (deductively) from something else, they are exposing the backbone of our self-ascribed ability to reason. This book ... More

## Understanding Proofs

*Jeremy Avigad*

### in The Philosophy of Mathematical Practice

- Published in print:
- 2008
- Published Online:
- February 2010
- ISBN:
- 9780199296453
- eISBN:
- 9780191711961
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199296453.003.0013
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

In informal speech, ascriptions of mathematical understanding are typically clarified by spelling out the kinds of abilities an agent is assumed to possess. This chapter argues that such an approach ... More

## Introduction to Part II

*Jody Azzouni*

### in Tracking Reason: Proof, Consequence, and Truth

- Published in print:
- 2006
- Published Online:
- October 2011
- ISBN:
- 9780195187137
- eISBN:
- 9780199850570
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195187137.003.0009
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

This part of the book provides information about the remainder of the book. The next two-thirds of the book take up, respectively, proof and consequence. The book now turns to the absolutely central ... More

## Computer Proof, Apriori Knowledge, and Other Minds *

*Tyler Burge*

### in Cognition Through Understanding: Self-Knowledge, Interlocution, Reasoning, Reflection: Philosophical Essays, Volume 3

- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199672028
- eISBN:
- 9780191751929
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199672028.003.0013
- Subject:
- Philosophy, Philosophy of Mind, Metaphysics/Epistemology

‘Computer Proof, Apriori Knowledge, and Other Minds’ (1998) discusses the question: Does the fact that a mathematical proof is carried out by a computer, and is too long for a mathematician to check, ... More

## Artificial Extensions of Natural Language

*Jody Azzouni*

### in Semantic Perception: How the Illusion of a Common Language Arises and Persists

- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199967407
- eISBN:
- 9780199346066
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199967407.003.0013
- Subject:
- Philosophy, Philosophy of Language

The history of the emergence of artificial languages is sketched. It’s shown how our experience of natural language makes the invention of artificial languages possible. The emergence of artificial ... More

## Wittgenstein on Necessity: Some Reflections

*Michael Dummett*

### in The Seas of Language

- Published in print:
- 1996
- Published Online:
- November 2003
- ISBN:
- 9780198236214
- eISBN:
- 9780191597350
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198236212.003.0019
- Subject:
- Philosophy, Philosophy of Language

Radical conventionalism treats every necessary truth as a linguistic convention, akin to the statement ’There are seven days in a week’. Wittgenstein's endorsement of this view was a result of his ... More

## The Perfection of Pure Logic

*Jan von Plato*

### in The Great Formal Machinery Works: Theories of Deduction and Computation at the Origins of the Digital Age

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691174174
- eISBN:
- 9781400885039
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174174.003.0010
- Subject:
- History, History of Science, Technology, and Medicine

This chapter explains how Gerard Gentzen presented the general logical structure of mathematical proofs as a system of rules of proof by which a path is built from the assumptions of a theorem to its ... More

## Two Flavours of Mathematical Explanation

*Mark Colyvan, John Cusbert, and Kelvin McQueen*

### in Explanation Beyond Causation: Philosophical Perspectives on Non-Causal Explanations

- Published in print:
- 2018
- Published Online:
- June 2018
- ISBN:
- 9780198777946
- eISBN:
- 9780191823404
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198777946.003.0012
- Subject:
- Philosophy, Philosophy of Science, Metaphysics/Epistemology

A proof of a mathematical theorem tells us that the theorem is true (or should be accepted), but some proofs go further and tell us why the theorem is true (or should be accepted). That is, some, but ... More

## Provability, Mechanism, and the Diagonal Problem

*Graham Leach-Krouse*

### in Gödel's Disjunction: The scope and limits of mathematical knowledge

- Published in print:
- 2016
- Published Online:
- November 2016
- ISBN:
- 9780198759591
- eISBN:
- 9780191820373
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198759591.003.0009
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

The primary goal in this chapter is to understand and to make a start at evaluating Gödel and Post’s conceptions of absolute provability. The starting point will be Gödel’s famous assertion from the ... More

