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Ideas for 'Parmenides', 'The Principles of Mathematics' and 'On Freedom'

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5 ideas

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism

16150	One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato]
	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.
	From: Plato (Parmenides [c.364 BCE], 144a)
	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.

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism

14162	Mathematics doesn't care whether its entities exist [Russell]
	Full Idea: Mathematics is throughout indifferent to the question whether its entities exist.
	From: Bertrand Russell (The Principles of Mathematics [1903], §434)
	A reaction: There is an 'if-thenist' attitude in this book, since he is trying to reduce mathematics to logic. Total indifference leaves the problem of why mathematics is applicable to the real world.

6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism

14103	Pure mathematics is the class of propositions of the form 'p implies q' [Russell]
	Full Idea: Pure mathematics is the class of all propositions of the form 'p implies q', where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants.
	From: Bertrand Russell (The Principles of Mathematics [1903], §001)
	A reaction: Linnebo calls Russell's view here 'deductive structuralism'. Russell gives (§5) as an example that Euclid is just whatever is deduced from his axioms.

6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory

21555	For 'x is a u' to be meaningful, u must be one range of individuals (or 'type') higher than x [Russell]
	Full Idea: In his 1903 theory of types he distinguished between individuals, ranges of individuals, ranges of ranges of individuals, and so on. Each level was a type, and it was stipulated that for 'x is a u' to be meaningful, u must be one type higher than x.
	From: Bertrand Russell (The Principles of Mathematics [1903], App)
	A reaction: Russell was dissatisfied because this theory could not deal with Cantor's Paradox. Is this the first time in modern philosophy that someone has offered a criterion for whether a proposition is 'meaningful'?

18003	In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless [Russell, by Magidor]
	Full Idea: Russell argues that in a statement of the form 'x is a u' (and correspondingly, 'x is a not-u'), 'x must be of different types', and hence that ''x is an x' must in general be meaningless'.
	From: report of Bertrand Russell (The Principles of Mathematics [1903], App B:524) by Ofra Magidor - Category Mistakes 1.2
	A reaction: " 'Word' is a word " comes to mind, but this would be the sort of ascent to a metalanguage (to distinguish the types) which Tarski exploited. It is the simple point that a classification can't be the same as a member of the classification.