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Ideas for 'fragments/reports', 'Replies on 'Limits of Abstraction'' and 'Causal Powers'

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5 ideas

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

15273	Points can be 'dense' by unending division, but must meet a tougher criterion to be 'continuous' [Harré/Madden]
	Full Idea: Points can be 'dense' by indefinitely prolonged division. To be 'continuous' is more stringent; the points must be cut into two sets, and meet the condition laid down by Boscovich and Dedekind.
	From: Harré,R./Madden,E.H. (Causal Powers [1975], 6.IV)
	A reaction: This idea goes with Idea 15274, which lays down the specification of the Dedekind Cut, which is the criterion for the real (and continuous) numbers. Harré and Madden are interested in whether time can support continuity of objects.

10573	Dedekind cuts lead to the bizarre idea that there are many different number 1's [Fine,K]
	Full Idea: Because of Dedekind's definition of reals by cuts, there is a bizarre modern doctrine that there are many 1's - the natural number 1, the rational number 1, the real number 1, and even the complex number 1.
	From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2)
	A reaction: See Idea 10572.

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts

15274	Points are 'continuous' if any 'cut' point participates in both halves of the cut [Harré/Madden]
	Full Idea: Divide points into left and right set. They're 'continuous' if that point is either last member of left set, and greatest lower bound of right (so no least member), or least upper bound of left set (so no last member) and first member of right set.
	From: Harré,R./Madden,E.H. (Causal Powers [1975], 6.IV)
	A reaction: The best attempt I have yet encountered to explain a Dedekind Cut for the layperson. I gather modern mathematicians no longer rely on this way of defining the real numbers.

10575	Why should a Dedekind cut correspond to a number? [Fine,K]
	Full Idea: By what right can Dedekind suppose that there is a number corresponding to any pair of irrationals that constitute an irrational cut?
	From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2)

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero

10574	Unless we know whether 0 is identical with the null set, we create confusions [Fine,K]
	Full Idea: What is the union of the singleton {0}, of zero, and the singleton {φ}, of the null set? Is it the one-element set {0}, or the two-element set {0, φ}? Unless the question of identity between 0 and φ is resolved, we cannot say.
	From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2)