Combining Texts

All the ideas for 'Chomsky on himself', 'Predest.,God's foreknowledge and contingents' 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.
7. Existence / E. Categories / 5. Category Anti-Realism
16654 Our words and concepts don't always correspond to what is out there [William of Ockham]
     Full Idea: It should not be said that as distinct words and intentions or concepts are distinct from one another, so too the corresponding things are distinct. Those distinctions do not always line up with distinctions among things that are signified.
     From: William of Ockham (Predest.,God's foreknowledge and contingents [1320], 7.1), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 12.2
     A reaction: [compressed] This is the great nominalist opponent of the idea that Aristotle's ten categories give an accurate map of reality. He proposed just substance and accidents, and based categorisation on the questions we ask.
18. Thought / D. Concepts / 2. Origin of Concepts / c. Nativist concepts
6649 Chomsky now says concepts are basically innate, as well as syntax [Chomsky, by Lowe]
     Full Idea: Chomsky now contends that not only the syntax of natural language but also the concepts expressible in it have an innate basis.
     From: report of Noam Chomsky (Chomsky on himself [1994]) by E.J. Lowe - Introduction to the Philosophy of Mind Ch.7 n25
     A reaction: This seems to follow Fodor, who has been mocked for implying that we have an innate idea of a screwdriver etc. Note that Chomsky says concepts have an innate 'basis'. This fits well with modern (cautious) rationalism, with which I am happy.