102 ideas
20110 | Hegel, Fichte and Schelling wanted to know Kant's thing-in-itself, as ego, or nature, or spirit [Safranski] |
19275 | You cannot understand what exists without understanding possibility and necessity [Hale] |
10308 | Questions about objects are questions about certain non-vacuous singular terms [Hale] |
19291 | A canonical defintion specifies the type of thing, and what distinguish this specimen [Hale] |
9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
10314 | An expression is a genuine singular term if it resists elimination by paraphrase [Hale] |
19297 | The two Barcan principles are easily proved in fairly basic modal logic [Hale] |
19301 | With a negative free logic, we can dispense with the Barcan formulae [Hale] |
10098 | The 'power set' of A is all the subsets of A [George/Velleman] |
10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman] |
10101 | Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B [George/Velleman] |
10103 | Grouping by property is common in mathematics, usually using equivalence [George/Velleman] |
10104 | 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman] |
10096 | Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman] |
10097 | Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman] |
10100 | Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman] |
17900 | The Axiom of Reducibility made impredicative definitions possible [George/Velleman] |
10109 | ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman] |
10108 | As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman] |
19296 | If second-order variables range over sets, those are just objects; properties and relations aren't sets [Hale] |
19289 | Maybe conventionalism applies to meaning, but not to the truth of propositions expressed [Hale] |
10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
10316 | We should decide whether singular terms are genuine by their usage [Hale] |
10312 | Often the same singular term does not ensure reliable inference [Hale] |
10313 | Plenty of clear examples have singular terms with no ontological commitment [Hale] |
10322 | If singular terms can't be language-neutral, then we face a relativity about their objects [Hale] |
19298 | Unlike axiom proofs, natural deduction proofs needn't focus on logical truths and theorems [Hale] |
10129 | A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman] |
10105 | Differences between isomorphic structures seem unimportant [George/Velleman] |
10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman] |
10126 | A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman] |
10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman] |
10127 | A 'complete' theory contains either any sentence or its negation [George/Velleman] |
10106 | Rational numbers give answers to division problems with integers [George/Velleman] |
10102 | The integers are answers to subtraction problems involving natural numbers [George/Velleman] |
10632 | The real numbers may be introduced by abstraction as ratios of quantities [Hale, by Hale/Wright] |
10107 | Real numbers provide answers to square root problems [George/Velleman] |
9946 | Logicists say mathematics is applicable because it is totally general [George/Velleman] |
10125 | The classical mathematician believes the real numbers form an actual set [George/Velleman] |
17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman] |
10128 | The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman] |
17902 | A successor is the union of a set with its singleton [George/Velleman] |
10133 | Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman] |
10130 | Set theory can prove the Peano Postulates [George/Velleman] |
10089 | Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman] |
10131 | If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman] |
10092 | In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman] |
10094 | The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman] |
10095 | Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman] |
17901 | Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman] |
19295 | Add Hume's principle to logic, to get numbers; arithmetic truths rest on the nature of the numbers [Hale] |
10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman] |
10134 | Much infinite mathematics can still be justified finitely [George/Velleman] |
10123 | The intuitionists are the idealists of mathematics [George/Velleman] |
10124 | Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman] |
19281 | Interesting supervenience must characterise the base quite differently from what supervenes on it [Hale] |
10512 | The abstract/concrete distinction is based on what is perceivable, causal and located [Hale] |
10517 | Colours and points seem to be both concrete and abstract [Hale] |
10519 | The abstract/concrete distinction is in the relations in the identity-criteria of object-names [Hale] |
10520 | Token-letters and token-words are concrete objects, type-letters and type-words abstract [Hale] |
10524 | There is a hierarchy of abstraction, based on steps taken by equivalence relations [Hale] |
19278 | There is no gap between a fact that p, and it is true that p; so we only have the truth-condtions for p [Hale] |
10521 | If F can't have location, there is no problem of things having F in different locations [Hale] |
10511 | It is doubtful if one entity, a universal, can be picked out by both predicates and abstract nouns [Hale] |
10318 | Realists take universals to be the referrents of both adjectives and of nouns [Hale] |
10310 | Objections to Frege: abstracta are unknowable, non-independent, unstatable, unindividuated [Hale] |
10518 | Shapes and directions are of something, but games and musical compositions are not [Hale] |
10513 | Many abstract objects, such as chess, seem non-spatial, but are not atemporal [Hale] |
10514 | If the mental is non-spatial but temporal, then it must be classified as abstract [Hale] |
10523 | Being abstract is based on a relation between things which are spatially separated [Hale] |
10307 | The modern Fregean use of the term 'object' is much broader than the ordinary usage [Hale] |
10315 | We can't believe in a 'whereabouts' because we ask 'what kind of object is it?' [Hale] |
19302 | If a chair could be made of slightly different material, that could lead to big changes [Hale] |
10522 | The relations featured in criteria of identity are always equivalence relations [Hale] |
10321 | We sometimes apply identity without having a real criterion [Hale] |
15086 | Absolute necessity might be achievable either logically or metaphysically [Hale] |
19290 | Absolute necessities are necessarily necessary [Hale] |
8261 | Maybe not-p is logically possible, but p is metaphysically necessary, so the latter is not absolute [Hale] |
15081 | A strong necessity entails a weaker one, but not conversely; possibilities go the other way [Hale] |
15080 | 'Relative' necessity is just a logical consequence of some statements ('strong' if they are all true) [Hale] |
19286 | 'Absolute necessity' is when there is no restriction on the things which necessitate p [Hale] |
19288 | Logical and metaphysical necessities differ in their vocabulary, and their underlying entities [Hale] |
15082 | Metaphysical necessity says there is no possibility of falsehood [Hale] |
15085 | 'Broadly' logical necessities are derived (in a structure) entirely from the concepts [Hale] |
15088 | Logical necessities are true in virtue of the nature of all logical concepts [Hale] |
19285 | Logical necessity is something which is true, no matter what else is the case [Hale] |
19287 | Maybe each type of logic has its own necessity, gradually becoming broader [Hale] |
12432 | Explanation of necessity must rest on something necessary or something contingent [Hale] |
12434 | Why is this necessary, and what is necessity in general; why is this necessary truth true, and why necessary? [Hale] |
12435 | The explanation of a necessity can be by a truth (which may only happen to be a necessary truth) [Hale] |
19282 | It seems that we cannot show that modal facts depend on non-modal facts [Hale] |
12433 | If necessity rests on linguistic conventions, those are contingent, so there is no necessity [Hale] |
15087 | Conceptual necessities are made true by all concepts [Hale] |
12436 | Concept-identities explain how we know necessities, not why they are necessary [Hale] |
19276 | The big challenge for essentialist views of modality is things having necessary existence [Hale] |
19293 | Essentialism doesn't explain necessity reductively; it explains all necessities in terms of a few basic natures [Hale] |
19294 | If necessity derives from essences, how do we explain the necessary existence of essences? [Hale] |
19279 | What are these worlds, that being true in all of them makes something necessary? [Hale] |
19299 | Possible worlds make every proposition true or false, which endorses classical logic [Hale] |
19300 | The molecules may explain the water, but they are not what 'water' means [Hale] |
10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |