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All the ideas for 'Letters to Edward Stillingfleet', 'Universals' and 'The Philosophy of Mathematics'

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12 ideas

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
9193 ZF set theory has variables which range over sets, 'equals' and 'member', and extensionality [Dummett]
     Full Idea: ZF set theory is a first-order axiomatization. Variables range over sets, there are no second-order variables, and primitive predicates are just 'equals' and 'member of'. The axiom of extensionality says sets with the same members are identical.
     From: Michael Dummett (The Philosophy of Mathematics [1998], 7)
     A reaction: If the eleven members of the cricket team are the same as the eleven members of the hockey team, is the cricket team the same as the hockey team? Our cricket team is better than our hockey team, so different predicates apply to them.
9194 The main alternative to ZF is one which includes looser classes as well as sets [Dummett]
     Full Idea: The main alternative to ZF is two-sorted theories, with some variables ranging over classes. Classes have more generous existence assumptions: there is a universal class, containing all sets, and a class containing all ordinals. Classes are not members.
     From: Michael Dummett (The Philosophy of Mathematics [1998], 7.1.1)
     A reaction: My intuition is to prefer strict systems when it comes to logical theories. The whole point is precision. Otherwise we could just think about things, and skip all this difficult symbolic stuff.
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
9195 Intuitionists reject excluded middle, not for a third value, but for possibility of proof [Dummett]
     Full Idea: It must not be concluded from the rejection of excluded middle that intuitionistic logic operates with three values: true, false, and neither true nor false. It does not make use of true and false, but only with a construction being a proof.
     From: Michael Dummett (The Philosophy of Mathematics [1998], 8.1)
     A reaction: This just sounds like verificationism to me, with all its problems. It seems to make speculative statements meaningless, which can't be right. Realism has lots of propositions which are assumed to be true or false, but also unknowable.
5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
9186 First-order logic concerns objects; second-order adds properties, kinds, relations and functions [Dummett]
     Full Idea: First-order logic is distinguished by generalizations (quantification) only over objects: second-order logic admits generalizations or quantification over properties or kinds of objects, and over relations between them, and functions defined over them.
     From: Michael Dummett (The Philosophy of Mathematics [1998], 3.1)
     A reaction: Second-order logic was introduced by Frege, but is (interestingly) rejected by Quine, because of the ontological commitments involved. I remain unconvinced that quantification entails ontological commitment, so I'm happy.
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
9187 Logical truths and inference are characterized either syntactically or semantically [Dummett]
     Full Idea: There are two ways of characterizing logical truths and correct inference. Proof-theoretic or syntactic characterizations, if the formalization admits of proof or derivation; and model-theoretic or semantic versions, being true in all interpretations.
     From: Michael Dummett (The Philosophy of Mathematics [1998], 3.1)
     A reaction: Dummett calls this distinction 'fundamental'. The second one involves truth, and hence meaning, where the first one just responds to rules. ..But how can you have a notion of correctly following a rule, without a notion of truth?
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
9191 Ordinals seem more basic than cardinals, since we count objects in sequence [Dummett]
     Full Idea: It can be argued that the notion of ordinal numbers is more fundamental than that of cardinals. To count objects, we must count them in sequence. ..The theory of ordinals forms the substratum of Cantor's theory of cardinals.
     From: Michael Dummett (The Philosophy of Mathematics [1998], 5)
     A reaction: Depends what you mean by 'fundamental'. I would take cardinality to be psychologically prior ('that is a lot of sheep'). You can't order people by height without first acquiring some people with differing heights. I vote for cardinals.
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
9192 The number 4 has different positions in the naturals and the wholes, with the same structure [Dummett]
     Full Idea: The number 4 cannot be characterized solely by its position in a system, because it has different positions in the system of natural numbers and that of the positive whole numbers, whereas these systems have the very same structure.
     From: Michael Dummett (The Philosophy of Mathematics [1998], 6.1)
     A reaction: Dummett seems to think this is fairly decisive against structuralism. There is also the structure of the real numbers. We will solve this by saying that the wholes are abstracted from the naturals, which are abstracted from the reals. Job done.
8. Modes of Existence / D. Universals / 1. Universals
8495 The distinction between particulars and universals is a mistake made because of language [Ramsey]
     Full Idea: The whole theory of particulars and universals is due to mistaking for a fundamental characteristic of reality what is merely a characteristic of language.
     From: Frank P. Ramsey (Universals [1925], p.13)
     A reaction: [Fraser MacBride has pursued this idea] It is rather difficult to deny the existence of particulars, in the sense of actual objects, so this appears to make Ramsey a straightforward nominalist, of some sort or other.
8493 We could make universals collections of particulars, or particulars collections of their qualities [Ramsey]
     Full Idea: The two obvious methods of abolishing the distinction between particulars and universals are by holding either that universals are collections of particulars, or that particulars are collections of their qualities.
     From: Frank P. Ramsey (Universals [1925], p.8)
     A reaction: Ramsey proposes an error theory, arising out of language. Quine seems to offer another attempt, making objects and predication unanalysable and basic. Abstract reference seems to make the strongest claim to separate out the universals.
8. Modes of Existence / E. Nominalism / 1. Nominalism / a. Nominalism
8494 Obviously 'Socrates is wise' and 'Socrates has wisdom' express the same fact [Ramsey]
     Full Idea: It seems to me as clear as anything can be in philosophy that the two sentences 'Socrates is wise' and 'wisdom is a characteristic of Socrates' assert the same fact and express the same proposition.
     From: Frank P. Ramsey (Universals [1925], p.12)
     A reaction: Could be challenged. One says Socrates is just the way he is, the other says he is attached to an abstract entity greater than himself. The squabble over universals has become a squabble over logical form. Finding logical form needs metaphysics!
9. Objects / D. Essence of Objects / 3. Individual Essences
15990 Every individual thing which exists has an essence, which is its internal constitution [Locke]
     Full Idea: I take essences to be in everything that internal constitution or frame for the modification of substance, which God in his wisdom gives to every particular creature, when he gives it a being; and such essences I grant there are in all things that exist.
     From: John Locke (Letters to Edward Stillingfleet [1695], Letter 1), quoted by Simon Blackburn - Quasi-Realism no Fictionalism
     A reaction: This is the clearest statement I have found of Locke's commitment to essences, for all his doubts about whether we can know such things. Alexander says (ch.13) Locke was reacting against scholastic essence, as pertaining to species.
11. Knowledge Aims / B. Certain Knowledge / 1. Certainty
15994 If it is knowledge, it is certain; if it isn't certain, it isn't knowledge [Locke]
     Full Idea: What reaches to knowledge, I think may be called certainty; and what comes short of certainty, I think cannot be knowledge.
     From: John Locke (Letters to Edward Stillingfleet [1695], Letter 2), quoted by Simon Blackburn - Quasi-Realism no Fictionalism
     A reaction: I much prefer that fallibilist approach offered by the pragmatists. Knowledge is well-supported belief which seems (and is agreed) to be true, but there is a small shadow of doubt hanging over all of it.