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Ideas for 'Individuals without Sortals', 'Thinking About Mathematics' and 'Pyrrhonian Arguments (frags)'

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4 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number

8763	The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
	Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2)
	A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed.

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy

18249	Cauchy gave a formal definition of a converging sequence. [Shapiro]
	Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε.
	From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4)
	A reaction: The sequence is 'Cauchy' if N exists.

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / d. Counting via concepts

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.