Combining Texts

Ideas for 'Frege's Theory of Numbers', 'Abstract Objects: a Case Study' and 'What is the Source of Knowledge of Modal Truths?'

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3 ideas

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 / 6. Logicism / c. Neo-logicism

10580	Mathematics is both necessary and a priori because it really consists of logical truths [Yablo]
	Full Idea: Mathematics seems necessary because the real contents of mathematical statements are logical truths, which are necessary, and it seems a priori because logical truths really are a priori.
	From: Stephen Yablo (Abstract Objects: a Case Study [2002], 10)
	A reaction: Yablo says his logicism has a Kantian strain, because numbers and sets 'inscribed on our spectacles', but he takes a different view (in the present Idea) from Kant about where the necessity resides. Personally I am tempted by an a posteriori necessity.

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

10579	Putting numbers in quantifiable position (rather than many quantifiers) makes expression easier [Yablo]
	Full Idea: Saying 'the number of Fs is 5', instead of using five quantifiers, puts the numeral in quantifiable position, which brings expressive advantages. 'There are more sheep in the field than cows' is an infinite disjunction, expressible in finite compass.
	From: Stephen Yablo (Abstract Objects: a Case Study [2002], 08)
	A reaction: See Hofweber with similar thoughts. This idea I take to be a key one in explaining many metaphysical confusions. The human mind just has a strong tendency to objectify properties, relations, qualities, categories etc. - for expression and for reasoning.