16150
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One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato]
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Full Idea:
If one is, there must also necessarily be number - Necessarily - But if there is number, there would be many, and an unlimited multitude of beings. ..So if all partakes of being, each part of number would also partake of it.
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From:
Plato (Parmenides [c.364 BCE], 144a)
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A reaction:
This seems to commit to numbers having being, then to too many numbers, and hence to too much being - but without backing down and wondering whether numbers had being after all. Aristotle disagreed.
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10047
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Russell's improvements blocked mathematics as well as paradoxes, and needed further axioms [Russell, by Musgrave]
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Full Idea:
Unfortunately, Russell's new logic, as well as preventing the deduction of paradoxes, also prevented the deduction of mathematics, so he supplemented it with additional axioms, of Infinity, of Choice, and of Reducibility.
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From:
report of Bertrand Russell (Mathematical logic and theory of types [1908]) by Alan Musgrave - Logicism Revisited §2
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A reaction:
The first axiom seems to be an empirical hypothesis, and the second has turned out to be independent of logic and set theory.
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21718
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Ramified types can be defended as a system of intensional logic, with a 'no class' view of sets [Russell, by Linsky,B]
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Full Idea:
A defence of the ramified theory of types comes in seeing it as a system of intensional logic which includes the 'no class' account of sets, and indeed the whole development of mathematics, as just a part.
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From:
report of Bertrand Russell (Mathematical logic and theory of types [1908]) by Bernard Linsky - Russell's Metaphysical Logic 6.1
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A reaction:
So Linsky's basic project is to save logicism, by resting on intensional logic (rather than extensional logic and set theory). I'm not aware that Linsky has acquired followers for this. Maybe Crispin Wright has commented?
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18124
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Vicious Circle says if it is expressed using the whole collection, it can't be in the collection [Russell, by Bostock]
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Full Idea:
The Vicious Circle Principle says, roughly, that whatever involves, or presupposes, or is only definable in terms of, all of a collection cannot itself be one of the collection.
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From:
report of Bertrand Russell (Mathematical logic and theory of types [1908], p.63,75) by David Bostock - Philosophy of Mathematics 8.1
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A reaction:
This is Bostock's paraphrase of Russell, because Russell never quite puts it clearly. The response is the requirement to be 'predicative'. Bostock emphasises that it mainly concerns definitions. The Principle 'always leads to hierarchies'.
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