16 ideas
| 12268 | Contradiction is impossible [Antisthenes (I), by Aristotle] |
| Full Idea: Antisthenes said that contradiction is impossible. | |
| From: report of Antisthenes (Ath) (fragments/reports [c.405 BCE]) by Aristotle - Topics 104b21 | |
| A reaction: Aristotle is giving an example of a 'thesis'. It should be taken seriously if a philosopher proposes it, but dismissed as rubbish if anyone else proposes it! No context is given for the remark. |
| 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) |
| 602 | Some fools think you cannot define anything, but only say what it is like [Antisthenes (I), by Aristotle] |
| Full Idea: There is an application of that old chestnut of the cynic Antisthenes' followers (and other buffoons of that kind). Their claim was that a definition of what something is is impossible. You cannot define silver, though you can say it is like tin. | |
| From: report of Antisthenes (Ath) (fragments/reports [c.405 BCE]) by Aristotle - Metaphysics 1043b |
| 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. |
| 18552 | Forget about beauty; just concentrate on the virtues of delicacy and discernment admired in critics [Hume, by Scruton] |
| Full Idea: Hume suggest we get away from the fruitless discussion of beauty, and simply concentrate on the qualities we admire, and ought to admire, in a critic - qualities such as delicacy and discernment. | |
| From: report of David Hume (Of the standard of taste [1757]) by Roger Scruton - Beauty: a very short introduction 6 | |
| A reaction: We might wonder how you can admire 'discernment' without some view of the thing being discern, which is in danger of being beauty. How do you judge delicacy and discernment without judging the objects of the critic's taste? Mere authority? |
| 6608 | Strong sense, delicate sentiment, practice, comparisons, and lack of prejudice, are all needed for good taste [Hume] |
| Full Idea: Strong sense, united to delicate sentiment, improved by practice, perfected by comparison, and cleared of all prejudice, can alone entitle critics to the valuable character of having 'taste'. | |
| From: David Hume (Of the standard of taste [1757]), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.6 | |
| A reaction: I agree entirely with this, but then I am a very politically incorrect elitist when it comes to taste. It just seems screamingly obvious that professional wine-tasters have a better appreciation of wine than me, and so on for the rest of the arts. |
| 1664 | I would rather go mad than experience pleasure [Antisthenes (I)] |
| Full Idea: I would rather go mad than experience pleasure. | |
| From: Antisthenes (Ath) (fragments/reports [c.405 BCE]), quoted by Diogenes Laertius - Lives of Eminent Philosophers 06.3 | |
| A reaction: Did he actually prefer pain? If both experiences would drive him mad, it seems like a desire for death. I cannot understand why anyone is opposed to harmless pleasures. |
| 21385 | Antisthenes said virtue is teachable and permanent, is life's goal, and is like universal wealth [Antisthenes (I), by Long] |
| Full Idea: The moral propositions of Antisthenes foreshadowed the Stoics: virtue can be taught and once acquired cannot be lost (fr.69,71); virtue is the goal of life (22); the sage is self-sufficient, since he has (by being wise) the wealth of all men (8o). | |
| From: report of Antisthenes (Ath) (fragments/reports [c.405 BCE]) by A.A. Long - Hellenistic Philosophy 1 | |
| A reaction: [He cites Caizzi for the fragments] The distinctive idea here is (I think) that once acquired virtue can never be lost. It sounds plausible, but I'm wondering why it should be true. Is it like riding a bicycle, or like learning to speak Russian? |
| 2631 | Antisthenes says there is only one god, which is nature [Antisthenes (I), by Cicero] |
| Full Idea: Antisthenes says there is only one god, which is nature. | |
| From: report of Antisthenes (Ath) (fragments/reports [c.405 BCE]) by M. Tullius Cicero - On the Nature of the Gods ('De natura deorum') I.32 |