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All the ideas for 'poems', 'Intro to III: Quantifiers' and 'works'

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6 ideas

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
3340 Von Neumann defines each number as the set of all smaller numbers [Neumann, by Blackburn]
     Full Idea: Von Neumann defines each number as the set of all smaller numbers.
     From: report of John von Neumann (works [1935]) by Simon Blackburn - Oxford Dictionary of Philosophy p.280
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
3355 Von Neumann wanted mathematical functions to replace sets [Neumann, by Benardete,JA]
     Full Idea: Von Neumann suggested that functions be pressed into service to replace sets.
     From: report of John von Neumann (works [1935]) by José A. Benardete - Metaphysics: the logical approach Ch.23
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
9465 Substitutional universal quantification retains truth for substitution of terms of the same type [Jacquette]
     Full Idea: The substitutional interpretation says the universal quantifier is true just in case it remains true for all substitutions of terms of the same type as that of the universally bound variable.
     From: Dale Jacquette (Intro to III: Quantifiers [2002], p.143)
     A reaction: This doesn't seem to tell us how it gets started with being true.
9466 Nominalists like substitutional quantification to avoid the metaphysics of objects [Jacquette]
     Full Idea: Some substitutional quantificationists in logic hope to avoid philosophical entanglements about the metaphysics of objects, ..and nominalists can find aid and comfort there.
     From: Dale Jacquette (Intro to III: Quantifiers [2002], p.143)
     A reaction: This has an appeal for me, particularly if it avoids abstract objects, but I don't see much problem with material objects, so we might as well have a view that admits those.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
22716 Von Neumann defined ordinals as the set of all smaller ordinals [Neumann, by Poundstone]
     Full Idea: At age twenty, Von Neumann devised the formal definition of ordinal numbers that is used today: an ordinal number is the set of all smaller ordinal numbers.
     From: report of John von Neumann (works [1935]) by William Poundstone - Prisoner's Dilemma 02 'Sturm'
     A reaction: I take this to be an example of an impredicative definition (not predicating something new), because it uses 'ordinal number' in the definition of ordinal number. I'm guessing the null set gets us started.
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / j. Ethics by convention
6017 Nomos is king [Pindar]
     Full Idea: Nomos is king.
     From: Pindar (poems [c.478 BCE], S 169), quoted by Thomas Nagel - The Philosophical Culture
     A reaction: This seems to be the earliest recorded shot in the nomos-physis wars (the debate among sophists about moral relativism). It sounds as if it carries the full relativist burden - that all that matters is what has been locally decreed.