Platonism and Anti-Platonism in MathematicsOxford University Press, 2001 - 217 páginas In this highly absorbing work, Balaguer demonstrates that no good arguments exist either for or against mathematical platonism-for example, the view that abstract mathematical objects do exist and that mathematical theories are descriptions of such objects. Balaguer does this by establishing that both platonism and anti-platonism are justifiable views. Introducing a form of platonism, called "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 arguments for or against platonism but that we could never have such an argument. This lucid and accessible book breaks new ground in its area of engagement and makes vital reading for both specialists and all those intrigued by the philosophy of mathematics, or metaphysics in general. |
Índice
The Epistemological Argument Against Platonism | 21 |
2 Formulating the Epistemological Argument | 22 |
3 A Taxonomy of Platonist Responses | 24 |
Godel | 25 |
Maddy | 28 |
6 Knowledge Without Contact | 35 |
A New Platonist Epistemology | 48 |
3 Intemalist vs Extemalist Explanations | 53 |
5 The Refutation of Realistic AntiPlatonism | 104 |
6 Platonism and the Issue of Applicability and Indispensability | 109 |
Denying the Existence of Indispensable Applications Toward a Nominalization of Quantum Mechanics | 113 |
2 How Field Nominalizes | 114 |
3 Malaments Objection | 117 |
4 The Strategy for Nominalizing QM | 120 |
5 The Nominalistic Status of Propensities | 126 |
Accounting for Indispensable Applications from a Fictionalist Point of View | 128 |
4 Defending and Motivating FBP | 58 |
5 Consistency | 69 |
NonUniqueness Embraced | 76 |
2 Trying to Salvage the Numbers | 77 |
3 Structuralism | 80 |
4 The Solution | 84 |
5 Two Loose Ends | 90 |
AntiPlatonism | 93 |
The Fregean Argument Against AntiPlatonism | 95 |
3 In Defense of Fictionalism | 98 |
4 NonFictionalistic Versions of AntiRealistic AntiPlatonism | 100 |
3 A Fictionalist Account of the Applicability of Mathematics | 130 |
4 Problems with Platonism Revisited | 142 |
Conclusions | 149 |
The Unsolvability of the Problem and a Kinder Gentler Positivism | 151 |
2 The Strong Epistemic Conclusion | 152 |
3 The Metaphysical Conclusion | 158 |
4 My Official View | 178 |
Notes | 181 |
| 207 | |
| 213 | |
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Términos y frases comunes
abstract mathematical objects abstract objects acquire knowledge anti-platonists anti-realism anti-realistic applications of mathematics argue axiom Benacerraf's Benacerrafian causally chapter claim consistent purely mathematical ematical empirical science empirical theories endorse epistemological epistemological argument explain externalist fact false FBP-ists and fictionalists Field's Gödel Hilbert spaces instance Kitcher Maddy math mathematical beliefs mathematical intuition mathematical knowledge mathematical platonism mathematical practice mathematical realm mathematical singular mathematicians matical metaphysical conclusion natural numbers no-contact nominalistic content nominalization non-spatiotemporal non-uniqueness notion of consistency Ockham's razor orthomodular lattice philosophy of mathematics physical objects physical world platonists possible worlds possible-world-style truth conditions propensities purely mathematical theories quantum system question real number reason Resnik response seems semantic sentence sequence set theory Shapiro simply sort spacetime standard model strategy structuralists structure sub-section theorem theories are true theories truly describe thesis things unique version of anti-platonism version of platonism w-sequences worry
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Página 7 - I have been saying the exact reverse: if the arbitrarily given axioms do not contradict one another with all their consequences, then they are true and the things defined by the axioms exist. This is for me the criterion of truth and existence.
Página 17 - I am not myself persuaded that this is right ; but even if we accept this, it does not follow that there is no fact of the matter as to which of two observationally equivalent theories, by Quine's criterion, is correct.
Página 19 - Somewhere over the rainbow Way up high There's a land that I heard of Once in a lullaby Somewhere over the rainbow Skies are blue, And the dreams that you dare to dream Really do come true...
Página 19 - Back in Kansas, Auntie Em is delivering the scolding that is the prelude to one of the cinema's immortal moments. You always get yourself into a fret about nothing ... find yourself a place where you won't get into any trouble! Some place where there isn't any trouble. Do you suppose there is such a place, Toto? There must be. Anybody who has swallowed the scriptwriters...