Combining Texts

All the ideas for 'Reportatio', 'The Philosophy of Mathematics' and 'The Morality of Happiness'

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17 ideas

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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.
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
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
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
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
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
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.
20. Action / C. Motives for Action / 3. Acting on Reason / a. Practical reason
'Phronesis' should translate as 'practical intelligence', not as prudence [Annas]
     Full Idea: The best translation of 'phronesis' is probably not 'prudence' (which implies a non-moral motive), or 'practical wisdom' (which makes it sound contemplative), but 'practical intelligence', or just 'intelligence'.
     From: Julia Annas (The Morality of Happiness [1993], 2.3)
22. Metaethics / C. The Good / 3. Pleasure / d. Sources of pleasure
Epicureans achieve pleasure through character development [Annas]
     Full Idea: Since having a virtue does not reduce to performing certain kinds of acts, the Epicurean will achieve pleasure only by aiming at being a certain kind of person.
     From: Julia Annas (The Morality of Happiness [1993], 2.4)
     A reaction: No Epicurean would want to merely possess virtues, without enacting them. I assume that virtues are sought as guides to finding the finest pleasures (such as friendship).
23. Ethics / A. Egoism / 3. Cyrenaic School
Cyrenaics pursue pleasure, but don't equate it with happiness [Annas]
     Full Idea: Cyrenaics claimed our final good was pleasure, best achieved by seeking maximum intensity of pleasurable experiences, but they explicitly admitted that this was not happiness.
     From: Julia Annas (The Morality of Happiness [1993], 1)
23. Ethics / C. Virtue Theory / 2. Elements of Virtue Theory / c. Motivation for virtue
Ancient ethics uses attractive notions, not imperatives [Annas]
     Full Idea: Instead of modern 'imperative' notions of ethics (involving obligation, duty and rule-following), ancient ethics uses 'attractive' notions like those of goodness and worth
     From: Julia Annas (The Morality of Happiness [1993], Intro)
23. Ethics / D. Deontological Ethics / 1. Deontology
Principles cover life as a whole, where rules just cover actions [Annas]
     Full Idea: Principles concern not just types of actions, but one's life as a whole, grasping truths about the nature of justice, and the like; they explain rules, giving the 'why' and not just the 'what'.
     From: Julia Annas (The Morality of Happiness [1993], 2.4)
23. Ethics / D. Deontological Ethics / 2. Duty
Virtue theory tries to explain our duties in terms of our character [Annas]
     Full Idea: An ethics of virtue moves from an initial interest in what we ought to do to an interest in the kinds of people we are and hope to be, because the latter is taken to be the best way of understanding the former.
     From: Julia Annas (The Morality of Happiness [1993], 2.5)
23. Ethics / D. Deontological Ethics / 6. Motivation for Duty
If excessively good actions are admirable but not required, then duty isn't basic [Annas]
     Full Idea: Supererogatory actions are admirable and valuable, and we praise people for doing them, but they do not generate obligations to perform them, which casts doubt on obligation as the basic notion in ethics.
     From: Julia Annas (The Morality of Happiness [1993], 2.6)
23. Ethics / E. Utilitarianism / 1. Utilitarianism
We should do good when necessary, not maximise it [Annas]
     Full Idea: Why should I want to maximise my acting courageously? I act courageously when it is required.
     From: Julia Annas (The Morality of Happiness [1993], 1)
28. God / A. Divine Nature / 3. Divine Perfections
God is not wise, but more-than-wise; God is not good, but more-than-good [William of Ockham]
     Full Idea: God is not wise, but more-than-wise; God is not good, but more-than-good.
     From: William of Ockham (Reportatio [1330], III Q viii)
     A reaction: [He is quoting 'Damascene'] I quote this for interest, but I very much doubt whether Damascene or William knew what it meant, and I certainly don't. There seems to have been a politically correct desire to invent super-powers for God.
28. God / C. Attitudes to God / 4. God Reflects Humanity
We could never form a concept of God's wisdom if we couldn't abstract it from creatures [William of Ockham]
     Full Idea: What we abstract is said to belong to perfection in so far as it can be predicated of God and can stand for Him. For if such a concept could not be abstracted from a creature, then in this life we could not arrive at a cognition of God's wisdom.
     From: William of Ockham (Reportatio [1330], III Q viii)
     A reaction: This seems to be the germ of an important argument. Without the ability to abstract from what is experienced, we would not be able to apply general concepts to things which are beyond experience. It is a key idea for empiricism.