87 ideas
| 5196 | Philosophy is a department of logic [Ayer] |
| 5189 | Philosophers should abandon speculation, as philosophy is wholly critical [Ayer] |
| 7919 | Humeans rejected the a priori synthetic, and so rejected even Kantian metaphysics [Ayer, by Macdonald,C] |
| 5195 | Critics say analysis can only show the parts, and not their distinctive configuration [Ayer] |
| 5179 | Philosophy deals with the questions that scientists do not wish to handle [Ayer] |
| 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] |
| 4749 | We cannot analyse the concept of 'truth', because it is simply a mark that a sentence is asserted [Ayer] |
| 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] |
| 5202 | Maths and logic are true universally because they are analytic or tautological [Ayer] |
| 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] |
| 6523 | Positivists regard ontology as either meaningless or stipulated [Ayer, by Robinson,H] |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| 5183 | Only tautologies can be certain; other propositions can only be probable [Ayer] |
| 6525 | Logical positivists could never give the sense-data equivalent of 'there is a table next door' [Robinson,H on Ayer] |
| 5170 | Material things are constructions from actual and possible occurrences of sense-contents [Ayer] |
| 5198 | We could verify 'a thing can't be in two places at once' by destroying one of the things [Ierubino on Ayer] |
| 2619 | Whether geometry can be applied to reality is an empirical question outside of geometry [Ayer] |
| 5197 | By changing definitions we could make 'a thing can't be in two places at once' a contradiction [Ayer] |
| 5204 | To say that a proposition is true a priori is to say that it is a tautology [Ayer] |
| 6524 | Positivists prefer sense-data to objects, because the vocabulary covers both illusions and perceptions [Ayer, by Robinson,H] |
| 5193 | Causal and representative theories of perception are wrong as they refer to unobservables [Ayer] |
| 5200 | The main claim of rationalism is that thought is an independent source of knowledge [Ayer] |
| 4729 | Empiricism lacked a decent account of the a priori, until Ayer said it was entirely analytic [O'Grady on Ayer] |
| 5180 | All propositions (especially 'metaphysics') must begin with the senses [Ayer] |
| 5169 | My empiricism logically distinguishes analytic and synthetic propositions, and metaphysical verbiage [Ayer] |
| 5185 | It is further sense-experience which informs us of the mistakes that arise out of sense-experience [Ayer] |
| 5199 | Empiricism, it is said, cannot account for our knowledge of necessary truths [Ayer] |
| 5190 | The induction problem is to prove generalisations about the future based on the past [Ayer] |
| 5191 | We can't use the uniformity of nature to prove induction, as that would be circular [Ayer] |
| 5177 | Other minds are 'metaphysical' objects, because I can never observe their experiences [Ayer] |
| 5178 | A conscious object is by definition one that behaves in a certain way, so behaviour proves consciousness [Ayer] |
| 5172 | If the self is meaningful, it must be constructed from sense-experiences [Ayer] |
| 5173 | Two experiences belong to one self if their contents belong with one body [Ayer] |
| 5176 | Empiricists can define personal identity as bodily identity, which consists of sense-contents [Ayer] |
| 5171 | The supposed 'gulf' between mind and matter is based on the senseless concept of 'substances' [Ayer] |
| 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] |
| 5181 | A sentence is factually significant to someone if they know how to verify its proposition [Ayer] |
| 5184 | Factual propositions imply (in conjunction with a few other premises) possible experiences [Ayer] |
| 5186 | Tautologies and empirical hypotheses form the entire class of significant propositions [Ayer] |
| 5205 | Moral intuition is worthless if there is no criterion to decide between intuitions [Ayer] |
| 23725 | Ayer defends the emotivist version of expressivism [Ayer, by Smith,M] |
| 5206 | To say an act is wrong makes no further statement about it, but merely expresses disapproval [Ayer] |
| 6613 | The natural kinds are objects, processes and properties/relations [Ellis] |
| 6616 | Least action is not a causal law, but a 'global law', describing a global essence [Ellis] |
| 6615 | A species requires a genus, and its essence includes the essence of the genus [Ellis] |
| 6614 | A hierarchy of natural kinds is elaborate ontology, but needed to explain natural laws [Ellis] |
| 6612 | Without general principles, we couldn't predict the behaviour of dispositional properties [Ellis] |
| 5208 | A person with non-empirical attributes is unintelligible. [Ayer] |
| 5187 | When we ascribe an attribute to a thing, we covertly assert that it exists [Ayer] |
| 5207 | If theism is non-sensical, then so is atheism. [Ayer] |
| 5209 | The 'truths' expressed by theists are not literally significant [Ayer] |