Combining Texts

All the ideas for 'Some Puzzles of Ground', 'works' and 'On Platonism in Mathematics'

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6 ideas

4. Formal Logic / E. Nonclassical Logics / 3. Many-Valued Logic
14263 Strong Kleene disjunction just needs one true disjunct; Weak needs the other to have some value [Fine,K]
     Full Idea: Under strong Kleene tables, a disjunction will be true if one of the disjuncts is true, regardless of whether or not the other disjunct has a truth-value; under the weak table it is required that the other disjunct also have a value. So for other cases.
     From: Kit Fine (Some Puzzles of Ground [2010], n7)
     A reaction: [see also p.111 of Fine's article] The Kleene tables seem to be the established form of modern three-valued logic, with the third value being indeterminate.
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
10304 Very few things in set theory remain valid in intuitionist mathematics [Bernays]
     Full Idea: Very few things in set theory remain valid in intuitionist mathematics.
     From: Paul Bernays (On Platonism in Mathematics [1934])
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
10303 Restricted Platonism is just an ideal projection of a domain of thought [Bernays]
     Full Idea: A restricted Platonism does not claim to be more than, so to speak, an ideal projection of a domain of thought.
     From: Paul Bernays (On Platonism in Mathematics [1934], p.261)
     A reaction: I have always found Platonism to be congenial when it talks of 'ideals', and ridiculous when it talks of a special form of 'existence'. Ideals only 'exist' because we idealise things. I may declare myself, after all, to be a Restricted Platonist.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
10306 Mathematical abstraction just goes in a different direction from logic [Bernays]
     Full Idea: Mathematical abstraction does not have a lesser degree than logical abstraction, but rather another direction.
     From: Paul Bernays (On Platonism in Mathematics [1934], p.268)
     A reaction: His point is that the logicists seem to think that if you increasingly abstract from mathematics, you end up with pure logic.
7. Existence / C. Structure of Existence / 1. Grounding / a. Nature of grounding
14262 Formal grounding needs transitivity of grounding, no self-grounding, and the existence of both parties [Fine,K]
     Full Idea: The general formal principles of grounding are Transitivity (A«B, B«C/A«C: if A helps ground B and B helps C, then A helps C), Irreflexivity (A«A/absurd: A can't ground itself) and Factivity (A«B/A; A«/B: for grounding both A and B must be the case).
     From: Kit Fine (Some Puzzles of Ground [2010], 4)
9. Objects / B. Unity of Objects / 2. Substance / b. Need for substance
10710 We accept substance, to avoid infinite backwards chains of meaning [Wittgenstein, by Potter]
     Full Idea: Wittgenstein is the most renowned modern proponent of substance, and argued that sense must be determinate ...and that any conceptual scheme which genuinely represents a world cannot contain infinite backward chains of meaning.
     From: report of Ludwig Wittgenstein (works [1935]) by Michael Potter - Set Theory and Its Philosophy 03.3
     A reaction: This is a key idea for explaining the somewhat surprising revival of the notion of substance in modern times, when it appeared to have been buried by atomism in the seventeenth century. The new argument is a semantic one.