Platonism and anti-Platonism in mathematics
Mathematics Platonists e-böcker
Oxford University Press
1998
EISBN 9780195143980
Cover.
Contents.
1 Introduction.
1. The Project of This Book.
2. Mathematical Platonism and Anti-Platonism.
3. Synopsis of the Book.
ONE: Platonism.
2 The Epistemological Argument Against Platonism.
1. Introduction.
2. Formulating the Epistemological Argument.
3. A Taxonomy of Platonist Responses.
4. Contact with Other Worlds: Gödel.
5. Contact in This World: Maddy.
6. Knowledge Without Contact.
3 A New Platonist Epistemology.
1. Introduction.
2. Skeleton of the Refutation of the Epistemological Argument.
3. Internalist vs. Externalist Explanations.
4. Defending and Motivating FBP.
5. Consistency.
4 Non-Uniqueness Embraced.
1. Introduction.
2. Trying to Salvage the Numbers.
3. Structuralism.
4. The Solution.
5. Two Loose Ends.
TWO: Anti-Platonism.
5 The Fregean Argument Against Anti-Platonism.
1. Introduction.
2. The Argument.
3. In Defense of Fictionalism.
4. Nonfictionalistic Versions of Anti-Realistic Anti-Platonism.
5. The Refutation of Realistic Anti-Platonism.
6. Platonism and the Issue of Applicability and Indispensability.
6 Denying the Existence of Indispensable Applications: Toward a Nominalization of Quantum Mechanics.
1. Introduction.
2. How Field Nominalizes.
3. Malament's Objection.
4. The Strategy for Nominalizing QM.
5. The Nominalistic Status of Propensities.
7 Accounting for Indispensable Applications from a Fictionalist Point of View.
1. Introduction.
2. What, Exactly, Needs to Be Accounted For?.
3. A Fictionalist Account of the Applicability of Mathematics.
4. Problems with Platonism Revisited.
THREE: Conclusions.
8 The Unsolvability of the Problem and a Kinder, Gentler Positivism.
1. Introduction.
2. The Strong Epistemic Conclusion.
3. The Metaphysical Conclusion.
4. My Official View.
Notes.
Bibliography.
Index.
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
K.
L.
M.
N.
O.
P.
Q.
R.
S.
T.
U.
V.
W.
Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks, most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good argument for or against platonism, but that we could never have such an argument and, indeed, that there is no fact of the matter as to whether platonism is correct.
Contents.
1 Introduction.
1. The Project of This Book.
2. Mathematical Platonism and Anti-Platonism.
3. Synopsis of the Book.
ONE: Platonism.
2 The Epistemological Argument Against Platonism.
1. Introduction.
2. Formulating the Epistemological Argument.
3. A Taxonomy of Platonist Responses.
4. Contact with Other Worlds: Gödel.
5. Contact in This World: Maddy.
6. Knowledge Without Contact.
3 A New Platonist Epistemology.
1. Introduction.
2. Skeleton of the Refutation of the Epistemological Argument.
3. Internalist vs. Externalist Explanations.
4. Defending and Motivating FBP.
5. Consistency.
4 Non-Uniqueness Embraced.
1. Introduction.
2. Trying to Salvage the Numbers.
3. Structuralism.
4. The Solution.
5. Two Loose Ends.
TWO: Anti-Platonism.
5 The Fregean Argument Against Anti-Platonism.
1. Introduction.
2. The Argument.
3. In Defense of Fictionalism.
4. Nonfictionalistic Versions of Anti-Realistic Anti-Platonism.
5. The Refutation of Realistic Anti-Platonism.
6. Platonism and the Issue of Applicability and Indispensability.
6 Denying the Existence of Indispensable Applications: Toward a Nominalization of Quantum Mechanics.
1. Introduction.
2. How Field Nominalizes.
3. Malament's Objection.
4. The Strategy for Nominalizing QM.
5. The Nominalistic Status of Propensities.
7 Accounting for Indispensable Applications from a Fictionalist Point of View.
1. Introduction.
2. What, Exactly, Needs to Be Accounted For?.
3. A Fictionalist Account of the Applicability of Mathematics.
4. Problems with Platonism Revisited.
THREE: Conclusions.
8 The Unsolvability of the Problem and a Kinder, Gentler Positivism.
1. Introduction.
2. The Strong Epistemic Conclusion.
3. The Metaphysical Conclusion.
4. My Official View.
Notes.
Bibliography.
Index.
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
K.
L.
M.
N.
O.
P.
Q.
R.
S.
T.
U.
V.
W.
Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks, most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good argument for or against platonism, but that we could never have such an argument and, indeed, that there is no fact of the matter as to whether platonism is correct.
