8297
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Numbers are universals, being sets whose instances are sets of appropriate cardinality [Lowe]
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Full Idea:
My view is that numbers are universals, beings kinds of sets (that is, kinds whose particular instances are individual sets of appropriate cardinality).
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From:
E.J. Lowe (The Possibility of Metaphysics [1998], 10)
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A reaction:
[That is, 12 is the set of all sets which have 12 members] This would mean, I take it, that if the number of objects in existence was reduced to 11, 12 would cease to exist, which sounds wrong. Or are we allowed imagined instances?
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8298
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Sets are instances of numbers (rather than 'collections'); numbers explain sets, not vice versa [Lowe]
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Full Idea:
I favour an account of sets which sees them as being instances of numbers, thereby avoiding the unhelpful metaphor which speaks of a set as being a 'collection' of things. This reverses the normal view, which explains numbers in terms of sets.
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From:
E.J. Lowe (The Possibility of Metaphysics [1998], 10)
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A reaction:
Cf. Idea 8297. Either a set is basic, or a number is. We might graft onto Lowe's view an account of numbers in terms of patterns, which would give an empirical basis to the picture, and give us numbers which could be used to explain sets.
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8311
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If 2 is a particular, then adding particulars to themselves does nothing, and 2+2=2 [Lowe]
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Full Idea:
If 2 is a particular, 'adding' it to itself can, it would seem, only leave us with 2, not another number. (If 'Socrates + Socrates' denotes anything, it most plausibly just denotes Socrates).
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From:
E.J. Lowe (The Possibility of Metaphysics [1998], 10.7)
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A reaction:
This suggest Kant's claim that arithmetical sums are synthetic (Idea 5558). It is a nice question why, when you put two 2s together, they come up with something new. Addition is movement. Among patterns, or along abstract sequences.
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8748
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Logical positivists incorporated geometry into logicism, saying axioms are just definitions [Carnap, by Shapiro]
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Full Idea:
The logical positivists brought geometry into the fold of logicism. The axioms of, say, Euclidean geometry are simply definitions of primitive terms like 'point' and 'line'.
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From:
report of Rudolph Carnap (Empiricism, Semantics and Ontology [1950]) by Stewart Shapiro - Thinking About Mathematics 5.3
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A reaction:
If the concept of 'line' is actually created by its definition, then we need to know exactly what (say) 'shortest' means. If we are merely describing a line, then our definition can be 'impredicative', using other accepted concepts.
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