20 ideas
| 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. |
| 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. |
| 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. |
| 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? |
| 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. |
| 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. |
| 23814 | Every human yearns for an unattainable transcendent good [Weil] |
| Full Idea: There is a reality outside the world …outside any sphere that is accessible to human faculties. Corresponding to this reality, at the centre of the human heart, is the longing for an absolute good, which is always there and never appeased by this world. | |
| From: Simone Weil (Draft Statement of Human Obligations [1943], p.221) | |
| A reaction: I don't believe in any sort of transcendent reality, but I can identify with this. Even if you have a highly naturalistic view of what is valuable (see late Philippa Foot), there is this indeterminate yearning for that value. |
| 23824 | Where human needs are satisfied we find happiness, friendship and beauty [Weil] |
| Full Idea: Any place where the needs of human beings are satisfied can be recognised by the fact that there is a flowering of fraternity, joy, beauty, and happiness. | |
| From: Simone Weil (Draft Statement of Human Obligations [1943], p.230) | |
| A reaction: Weil writes a lengthy analysis of what she sees as the basic human needs, beyond the obvious food, water etc. An excellent place to start a line of political thought. |
| 23815 | We cannot equally respect what is unequal, so equal respect needs a shared ground [Weil] |
| Full Idea: It is impossible to feel equal respect for things that are in fact unequal unless the respect is given to something that is identical in all of them. Men are all unequal in all their relations with things of this world. | |
| From: Simone Weil (Draft Statement of Human Obligations [1943], p.223) | |
| A reaction: Weil votes for some link to transcendence in each of us, but I would prefer some more naturalistic proposal for what we all have in common. There are plenty of aspects which unite all human beings, which grounds this unconditional respect. |
| 23823 | Life needs risks to avoid sickly boredom [Weil] |
| Full Idea: The boredom produced by a complete absence of risk is a sickness of the human soul. | |
| From: Simone Weil (Draft Statement of Human Obligations [1943], p.229) | |
| A reaction: An unusual analysis of boredom. I think it is probably purposeful activity that we need, rather than actual risk, with all the stresses that involves. Risks are justified by their rewards. |
| 23822 | We all need to partipate in public tasks, and take some initiative [Weil] |
| Full Idea: The human soul has need of disciplined participation in a common task of public value, and it has need of personal initiative within this participation. | |
| From: Simone Weil (Draft Statement of Human Obligations [1943], p.229) | |
| A reaction: The intrusion of competitive capitalism into almost every area of modern life has more or less eliminated such activities. Only state employees now have such satisfactions, on the whole. I admire Weil's approach here. |
| 23817 | We need both equality (to attend to human needs) and hierarchy (as a scale of responsibilities) [Weil] |
| Full Idea: The human soul has need of equality and of hierarchy. Equality is the public recognition …of the principal that an equal degree of attention is due to the needs of all human beings. Hierarchy is the scale of responsibilities. | |
| From: Simone Weil (Draft Statement of Human Obligations [1943], p.228) | |
| A reaction: This is the conservative aspect of Weil's largely radical political thinking. Presumably what we respect in these people is their responsibilies, and not their mere rank. Idle members of the British House of Lords have no rank in this hierarchy. |
| 23819 | Deliberate public lying should be punished [Weil] |
| Full Idea: Every avoidable material falsehood publicly asserted should become a punishable offence. | |
| From: Simone Weil (Draft Statement of Human Obligations [1943], p.228) | |
| A reaction: Yes please! The early 21st century has become the time when truth lost all value in public life. Lying to the House of Commons in the UK required instant resignation 50 years ago. Now it is just a source of laughter. No freedom to lie! |
| 23818 | We have liberty in the space between nature and accepted authority [Weil] |
| Full Idea: Liberty is the power of choice within the latitude left between the direct constraint of natural forces and the authority accepted as legitimate. | |
| From: Simone Weil (Draft Statement of Human Obligations [1943], p.228) | |
| A reaction: Accepting legitimate authority is a nicely softened version of the social contract. We often find that the office and rank are accepted as legitimate, but then are unable to accept the appalling individual who holds the office. |
| 23820 | People need personal and collective property, and a social class lacking property is shameful [Weil] |
| Full Idea: The human soul has need of both personal property and collective property. …The existence of a social class defined by the lack of personal and collective property is as shameful as slavery. | |
| From: Simone Weil (Draft Statement of Human Obligations [1943], p.229) | |
| A reaction: Nice. Particularly the idea that we all need collective property, such as parks and beaches and public buildings. |
| 23821 | Crime should be punished, to bring the perpetrator freely back to morality [Weil] |
| Full Idea: The human soul needs punishment and honour. A committer of crime has become exiled from good, and needs to be reintegrated with it through suffering. This aims to bring the soul to recognise freely some day that is infliction was just. | |
| From: Simone Weil (Draft Statement of Human Obligations [1943], p.229) | |
| A reaction: The Scanlon contractualist approach to punishment - that the victim of it accepts its justice. Given her saintly character, Simone had a very tough view of this issue. |
| 9111 | 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. |
| 23816 | Attention to a transcendent reality motivates a duty to foster the good of humanity [Weil] |
| Full Idea: Anyone whose attention and love are directed towards the reality outside the world recognises that he is bound by the permanent obligation to remedy …all the privations of soul and body which are liable to destroy or damage any human being whatsoever. | |
| From: Simone Weil (Draft Statement of Human Obligations [1943], p.225) | |
| A reaction: [abridged] An interesting attempt to articulate the religious motivation of morality. The Euthyphro question remains - of why this vision of a wholly good higher morality should motivate anyone, unless they already possess a desire for that good. |
| 9112 | 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. |