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Set Theory and The Philosophy of Mathematics

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/9780199263301.003.0011
Subject:
Philosophy, Logic/Philosophy of Mathematics

This chapter discusses certain aspects of naive set theory, that is, set theory in which the abstraction scheme holds for all formulas of the language of set theory, and some of its implications for ... More


Naturalism and Mathematical Practice

Mary Leng

in Mathematics and Reality

Published in print:
2010
Published Online:
May 2010
ISBN:
9780199280797
eISBN:
9780191723452
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199280797.003.0004
Subject:
Philosophy, Logic/Philosophy of Mathematics, Metaphysics/Epistemology

‘Naturalism’, as we have used the term, is the commitment to look to science to discover what we ought to believe. If one reads ‘science’ broadly, to include pure mathematics, then this commitment ... More


Anti-Realism in Mathematics

Charles S. Chihara

in The Worlds of Possibility: Modal Realism and the Semantics of Modal Logic

Published in print:
2001
Published Online:
October 2011
ISBN:
9780199246557
eISBN:
9780191697609
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199246557.003.0010
Subject:
Philosophy, Logic/Philosophy of Mathematics, Metaphysics/Epistemology

This chapter discusses the implications of the modal theory developed in this book in terms of mathematics. It focuses on Mathematical Realism, because its own position regarding the ontology of ... More


What Is Mathematical Realism?

Michael D. Resnik

in Mathematics as a Science of Patterns

Published in print:
1999
Published Online:
November 2003
ISBN:
9780198250142
eISBN:
9780191598296
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198250142.003.0002
Subject:
Philosophy, Logic/Philosophy of Mathematics

Talk of truth plays a major role in formulating realism, to the point that realist theories are often criticized by attacking the correspondence theory of truth that they are presumed to defend. In ... More


Mathematics and Its Applications

Mark Colyvan

in The Indispensability of Mathematics

Published in print:
2001
Published Online:
November 2003
ISBN:
9780195137545
eISBN:
9780199833139
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/019513754X.003.0001
Subject:
Philosophy, Logic/Philosophy of Mathematics

Indispensability arguments are introduced and their connection to scientific realism is discussed. The Quine–Putnam indispensability argument for mathematical realism is then outlined. In particular, ... More


The Case for Mathematical Realism

Michael D. Resnik

in Mathematics as a Science of Patterns

Published in print:
1999
Published Online:
November 2003
ISBN:
9780198250142
eISBN:
9780191598296
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198250142.003.0003
Subject:
Philosophy, Logic/Philosophy of Mathematics

The application of mathematics to science and the enormous success that derives from it is, perhaps, the strongest evidence in favour of mathematical realism. Quine and Putnam have taken the ... More


Doubts About Realism

Michael D. Resnik

in Mathematics as a Science of Patterns

Published in print:
1999
Published Online:
November 2003
ISBN:
9780198250142
eISBN:
9780191598296
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198250142.003.0005
Subject:
Philosophy, Logic/Philosophy of Mathematics

One of the strongest motivations for being an anti‐realist with regard to mathematics is the difficulty in formulating a plausible realist epistemology, given that there seems to be a lack of ties ... More


Modal Realism

Charles S. Chihara

in The Worlds of Possibility: Modal Realism and the Semantics of Modal Logic

Published in print:
2001
Published Online:
October 2011
ISBN:
9780199246557
eISBN:
9780191697609
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199246557.003.0004
Subject:
Philosophy, Logic/Philosophy of Mathematics, Metaphysics/Epistemology

The models studied in modal logic are set theoretical structures. So one can regard the model theory of modal logic as a branch of set theory. The philosophical question arises: what is the ... More


Conclusion

Mark Colyvan

in The Indispensability of Mathematics

Published in print:
2001
Published Online:
November 2003
ISBN:
9780195137545
eISBN:
9780199833139
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/019513754X.003.0007
Subject:
Philosophy, Logic/Philosophy of Mathematics

The varieties of mathematical realism that are consistent with the indispensability argument are outlined. It is shown that the argument is silent on many issues such as whether numbers are sets, and ... More


Recent Attempts at Blunting the Indispensability Thesis

Michael D. Resnik

in Mathematics as a Science of Patterns

Published in print:
1999
Published Online:
November 2003
ISBN:
9780198250142
eISBN:
9780191598296
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198250142.003.0004
Subject:
Philosophy, Logic/Philosophy of Mathematics

The indispensability thesis maintains both that using mathematical terms and assertions is an indispensable part of scientific practice and that this practice commits science to mathematical objects ... More


Positing Mathematical Objects

Michael D. Resnik

in Mathematics as a Science of Patterns

Published in print:
1999
Published Online:
November 2003
ISBN:
9780198250142
eISBN:
9780191598296
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198250142.003.0009
Subject:
Philosophy, Logic/Philosophy of Mathematics

If, as I grant, mathematical objects are abstract entities existing outside of space and time, and if the idea of supernaturally grasping abstract entities is scientifically unacceptable, then we ... More


The Elusive Distinction Between Mathematics and Natural Science

Michael D. Resnik

in Mathematics as a Science of Patterns

Published in print:
1999
Published Online:
November 2003
ISBN:
9780198250142
eISBN:
9780191598296
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198250142.003.0006
Subject:
Philosophy, Logic/Philosophy of Mathematics

It is commonly believed that the epistemology of mathematics must be different from the epistemology of science because their objects are different in kind, i.e. metaphysically different. In this ... More


Holism: Evidence in Science and Mathematics

Michael D. Resnik

in Mathematics as a Science of Patterns

Published in print:
1999
Published Online:
November 2003
ISBN:
9780198250142
eISBN:
9780191598296
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198250142.003.0007
Subject:
Philosophy, Logic/Philosophy of Mathematics

I present a theory of justification for mathematical beliefs that is both non‐foundationalist, in that it claims that some mathematics must be justified indirectly in terms of its consequences, and ... More


Justification and Explanation in Mathematics and Morality

Justin Clarke-Doane

in Oxford Studies in Metaethics, Volume 10

Published in print:
2015
Published Online:
August 2015
ISBN:
9780198738695
eISBN:
9780191802515
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198738695.003.0004
Subject:
Philosophy, Moral Philosophy

In his influential book, The Nature of Morality, Gilbert Harman writes: “In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the ... More


Sums in verse or a mathematical aesthetic?

Duncan F. Kennedy

in Forgotten Stars: Rediscovering Manilius' Astronomica

Published in print:
2011
Published Online:
May 2011
ISBN:
9780199586462
eISBN:
9780191724961
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199586462.003.0010
Subject:
Classical Studies, Literary Studies: Classical, Early, and Medieval

This chapter sets out to offer a more sympathetic characterization of the role of mathematics in the Astronomica than A. E. Housman’s famous but dismissive quip about Manilius’ ‘eminent aptitude for ... More


Random Numbers and God’s Nature

James Bradley

in Abraham's Dice: Chance and Providence in the Monotheistic Traditions

Published in print:
2016
Published Online:
May 2016
ISBN:
9780190277154
eISBN:
9780190277185
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780190277154.003.0004
Subject:
Religion, Theology

“Randomness,” despite being a common term, does not unfortunately have a common definition. Disciplines have strongly varying intuitions about its meaning. Researchers in a branch of theoretical ... More


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