display all the ideas for this combination of texts
6 ideas
| 18199 | Indispensability strongly supports predicative sets, and somewhat supports impredicative sets [Putnam] |
| Full Idea: We may say that indispensability is a pretty strong argument for the existence of at least predicative sets, and a pretty strong, but not as strong, argument for the existence of impredicative sets. | |
| From: Hilary Putnam (The Philosophy of Logic [1971], p.346), quoted by Penelope Maddy - Naturalism in Mathematics II.2 |
| 8857 | We must quantify over numbers for science; but that commits us to their existence [Putnam] |
| Full Idea: Quantification over mathematical entities is indispensable for science..., therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question. | |
| From: Hilary Putnam (The Philosophy of Logic [1971], p.57), quoted by Stephen Yablo - Apriority and Existence | |
| A reaction: I'm not surprised that Hartry Field launched his Fictionalist view of mathematics in response to such a counterintuitive claim. I take it we use numbers to slice up reality the way we use latitude to slice up the globe. No commitment to lines! |
| 1613 | Logicists cheerfully accept reference to bound variables and all sorts of abstract entities [Quine] |
| Full Idea: The logicism of Frege, Russell, Whitehead, Church and Carnap condones the use of bound variables or reference to abstract entities known and unknown, specifiable and unspecifiable, indiscriminately. | |
| From: Willard Quine (On What There Is [1948], p.14) |
| 1616 | Formalism says maths is built of meaningless notations; these build into rules which have meaning [Quine] |
| Full Idea: The formalism of Hilbert keeps classical maths as a play of insignificant notations. Agreement is found among the rules which, unlike the notations, are quite significant and intelligible. | |
| From: Willard Quine (On What There Is [1948], p.15) |
| 1615 | Intuitionism says classes are invented, and abstract entities are constructed from specified ingredients [Quine] |
| Full Idea: The intuitionism of Poincaré, Brouwer, Weyl and others holds that classes are invented, and accepts reference to abstract entities only if they are constructed from pre-specified ingredients. | |
| From: Willard Quine (On What There Is [1948], p.14) |
| 1614 | Conceptualism holds that there are universals but they are mind-made [Quine] |
| Full Idea: Conceptualism holds that there are universals but they are mind-made. | |
| From: Willard Quine (On What There Is [1948], p.14) |