71 ideas
10468 | A metaphysics has an ontology (objects) and an ideology (expressed ideas about them) [Oliver] |
10471 | Ockham's Razor has more content if it says believe only in what is causal [Oliver] |
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] |
10749 | Necessary truths seem to all have the same truth-maker [Oliver] |
10750 | Slingshot Argument: seems to prove that all sentences have the same truth-maker [Oliver] |
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] |
10747 | Accepting properties by ontological commitment tells you very little about them [Oliver] |
10748 | Reference is not the only way for a predicate to have ontological commitment [Oliver] |
10721 | If properties are sui generis, are they abstract or concrete? [Oliver] |
10719 | There are four conditions defining the relations between particulars and properties [Oliver] |
10716 | There are just as many properties as the laws require [Oliver] |
10720 | We have four options, depending whether particulars and properties are sui generis or constructions [Oliver] |
10714 | The expressions with properties as their meanings are predicates and abstract singular terms [Oliver] |
10715 | There are five main semantic theories for properties [Oliver] |
10739 | The property of redness is the maximal set of the tropes of exactly similar redness [Oliver] |
10738 | Tropes are not properties, since they can't be instantiated twice [Oliver] |
10740 | The orthodox view does not allow for uninstantiated tropes [Oliver] |
10741 | Maybe concrete particulars are mereological wholes of abstract particulars [Oliver] |
10742 | Tropes can overlap, and shouldn't be splittable into parts [Oliver] |
10472 | 'Structural universals' methane and butane are made of the same universals, carbon and hydrogen [Oliver] |
10724 | Located universals are wholly present in many places, and two can be in the same place [Oliver] |
10730 | If universals ground similarities, what about uniquely instantiated universals? [Oliver] |
7963 | Aristotle's instantiated universals cannot account for properties of abstract objects [Oliver] |
7962 | Uninstantiated properties are useful in philosophy [Oliver] |
10727 | Uninstantiated universals seem to exist if they themselves have properties [Oliver] |
10722 | Instantiation is set-membership [Oliver] |
10744 | Nominalism can reject abstractions, or universals, or sets [Oliver] |
8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
10726 | Things can't be fusions of universals, because two things could then be one thing [Oliver] |
10725 | Abstract sets of universals can't be bundled to make concrete things [Oliver] |
10745 | Science is modally committed, to disposition, causation and law [Oliver] |
3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
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] |
10746 | Conceptual priority is barely intelligible [Oliver] |