73 ideas
| 2463 | A standard naturalist view is realist, externalist, and computationalist, and believes in rationality [Fodor] |
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 2435 | Psychology has to include the idea that mental processes are typically truth-preserving [Fodor] |
| 9987 | An aggregate in which order does not matter I call a 'set' [Bolzano] |
| 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 |
| 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] |
| 2442 | Inferences are surely part of the causal structure of the world [Fodor] |
| 10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
| 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] |
| 10107 | Real numbers provide answers to square root problems [George/Velleman] |
| 9946 | Logicists say mathematics is applicable because it is totally general [George/Velleman] |
| 10856 | A truly infinite quantity does not need to be a variable [Bolzano] |
| 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] |
| 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] |
| 2462 | Control of belief is possible if you know truth conditions and what causes beliefs [Fodor] |
| 2461 | An experiment is a deliberate version of what informal thinking does all the time [Fodor] |
| 2454 | We can deliberately cause ourselves to have true thoughts - hence the value of experiments [Fodor] |
| 2455 | Interrogation and experiment submit us to having beliefs caused [Fodor] |
| 2460 | Participation in an experiment requires agreement about what the outcome will mean [Fodor] |
| 2458 | Theories are links in the causal chain between the environment and our beliefs [Fodor] |
| 2443 | I say psychology is intentional, semantics is informational, and thinking is computation [Fodor] |
| 2453 | We are probably the only creatures that can think about our own thoughts [Fodor] |
| 2446 | Cartesians consider interaction to be a miracle [Fodor] |
| 2445 | Semantics v syntax is the interaction problem all over again [Fodor] |
| 2464 | Type physicalism equates mental kinds with physical kinds [Fodor] |
| 2447 | Hume has no theory of the co-ordination of the mind [Fodor] |
| 2440 | Propositional attitudes are propositions presented in a certain way [Fodor] |
| 2450 | Rationality has mental properties - autonomy, productivity, experiment [Fodor] |
| 2437 | XYZ (Twin Earth 'water') is an impossibility [Fodor] |
| 2441 | Truth conditions require a broad concept of content [Fodor] |
| 3114 | Concepts aren't linked to stuff; they are what is caused by stuff [Fodor] |
| 2452 | Knowing the cause of a thought is almost knowing its content [Fodor] |
| 2432 | Is content basically information, fixed externally? [Fodor] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 2438 | In the information view, concepts are potentials for making distinctions [Fodor] |
| 2439 | Semantic externalism says the concept 'elm' needs no further beliefs or inferences [Fodor] |
| 2457 | If meaning is information, that establishes the causal link between the state of the world and our beliefs [Fodor] |
| 2451 | To know the content of a thought is to know what would make it true [Fodor] |
| 2433 | For holists no two thoughts are ever quite the same, which destroys faith in meaning [Fodor] |
| 2436 | It is claimed that reference doesn't fix sense (Jocasta), and sense doesn't fix reference (Twin Earth) [Fodor] |
| 2434 | Broad semantics holds that the basic semantic properties are truth and denotation [Fodor] |
| 2459 | Externalist semantics are necessary to connect the contents of beliefs with how the world is [Fodor] |