Combining Texts

All the ideas for 'Frege's Theory of Numbers', 'Quine on Quantifying In' and 'Realism, Mathematics and Modality'

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4 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 / C. Sources of Mathematics / 9. Fictional Mathematics

8714	Fictionalists say 2+2=4 is true in the way that 'Oliver Twist lived in London' is true [Field,H]
	Full Idea: The fictionalist can say that the sense in which '2+2=4' is true is pretty much the same as the sense in which 'Oliver Twist lived in London' is true. They are true 'according to a well-known story', or 'according to standard mathematics'.
	From: Hartry Field (Realism, Mathematics and Modality [1989], 1.1.1), quoted by Michèle Friend - Introducing the Philosophy of Mathematics 6.3
	A reaction: The roots of this idea are in Carnap. Fictionalism strikes me as brilliant, but poisonous in large doses. Novels can aspire to artistic truth, or to documentary truth. We invent a fiction, and nudge it slowly towards reality.