## Logic, 1900–1940

*John A. Goldsmith and Bernard Laks*

### in Battle in the Mind Fields

- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780226550800
- eISBN:
- 9780226550947
- Item type:
- chapter

- Publisher:
- University of Chicago Press
- DOI:
- 10.7208/chicago/9780226550947.003.0008
- Subject:
- History, History of Ideas

This chapter looks at the ways in which mathematicians and logicians rethought the nature of mathematical proof during the first four decades of the twentieth century. Three styles of thinking about ... More

## Husserl on the Origins of Geometry

*Ian Hacking*

### in Science and the Life-World: Essays on Husserl's Crisis of European Sciences

- Published in print:
- 2009
- Published Online:
- June 2013
- ISBN:
- 9780804756044
- eISBN:
- 9780804772945
- Item type:
- chapter

- Publisher:
- Stanford University Press
- DOI:
- 10.11126/stanford/9780804756044.003.0004
- Subject:
- Philosophy, Philosophy of Science

This chapter examines the foundations of mathematics, focusing on The Origin of Geometry, an appendix to the Crisis. It notes that Husserl is one of those philosophers who is awed by mathematics, ... More

## The Deep Question: P vs. NP

*Cristopher Moore and Stephan Mertens*

### in The Nature of Computation

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199233212
- eISBN:
- 9780191775079
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199233212.003.0006
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter considers the consequences of the possibility that P is equal to NP. It examines why it is extremely difficult to prove that P is not equal to NP by focusing on a set of ‘metatheorems’ ... More

## Absolute Provability and Safe Knowledge of Axioms

*Timothy Williamson*

### in Gödel's Disjunction: The scope and limits of mathematical knowledge

- Published in print:
- 2016
- Published Online:
- November 2016
- ISBN:
- 9780198759591
- eISBN:
- 9780191820373
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198759591.003.0010
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

Are there any in-principle (or absolutely) undecidable hypotheses? In order to make progress with this question, its terms must be first clarified. Against the background of a proposed clarification, ... More

## Gödel's Disjunction: The scope and limits of mathematical knowledge

*Leon Horsten and Philip Welch (eds)*

- Published in print:
- 2016
- Published Online:
- November 2016
- ISBN:
- 9780198759591
- eISBN:
- 9780191820373
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198759591.001.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

The logician Kurt Gödel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is equivalent to a Turing machine (i.e., a ... More

##
Do We Have Reason to Doubt the Importance of the Distinction between A Priori and A Posteriori Knowledge?: A Reply to Williamson^{1} Paul Boghossian

*Paul Boghossian and Timothy Williamson*

### in Debating the A Priori

- Published in print:
- 2020
- Published Online:
- October 2020
- ISBN:
- 9780198851707
- eISBN:
- 9780191886317
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198851707.003.0009
- Subject:
- Philosophy, General

This essay defends the a priori–a posteriori distinction against two skeptical challenges posed by Williamson in Chapter 8. Against the argument that no top-down characterization of the distinction ... More

## Intellectual Gestalts

*Elijah Chudnoff*

### in Phenomenal Intentionality

- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199764297
- eISBN:
- 9780199932191
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199764297.003.0010
- Subject:
- Philosophy, Philosophy of Mind

Phenomenal holism is the thesis that some phenomenal characters can only be instantiated by experiences that are parts of certain wholes. The first aim of this paper is to defend phenomenal holism. ... More

##
Williamson on the Distinction between the A Priori and the A Posteriori Once Again^{1}: Paul Boghossian

*Paul Boghossian and Timothy Williamson*

### in Debating the A Priori

- Published in print:
- 2020
- Published Online:
- October 2020
- ISBN:
- 9780198851707
- eISBN:
- 9780191886317
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198851707.003.0011
- Subject:
- Philosophy, General

This essay responds to Williamson’s reformulated argument against the feasibility of a top-down characterization of the a priori–a posteriori distinction, arguing that Williamson fails to show that ... More

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