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3 ideas
9191 | Ordinals seem more basic than cardinals, since we count objects in sequence [Dummett] |
Full Idea: It can be argued that the notion of ordinal numbers is more fundamental than that of cardinals. To count objects, we must count them in sequence. ..The theory of ordinals forms the substratum of Cantor's theory of cardinals. | |
From: Michael Dummett (The Philosophy of Mathematics [1998], 5) | |
A reaction: Depends what you mean by 'fundamental'. I would take cardinality to be psychologically prior ('that is a lot of sheep'). You can't order people by height without first acquiring some people with differing heights. I vote for cardinals. |
17518 | Counting 'coin in this box' may have coin as the unit, with 'in this box' merely as the scope [Ayers] |
Full Idea: If we count the concept 'coin in this box', we could regard coin as the 'unit', while taking 'in this box' to limit the scope. Counting coins in two boxes would be not a difference in unit (kind of object), but in scope. | |
From: M.R. Ayers (Individuals without Sortals [1974], 'Counting') | |
A reaction: This is a very nice alternative to the Fregean view of counting, depending totally on the concept, and rests more on a natural concept of object. I prefer Ayers. Compare 'count coins till I tell you to stop'. |
17516 | If counting needs a sortal, what of things which fall under two sortals? [Ayers] |
Full Idea: If we accepted that counting objects always presupposes some sortal, it is surely clear that the class of objects to be counted could be designated by two sortals rather than one. | |
From: M.R. Ayers (Individuals without Sortals [1974], 'Realist' vii) | |
A reaction: His nice example is an object which is both 'a single piece of wool' and a 'sweater', which had better not be counted twice. Wiggins struggles to argue that there is always one 'substance sortal' which predominates. |