82 ideas
15477 | Ontology is highly abstract physics, containing placeholders and exclusions [Martin,CB] |
22024 | Fichte's subjectivity struggles to then give any account of objectivity [Pinkard on Fichte] |
9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
15471 | Truth is a relation between a representation ('bearer') and part of the world ('truthmaker') [Martin,CB] |
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] |
10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
22017 | Normativity needs the possibility of negation, in affirmation and denial [Fichte, by Pinkard] |
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] |
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] |
17901 | Type theory prohibits (oddly) a set containing an individual and a set of individuals [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] |
10134 | Much infinite mathematics can still be justified finitely [George/Velleman] |
10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions [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] |
15484 | A property is a combination of a disposition and a quality [Martin,CB] |
15478 | Properties are the respects in which objects resemble, which places them in classes [Martin,CB] |
15483 | Properties are ways particular things are, and so they are tied to the identity of their possessor [Martin,CB] |
15480 | Objects are not bundles of tropes (which are ways things are, not parts of things) [Martin,CB] |
15489 | A property that cannot interact is worse than inert - it isn't there at all [Martin,CB] |
15487 | If unmanifested partnerless dispositions are still real, and are not just qualities, they can explain properties [Martin,CB] |
15479 | Properties endow a ball with qualities, and with powers or dispositions [Martin,CB] |
15488 | Qualities and dispositions are aspects of properties - what it exhibits, and what it does [Martin,CB] |
15469 | Dispositions in action can be destroyed, be recovered, or remain unchanged [Martin,CB] |
15467 | Powers depend on circumstances, so can't be given a conditional analysis [Martin,CB] |
15466 | 'The wire is live' can't be analysed as a conditional, because a wire can change its powers [Martin,CB] |
15465 | Structures don't explain dispositions, because they consist of dispositions [Martin,CB] |
15476 | Structural properties involve dispositionality, so cannot be used to explain it [Martin,CB] |
15481 | I favour the idea of a substratum for properties; spacetime seems to be just a bearer of properties [Martin,CB] |
15474 | Properly understood, wholes do no more causal work than their parts [Martin,CB] |
15486 | Only abstract things can have specific and full identity specifications [Martin,CB] |
15475 | The concept of 'identity' must allow for some changes in properties or parts [Martin,CB] |
22018 | Necessary truths derive from basic assertion and negation [Fichte, by Pinkard] |
15472 | It is pointless to say possible worlds are truthmakers, and then deny that possible worlds exist [Martin,CB] |
22064 | Fichte's logic is much too narrow, and doesn't deduce ethics, art, society or life [Schlegel,F on Fichte] |
22032 | Fichte's key claim was that the subjective-objective distinction must itself be subjective [Fichte, by Pinkard] |
15492 | Explanations are mind-dependent, theory-laden, and interest-relative [Martin,CB] |
22020 | We only see ourselves as self-conscious and rational in relation to other rationalities [Fichte] |
15495 | Analogy works, as when we eat food which others seem to be relishing [Martin,CB] |
15493 | Memory requires abstraction, as reminders of what cannot be fully remembered [Martin,CB] |
22060 | The Self is the spontaneity, self-relatedness and unity needed for knowledge [Fichte, by Siep] |
22066 | Novalis sought a much wider concept of the ego than Fichte's proposal [Novalis on Fichte] |
22016 | The self is not a 'thing', but what emerges from an assertion of normativity [Fichte, by Pinkard] |
22019 | Consciousness of an object always entails awareness of the self [Fichte] |
22061 | Judgement is distinguishing concepts, and seeing their relations [Fichte, by Siep] |
10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
22023 | Fichte's idea of spontaneity implied that nothing counts unless we give it status [Fichte, by Pinkard] |
22065 | Fichte reduces nature to a lifeless immobility [Schlegel,F on Fichte] |
15485 | Instead of a cause followed by an effect, we have dispositions in reciprocal manifestation [Martin,CB] |
15491 | Causation should be explained in terms of dispositions and manifestations [Martin,CB] |
15468 | Causal counterfactuals are just clumsy linguistic attempts to indicate dispositions [Martin,CB] |
15470 | Causal laws are summaries of powers [Martin,CB] |
15482 | We can't think of space-time as empty and propertyless, and it seems to be a substratum [Martin,CB] |