Combining Texts

All the ideas for 'Notes on John Wilkins', 'On the Very Idea of a Third Dogma' and 'An Axiomatization of Set Theory'

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4 ideas

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
15943 Limitation of Size is not self-evident, and seems too strong [Lavine on Neumann]
     Full Idea: Von Neumann's Limitation of Size axiom is not self-evident, and he himself admitted that it seemed too strong.
     From: comment on John von Neumann (An Axiomatization of Set Theory [1925]) by Shaughan Lavine - Understanding the Infinite VII.1
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
13672 All the axioms for mathematics presuppose set theory [Neumann]
     Full Idea: There is no axiom system for mathematics, geometry, and so forth that does not presuppose set theory.
     From: John von Neumann (An Axiomatization of Set Theory [1925]), quoted by Stewart Shapiro - Foundations without Foundationalism 8.2
     A reaction: Von Neumann was doubting whether set theory could have axioms, and hence the whole project is doomed, and we face relativism about such things. His ally was Skolem in this.
9. Objects / D. Essence of Objects / 14. Knowledge of Essences
12694 Essence is the distinct thinkability of anything [Leibniz]
     Full Idea: (Essence) is the distinct thinkability (cogitabilitas) of anything.
     From: Gottfried Leibniz (Notes on John Wilkins [1672], A6.2.487-8), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 1
     A reaction: A very original remark from the young Leibniz. It is neutral as to whether this is a real feature of objects, or a feature of human mental capacities. Presumably accidental features are thinkable, so 'distinct' is the key word.
19. Language / F. Communication / 6. Interpreting Language / a. Translation
19045 Translation is too flimsy a notion to support theories of cultural incommensurability [Quine]
     Full Idea: Translation is a flimsy notion, unfit to bear the weight of the theories of cultural incommensurability that Davidson effectively and justly criticises.
     From: Willard Quine (On the Very Idea of a Third Dogma [1981], p.42)
     A reaction: I presume he means that a claim to accurately translate something is false, because there is no clear idea of what a good translation looks like it. I just don't believe him. The practice of daily life belies Quine's theories on this.