45 ideas
| 17621 | What matters in mathematics is its objectivity, not the existence of the objects [Dummett] |
| 10041 | Impredicative Definitions refer to the totality to which the object itself belongs [Gödel] |
| 18806 | Frege thought traditional categories had psychological and linguistic impurities [Frege, by Rumfitt] |
| 10537 | The ordered pairs <x,y> can be reduced to the class of sets of the form {{x},{x,y}} [Dummett] |
| 10542 | To associate a cardinal with each set, we need the Axiom of Choice to find a representative [Dummett] |
| 21716 | In simple type theory the axiom of Separation is better than Reducibility [Gödel, by Linsky,B] |
| 10035 | Mathematical Logic is a non-numerical branch of mathematics, and the supreme science [Gödel] |
| 8490 | First-level functions have objects as arguments; second-level functions take functions as arguments [Frege] |
| 8492 | Relations are functions with two arguments [Frege] |
| 10042 | Reference to a totality need not refer to a conjunction of all its elements [Gödel] |
| 10038 | A logical system needs a syntactical survey of all possible expressions [Gödel] |
| 10046 | The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers [Gödel] |
| 10039 | Some arithmetical problems require assumptions which transcend arithmetic [Gödel] |
| 10554 | Intuitionists find the Incompleteness Theorem unsurprising, since proof is intuitive, not formal [Dummett] |
| 10043 | Mathematical objects are as essential as physical objects are for perception [Gödel] |
| 8487 | Arithmetic is a development of logic, so arithmetical symbolism must expand into logical symbolism [Frege] |
| 10552 | Intuitionism says that totality of numbers is only potential, but is still determinate [Dummett] |
| 10045 | Impredicative definitions are admitted into ordinary mathematics [Gödel] |
| 18899 | Frege takes the existence of horses to be part of their concept [Frege, by Sommers] |
| 10515 | Ostension is possible for concreta; abstracta can only be referred to via other objects [Dummett, by Hale] |
| 10544 | The concrete/abstract distinction seems crude: in which category is the Mistral? [Dummett] |
| 10546 | We don't need a sharp concrete/abstract distinction [Dummett] |
| 10540 | We can't say that light is concrete but radio waves abstract [Dummett] |
| 10548 | The context principle for names rules out a special philosophical sense for 'existence' [Dummett] |
| 10281 | The objects we recognise the world as containing depends on the structure of our language [Dummett] |
| 4028 | Frege allows either too few properties (as extensions) or too many (as predicates) [Mellor/Oliver on Frege] |
| 10532 | We can understand universals by studying predication [Dummett] |
| 10534 | 'Nominalism' used to mean denial of universals, but now means denial of abstract objects [Dummett] |
| 10541 | Concrete objects such as sounds and smells may not be possible objects of ostension [Dummett] |
| 10545 | Abstract objects may not cause changes, but they can be the subject of change [Dummett] |
| 10555 | If we can intuitively apprehend abstract objects, this makes them observable and causally active [Dummett] |
| 10543 | Abstract objects must have names that fall within the range of some functional expression [Dummett] |
| 10320 | If a genuine singular term needs a criterion of identity, we must exclude abstract nouns [Dummett, by Hale] |
| 10547 | Abstract objects can never be confronted, and need verbal phrases for reference [Dummett] |
| 8489 | The concept 'object' is too simple for analysis; unlike a function, it is an expression with no empty place [Frege] |
| 10531 | There is a modern philosophical notion of 'object', first introduced by Frege [Dummett] |
| 9947 | Concepts are the ontological counterparts of predicative expressions [Frege, by George/Velleman] |
| 19168 | Concepts only have a 'functional character', because they map to truth values, not objects [Dummett, by Davidson] |
| 10319 | An assertion about the concept 'horse' must indirectly speak of an object [Frege, by Hale] |
| 8488 | A concept is a function whose value is always a truth-value [Frege] |
| 9948 | Unlike objects, concepts are inherently incomplete [Frege, by George/Velleman] |
| 10549 | Since abstract objects cannot be picked out, we must rely on identity statements [Dummett] |
| 10516 | A realistic view of reference is possible for concrete objects, but not for abstract objects [Dummett, by Hale] |
| 4972 | I may regard a thought about Phosphorus as true, and the same thought about Hesperus as false [Frege] |
| 8491 | The Ontological Argument fallaciously treats existence as a first-level concept [Frege] |