18 ideas
| 10041 | Impredicative Definitions refer to the totality to which the object itself belongs [Gödel] |
| Full Idea: Impredicative Definitions are definitions of an object by reference to the totality to which the object itself (and perhaps also things definable only in terms of that object) belong. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], n 13) |
| 21716 | In simple type theory the axiom of Separation is better than Reducibility [Gödel, by Linsky,B] |
| Full Idea: In the superior realist and simple theory of types, the place of the axiom of reducibility is not taken by the axiom of classes, Zermelo's Aussonderungsaxiom. | |
| From: report of Kurt Gödel (Russell's Mathematical Logic [1944], p.140-1) by Bernard Linsky - Russell's Metaphysical Logic 6.1 n3 | |
| A reaction: This is Zermelo's Axiom of Separation, but that too is not an axiom of standard ZFC. |
| 10035 | Mathematical Logic is a non-numerical branch of mathematics, and the supreme science [Gödel] |
| Full Idea: 'Mathematical Logic' is a precise and complete formulation of formal logic, and is both a section of mathematics covering classes, relations, symbols etc, and also a science prior to all others, with ideas and principles underlying all sciences. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.447) | |
| A reaction: He cites Leibniz as the ancestor. In this database it is referred to as 'theory of logic', as 'mathematical' seems to be simply misleading. The principles of the subject are standardly applied to mathematical themes. |
| 10042 | Reference to a totality need not refer to a conjunction of all its elements [Gödel] |
| Full Idea: One may, on good grounds, deny that reference to a totality necessarily implies reference to all single elements of it or, in other words, that 'all' means the same as an infinite logical conjunction. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.455) |
| 10038 | A logical system needs a syntactical survey of all possible expressions [Gödel] |
| Full Idea: In order to be sure that new expression can be translated into expressions not containing them, it is necessary to have a survey of all possible expressions, and this can be furnished only by syntactical considerations. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.448) | |
| A reaction: [compressed] |
| 10046 | The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers [Gödel] |
| Full Idea: The generalized Continuum Hypothesis says that there exists no cardinal number between the power of any arbitrary set and the power of the set of its subsets. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.464) |
| 10039 | Some arithmetical problems require assumptions which transcend arithmetic [Gödel] |
| Full Idea: It has turned out that the solution of certain arithmetical problems requires the use of assumptions essentially transcending arithmetic. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.449) | |
| A reaction: A nice statement of the famous result, from the great man himself, in the plainest possible English. |
| 10043 | Mathematical objects are as essential as physical objects are for perception [Gödel] |
| Full Idea: Classes and concepts may be conceived of as real objects, ..and are as necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions, with neither case being about 'data'. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.456) | |
| A reaction: Note that while he thinks real objects are essential for mathematics, be may not be claiming the same thing for our knowledge of logic. If logic contains no objects, then how could mathematics be reduced to it, as in logicism? |
| 10045 | Impredicative definitions are admitted into ordinary mathematics [Gödel] |
| Full Idea: Impredicative definitions are admitted into ordinary mathematics. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.464) | |
| A reaction: The issue is at what point in building an account of the foundations of mathematics (if there be such, see Putnam) these impure definitions should be ruled out. |
| 6479 | Noninterference requires justification as much as interference does [Nagel] |
| Full Idea: Noninterference requires justification as much as interference does. | |
| From: Thomas Nagel (Equality and Partiality [1991], Ch.10) | |
| A reaction: I'm not convinced by this, as a simple rule. If I spend my whole life doing just the minimum for my own survival, I don't see why I should have to justify that, and I don't see a state is obliged to justify it either. |
| 6450 | Morality must be motivating, and not because of pre-moral motives [Nagel] |
| Full Idea: My own view is that moral justification must be capable of motivating, but not in virtue of reliance on pre-moral motives. | |
| From: Thomas Nagel (Equality and Partiality [1991], Ch.5) | |
| A reaction: This may well be the core and essence of Kantian moral theory. I'm inclined to think of it as 'Kant's dream', which is of ultra-rational beings who are driven by pure rationality as a motivator. People who fit this bill tend to be academics. |
| 6447 | Game theory misses out the motivation arising from the impersonal standpoint [Nagel] |
| Full Idea: I do not favour the route taken by Hobbes's modern descendants, using game theory, since I believe the impersonal standpoint makes an essential contribution to individual motivation which must be addressed by any ethically acceptable theory. | |
| From: Thomas Nagel (Equality and Partiality [1991], Ch.4) | |
| A reaction: The assumption of self-seeking at the core of game theory seems very bizarre, and leads to moral approval of free riders. Nagel offers the best response, which is the Kantian impersonal view. Nagel may be optimistic about motivation, though. |
| 6446 | In ethics we abstract from our identity, but not from our humanity [Nagel] |
| Full Idea: In pursuit of the kind of objectivity needed in the physical sciences, we abstract even from our humanity; but nothing further than abstraction from our identity (that is, who we are) enters into ethical theory. | |
| From: Thomas Nagel (Equality and Partiality [1991], Ch.2) | |
| A reaction: The 'brief' summary of this boils down to a nice and interesting slogan. It epitomises the modern Kantian approach to ethics. But compare Idea 4122, from Bernard Williams. |
| 6477 | I can only universalise a maxim if everyone else could also universalise it [Nagel] |
| Full Idea: It is implicit in the categorical imperative that I can will that everyone should adopt as a maxim only what everyone else can also will that everyone should adopt as a maxim. | |
| From: Thomas Nagel (Equality and Partiality [1991], Ch.5) | |
| A reaction: This is a nice move, because it shifts the theory away from a highly individualistic Cartesian view of morality towards the idea that morality is a community activity. |
| 6448 | A legitimate system is one accepted as both impartial and reasonably partial [Nagel] |
| Full Idea: A legitimate system is one which reconciles the two universal principles of impartiality and reasonable partiality so that no one can object that his interests are not being accorded sufficient weight or that the demands on him are excessive. | |
| From: Thomas Nagel (Equality and Partiality [1991], Ch.4) | |
| A reaction: This seems an appealing principle, and a nice attempt at stating the core of Kantian liberalism. It is obviously influenced by Scanlon's contractualist view, in the idea that 'no one can object', because everyone sees the justification. |
| 6478 | Democracy is opposed to equality, if the poor are not a majority [Nagel] |
| Full Idea: As things are, democracy is the enemy of comprehensive equality, once the poor cease to be a majority. | |
| From: Thomas Nagel (Equality and Partiality [1991], Ch.9) | |
| A reaction: This is obvious once you think about it, but it is well worth saying, because it is tempting to think that we live in an 'equal' society, merely because we are equal in things such as voting rights and equality before the law. |
| 16713 | Philosophers are the forefathers of heretics [Tertullian] |
| Full Idea: Philosophers are the forefathers of heretics. | |
| From: Tertullian (works [c.200]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.2 |
| 6610 | I believe because it is absurd [Tertullian] |
| Full Idea: I believe because it is absurd ('Credo quia absurdum est'). | |
| From: Tertullian (works [c.200]), quoted by Robert Fogelin - Walking the Tightrope of Reason n4.2 | |
| A reaction: This seems to be a rather desperate remark, in response to what must have been rather good hostile arguments. No one would abandon the support of reason if it was easy to acquire. You can't deny its engaging romantic defiance, though. |