69 ideas
4456 | Epistemological Ockham's Razor demands good reasons, but the ontological version says reality is simple [Moreland] |
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] |
4474 | Existence theories must match experience, possibility, logic and knowledge, and not be self-defeating [Moreland] |
4461 | Tropes are like Hume's 'impressions', conceived as real rather than as ideal [Moreland] |
4462 | A colour-trope cannot be simple (as required), because it is spread in space, and so it is complex [Moreland] |
4463 | In 'four colours were used in the decoration', colours appear to be universals, not tropes [Moreland] |
4453 | One realism is one-over-many, which may be the model/copy view, which has the Third Man problem [Moreland] |
4464 | Realists see properties as universals, which are single abstract entities which are multiply exemplifiable [Moreland] |
4451 | If properties are universals, what distinguishes two things which have identical properties? [Moreland] |
4450 | The traditional problem of universals centres on the "One over Many", which is the unity of natural classes [Moreland] |
4449 | Evidence for universals can be found in language, communication, natural laws, classification and ideals [Moreland] |
4454 | The One-In-Many view says universals have abstract existence, but exist in particulars [Moreland] |
4468 | How could 'being even', or 'being a father', or a musical interval, exist naturally in space? [Moreland] |
4452 | Maybe universals are real, if properties themselves have properties, and relate to other properties [Moreland] |
4467 | A naturalist and realist about universals is forced to say redness can be both moving and stationary [Moreland] |
4469 | There are spatial facts about red particulars, but not about redness itself [Moreland] |
4472 | Redness is independent of red things, can do without them, has its own properties, and has identity [Moreland] |
4459 | Moderate nominalism attempts to embrace the existence of properties while avoiding universals [Moreland] |
4458 | Unlike Class Nominalism, Resemblance Nominalism can distinguish natural from unnatural classes [Moreland] |
4457 | There can be predicates with no property, and there are properties with no predicate [Moreland] |
4471 | We should abandon the concept of a property since (unlike sets) their identity conditions are unclear [Moreland] |
8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
4476 | Most philosophers think that the identity of indiscernibles is false [Moreland] |
4460 | Abstractions are formed by the mind when it concentrates on some, but not all, the features of a thing [Moreland] |
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] |
4455 | It is always open to a philosopher to claim that some entity or other is unanalysable [Moreland] |
23601 | Leaders can only attack foreigners who have done wrong (as with their own subjects) [Vitoria] |
23569 | The only just cause for a war is a wrong received [Vitoria] |
23566 | Princes should not justify a war to their subjects, and doing so would undermine the state [Vitoria] |
23576 | Sacking a city is lawful if it motivates the attacking troops [Vitoria] |
4473 | 'Presentism' is the view that only the present moment exists [Moreland] |