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Philosophy of Mathematics 7 Luca Incurvati L.Incurvati@uva.nl Luca Incurvati (L.Incurvati@uva.nl) Phil of Maths 7 1 / 27 Outline 1 The indispensability argument 2 Maddy Luca Incurvati (L.Incurvati@uva.nl) Phil of Maths 7 2 / 27 The indispensability argument Quine and Putnam So far I have been developing an argument for realism along roughly the following lines: quantification over mathematical entities is indispensable for science, both formal and physical: therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question. This type of argument stems, of course, from Quine, who has for years stressed both the indispensability of quantification over mathematical entities and the intellectual dishonesty of denying the existence of what one daily presupposes. (Putnam 1971: 57) Luca Incurvati (L.Incurvati@uva.nl) Phil of Maths 7 3 / 27 The indispensability argument Quinean Naturalism Quine describes naturalism as the recognition that it is within science itself, and not in some prior philosophy, that reality is to be identified and described. We will take Quinean naturalism to be the following: Quinean Naturalism We have to look at science to discover what we ought to believe. Luca Incurvati (L.Incurvati@uva.nl) Phil of Maths 7 4 / 27 The indispensability argument Quinean Naturalism Quine describes naturalism as the recognition that it is within science itself, and not in some prior philosophy, that reality is to be identified and described. We will take Quinean naturalism to be the following: Quinean Naturalism We have to look at science to discover what we ought to believe. Luca Incurvati (L.Incurvati@uva.nl) Phil of Maths 7 4 / 27 The indispensability argument The Quine-Putnam indispensability argument (P1) If a statement is confirmed according to our ordinary scientific standards, then we ought to believe it (from Quinean Naturalism). (P2) If a theory T is confirmed according to our ordinary scien