Combining Texts

All the ideas for '67: Platonic Questions', 'Constructibility and Mathematical Existence' and 'The Nature of Necessity'

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8 ideas

4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
8758 We could talk of open sentences, instead of sets [Chihara, by Shapiro]
     Full Idea: Chihara's programme is to replace talk of sets with talk of open sentences. Instead of speaking of the set of all cats, we talk about the open sentence 'x is a cat'.
     From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Thinking About Mathematics 9.2
     A reaction: As Shapiro points out, this is following up Russell's view that sets should be replaced with talk of properties. Chihara is expressing it more linguistically. I'm in favour of any attempt to get rid of sets.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
10265 Chihara's system is a variant of type theory, from which he can translate sentences [Chihara, by Shapiro]
     Full Idea: Chihara's system is a version of type theory. Translate thus: replace variables of sets of type n with level n variables over open sentences, replace membership/predication with satisfaction, and high quantifiers with constructability quantifiers.
     From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Philosophy of Mathematics 7.4
8759 We can replace type theory with open sentences and a constructibility quantifier [Chihara, by Shapiro]
     Full Idea: Chihara's system is similar to simple type theory; he replaces each type with variables over open sentences, replaces membership (or predication) with satisfaction, and replaces quantifiers over level 1+ variables with constructability quantifiers.
     From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Thinking About Mathematics 9.2
     A reaction: This is interesting for showing that type theory may not be dead. The revival of supposedly dead theories is the bread-and-butter of modern philosophy.
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
10264 Introduce a constructibility quantifiers (Cx)Φ - 'it is possible to construct an x such that Φ' [Chihara, by Shapiro]
     Full Idea: Chihara has proposal a modal primitive, a 'constructability quantifier'. Syntactically it behaves like an ordinary quantifier: Φ is a formula, and x a variable. Then (Cx)Φ is a formula, read as 'it is possible to construct an x such that Φ'.
     From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Philosophy of Mathematics 7.4
     A reaction: We only think natural numbers are infinite because we see no barrier to continuing to count, i.e. to construct new numbers. We accept reals when we know how to construct them. Etc. Sounds promising to me (though not to Shapiro).
10. Modality / E. Possible worlds / 1. Possible Worlds / d. Possible worlds actualism
18383 Plantinga says there is just this world, with possibilities expressed in propositions [Plantinga, by Armstrong]
     Full Idea: Plantinga rejects other possible worlds, but adds to our world an uncountable multitude of sets of propositions, each set a way that the world might have been, but is in fact not. (Roughly, for each Lewis world, Plantinga has such a set).
     From: report of Alvin Plantinga (The Nature of Necessity [1974]) by David M. Armstrong - Truth and Truthmakers 07.2
     A reaction: To me it seems as ontologically extravagant to postulate unexpressed propositions as to postulate concrete possible worlds. I think the best line is that there is just the actual world, with the possibilities implied in its dispositions.
10. Modality / E. Possible worlds / 3. Transworld Objects / b. Rigid designation
11891 Possibilities for an individual can only refer to that individual, in some possible world [Plantinga, by Mackie,P]
     Full Idea: Plantinga says for an individual to exist with certain properties in some possible world is simply for it to be true that, had that possible world obtained, that individual would have existed with those properties.
     From: report of Alvin Plantinga (The Nature of Necessity [1974]) by Penelope Mackie - How Things Might Have Been 5.1
     A reaction: This is intended to dissolve the problem of transworld identity, and is certainly a flat rejection of counterparts. I take the point to be that the individual is the key element in defining the possible world, so can't possibly be different.
15. Nature of Minds / A. Nature of Mind / 2. Psuche
5960 When the soul is intelligent and harmonious, it is part of god and derives from god [Plutarch]
     Full Idea: The soul, when it has partaken of intelligence and reason and concord, is not merely a work but also a part of god and has come to be not by his agency but both from him as source and out of his substance.
     From: Plutarch (67: Platonic Questions [c.85], II.1001)
     A reaction: A most intriguing shift of view from earlier concepts of the psuché. How did this come about? This man is a pagan. The history is in the evolution of Platonism. See 'The Middle Platonists' by John Dillon. Davidson is also very impressed by reason.
28. God / B. Proving God / 2. Proofs of Reason / a. Ontological Proof
20704 A possible world contains a being of maximal greatness - which is existence in all worlds [Plantinga, by Davies,B]
     Full Idea: Plantinga reformulates Malcolm's argument thus: 1) There is a possible world in which there exists a being with maximal greatness, 2) A being has maximal greatness in a world only if it exists in every world.
     From: report of Alvin Plantinga (The Nature of Necessity [1974], p.213) by Brian Davies - Introduction to the Philosophy of Religion 4 'b Descartes'
     A reaction: This is only Plantinga's starting point, which says nothing about the nature of God, but only that this 'great' being exists in all worlds. I would like to know why it is a 'being' rather than a 'thing'. Malcolm says if it is possible it is necessary.