Combining Texts

Ideas for 'Mahaprajnaparamitashastra', 'The Philosophy of Logic' and 'Plurals and Complexes'

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8 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
10680 The theory of the transfinite needs the ordinal numbers [Hossack]
     Full Idea: The theory of the transfinite needs the ordinal numbers.
     From: Keith Hossack (Plurals and Complexes [2000], 8)
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10684 I take the real numbers to be just lengths [Hossack]
     Full Idea: I take the real numbers to be just lengths.
     From: Keith Hossack (Plurals and Complexes [2000], 9)
     A reaction: I love it. Real numbers are beginning to get on my nerves. They turn up to the party with no invitation and improperly dressed, and then refuse to give their names when challenged.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
18200 Very large sets should be studied in an 'if-then' spirit [Putnam]
     Full Idea: Sets of a very high type or very high cardinality (higher than the continuum, for example), should today be investigated in an 'if-then' spirit.
     From: Hilary Putnam (The Philosophy of Logic [1971], p.347), quoted by Penelope Maddy - Naturalism in Mathematics
     A reaction: Quine says the large sets should be regarded as 'uninterpreted'.
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
10674 A plural language gives a single comprehensive induction axiom for arithmetic [Hossack]
     Full Idea: A language with plurals is better for arithmetic. Instead of a first-order fragment expressible by an induction schema, we have the complete truth with a plural induction axiom, beginning 'If there are some numbers...'.
     From: Keith Hossack (Plurals and Complexes [2000], 4)
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
10681 In arithmetic singularists need sets as the instantiator of numeric properties [Hossack]
     Full Idea: In arithmetic singularists need sets as the instantiator of numeric properties.
     From: Keith Hossack (Plurals and Complexes [2000], 8)
10685 Set theory is the science of infinity [Hossack]
     Full Idea: Set theory is the science of infinity.
     From: Keith Hossack (Plurals and Complexes [2000], 10)
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
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!