Combining Texts

All the ideas for 'Frege's Theory of Numbers', 'The Philosophy of Logic' and 'Quine on Quantifying In'

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6 ideas

5. Theory of Logic / E. Structures of Logic / 1. Logical Form
12220 Is it the sentence-token or the sentence-type that has a logical form? [Fine,K]
     Full Idea: Do we attribute a logical form to a sentence token because it is a token of a type with that form, or do we attribute a logical form to a sentence type because it is a type of a token with that form?
     From: Kit Fine (Quine on Quantifying In [1990], p.110)
     A reaction: Since I believe in propositions (as the unambiguous thought that lies behind a sentence), I take it that logical form concerns propositions, though strict logicians don't like this, for fear that logic spills into psychology.
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
12222 Substitutional quantification is referential quantification over expressions [Fine,K]
     Full Idea: Substitutional quantification may be regarded as referential quantification over expressions.
     From: Kit Fine (Quine on Quantifying In [1990], p.124)
     A reaction: This is an illuminating gloss. Does such quantification involve some ontological commitment to expressions? I feel an infinite regress looming.
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
17447 Parsons says counting is tagging as first, second, third..., and converting the last to a cardinal [Parsons,C, by Heck]
     Full Idea: In Parsons's demonstrative model of counting, '1' means the first, and counting says 'the first, the second, the third', where one is supposed to 'tag' each object exactly once, and report how many by converting the last ordinal into a cardinal.
     From: report of Charles Parsons (Frege's Theory of Numbers [1965]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 3
     A reaction: This sounds good. Counting seems to rely on that fact that numbers can be both ordinals and cardinals. You don't 'convert' at the end, though, because all the way you mean 'this cardinality in this order'.
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 / 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!