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Ideas for 'Deflationary Metaontology of Thomasson', 'Concepts and Counting' and 'Cardinality, Counting and Equinumerosity'

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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic

17459	Frege's Theorem explains why the numbers satisfy the Peano axioms [Heck]
	Full Idea: The interest of Frege's Theorem is that it offers us an explanation of the fact that the numbers satisfy the Dedekind-Peano axioms.
	From: Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 6)
	A reaction: He says 'explaining' does not make it more fundamental, since all proofs explain why their conclusions hold.

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers

17461	Some 'how many?' answers are not predications of a concept, like 'how many gallons?' [Rumfitt]
	Full Idea: We hit trouble if we hear answers to some 'How many?' questions as predications about concepts. The correct answer to 'how many gallons of water are in the tank?' may be 'ten', but that doesn''t mean ten things instantiate 'gallon of water in the tank'.
	From: Ian Rumfitt (Concepts and Counting [2002], I)
	A reaction: Rumfitt makes the point that a huge number of things instantiate that concept in a ten gallon tank of water. No problem, says Rumfitt, because Frege wouldn't have counted that as a statement of number.