80 ideas
11912 | Substantive metaphysics says what a property is, not what a predicate means [Molnar] |
11920 | A real definition gives all the properties that constitute an identity [Molnar] |
8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
8694 | Free logic was developed for fictional or non-existent objects [Friend] |
8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
8672 | A 'powerset' is all the subsets of a set [Friend] |
8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
8666 | Infinite sets correspond one-to-one with a subset [Friend] |
8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
8668 | The 'rational' numbers are those representable as fractions [Friend] |
8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
8706 | Constructivism rejects too much mathematics [Friend] |
8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
11919 | Ontological dependence rests on essential connection, not necessary connection [Molnar] |
11929 | The three categories in ontology are objects, properties and relations [Molnar] |
11927 | Reflexive relations are syntactically polyadic but ontologically monadic [Molnar] |
11915 | If atomism is true, then all properties derive from ultimate properties [Molnar] |
11916 | 'Being physical' is a second-order property [Molnar] |
11956 | 'Categorical properties' are those which are not powers [Molnar] |
11928 | Are tropes transferable? If they are, that is a version of Platonism [Molnar] |
11933 | A power's type-identity is given by its definitive manifestation [Molnar] |
11932 | Powers have Directedness, Independence, Actuality, Intrinsicality and Objectivity [Molnar] |
11934 | The physical world has a feature very like mental intentionality [Molnar] |
11947 | Dispositions and external powers arise entirely from intrinsic powers in objects [Molnar] |
11952 | The Standard Model suggest that particles are entirely dispositional, and hence are powers [Molnar] |
11953 | Some powers are ungrounded, and others rest on them, and are derivative [Molnar] |
11943 | Dispositions can be causes, so they must be part of the actual world [Molnar] |
11939 | If powers only exist when actual, they seem to be nomadic, and indistinguishable from non-powers [Molnar] |
11914 | Platonic explanations of universals actually diminish our understanding [Molnar] |
11913 | For nominalists, predicate extensions are inexplicable facts [Molnar] |
11962 | Nominalists only accept first-order logic [Molnar] |
8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
11917 | Structural properties are derivate properties [Molnar] |
11955 | There are no 'structural properties', as properties with parts [Molnar] |
11918 | The essence of a thing need not include everything that is necessarily true of it [Molnar] |
11963 | What is the truthmaker for a non-existent possible? [Molnar] |
11951 | Hume allows interpolation, even though it and extrapolation are not actually valid [Molnar] |
11936 | The two ways proposed to distinguish mind are intentionality or consciousness [Molnar] |
11935 | Physical powers like solubility and charge also have directedness [Molnar] |
11944 | Rule occasionalism says God's actions follow laws, not miracles [Molnar] |
8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
11960 | Singular causation is prior to general causation; each aspirin produces the aspirin generalization [Molnar] |
11937 | We should analyse causation in terms of powers, not vice versa [Molnar] |
11954 | We should analyse causation in terms of powers [Molnar] |
11961 | Causal dependence explains counterfactual dependence, not vice versa [Molnar] |
11959 | Science works when we assume natural kinds have essences - because it is true [Molnar] |
9448 | Location in space and time are non-power properties [Molnar, by Mumford] |
11930 | One essential property of a muon doesn't entail the others [Molnar] |
11957 | It is contingent which kinds and powers exist in the world [Molnar] |
11921 | The laws of nature depend on the powers, not the other way round [Molnar] |
11931 | Energy fields are discontinuous at the very small [Molnar] |