86 ideas
18005 | Philosophy aims to become more disciplined about categories [Ryle] |
18004 | We can't do philosophy without knowledge of types and categories [Ryle] |
13985 | A true proposition seems true of one fact, but a false proposition seems true of nothing at all. [Ryle] |
13984 | Two maps might correspond to one another, but they are only 'true' of the country they show [Ryle] |
18194 | 'Forcing' can produce new models of ZFC from old models [Maddy] |
18195 | A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy [Maddy] |
13011 | New axioms are being sought, to determine the size of the continuum [Maddy] |
13014 | Extensional sets are clearer, simpler, unique and expressive [Maddy] |
13013 | The Axiom of Extensionality seems to be analytic [Maddy] |
13021 | The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics [Maddy] |
13022 | Infinite sets are essential for giving an account of the real numbers [Maddy] |
18191 | Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy] |
13023 | The Power Set Axiom is needed for, and supported by, accounts of the continuum [Maddy] |
18193 | The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy] |
13024 | Efforts to prove the Axiom of Choice have failed [Maddy] |
13025 | Modern views say the Choice set exists, even if it can't be constructed [Maddy] |
13026 | A large array of theorems depend on the Axiom of Choice [Maddy] |
17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
18169 | Axiom of Reducibility: propositional functions are extensionally predicative [Maddy] |
13019 | The Iterative Conception says everything appears at a stage, derived from the preceding appearances [Maddy] |
13018 | Limitation of Size is a vague intuition that over-large sets may generate paradoxes [Maddy] |
17824 | The master science is physical objects divided into sets [Maddy] |
8755 | Maddy replaces pure sets with just objects and perceived sets of objects [Maddy, by Shapiro] |
13979 | Logic studies consequence, compatibility, contradiction, corroboration, necessitation, grounding.... [Ryle] |
10594 | Henkin semantics is more plausible for plural logic than for second-order logic [Maddy] |
17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
18168 | 'Propositional functions' are propositions with a variable as subject or predicate [Maddy] |
10800 | The values of variables can't determine existence, because they are just expressions [Ryle, by Quine] |
17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
18171 | Cantor and Dedekind brought completed infinities into mathematics [Maddy] |
18190 | Completed infinities resulted from giving foundations to calculus [Maddy] |
17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
18175 | For any cardinal there is always a larger one (so there is no set of all sets) [Maddy] |
18196 | An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy] |
18172 | Infinity has degrees, and large cardinals are the heart of set theory [Maddy] |
18187 | Theorems about limits could only be proved once the real numbers were understood [Maddy] |
18182 | The extension of concepts is not important to me [Maddy] |
18177 | In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy] |
18164 | Frege solves the Caesar problem by explicitly defining each number [Maddy] |
18163 | Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy] |
17825 | Set theory (unlike the Peano postulates) can explain why multiplication is commutative [Maddy] |
17826 | Standardly, numbers are said to be sets, which is neat ontology and epistemology [Maddy] |
17828 | Numbers are properties of sets, just as lengths are properties of physical objects [Maddy] |
10718 | A natural number is a property of sets [Maddy, by Oliver] |
18185 | Unified set theory gives a final court of appeal for mathematics [Maddy] |
18183 | Set theory brings mathematics into one arena, where interrelations become clearer [Maddy] |
18186 | Identifying geometric points with real numbers revealed the power of set theory [Maddy] |
18184 | Making set theory foundational to mathematics leads to very fruitful axioms [Maddy] |
18188 | The line of rationals has gaps, but set theory provided an ordered continuum [Maddy] |
17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
17827 | Sets exist where their elements are, but numbers are more like universals [Maddy] |
17830 | Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy] |
17823 | If mathematical objects exist, how can we know them, and which objects are they? [Maddy] |
8756 | Intuition doesn't support much mathematics, and we should question its reliability [Maddy, by Shapiro] |
17733 | We know mind-independent mathematical truths through sets, which rest on experience [Maddy, by Jenkins] |
18204 | Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy] |
18207 | Maybe applications of continuum mathematics are all idealisations [Maddy] |
17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
17829 | Number words are unusual as adjectives; we don't say 'is five', and numbers always come first [Maddy] |
18167 | We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy] |
13988 | Many sentences do not state facts, but there are no facts which could not be stated [Ryle] |
18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
14297 | A dispositional property is not a state, but a liability to be in some state, given a condition [Ryle] |
14300 | No physical scientist now believes in an occult force-exerting agency [Ryle] |
13983 | Representation assumes you know the ideas, and the reality, and the relation between the two [Ryle] |
6455 | Maybe 'sense-data' just help us to talk about unusual perceptual situations [Lacey] |
6454 | Where do sense-data begin or end? Can they change? What sort of thing are they? [Lacey] |
6453 | Some claim sense-data are public, and are parts of objects [Lacey] |
2622 | Can one movement have a mental and physical cause? [Ryle] |
18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
1354 | We cannot introspect states of anger or panic [Ryle] |
1353 | Reporting on myself has the same problems as reporting on you [Ryle] |
2624 | I cannot prepare myself for the next thought I am going to think [Ryle] |
2620 | Dualism is a category mistake [Ryle] |
2388 | Behaviour depends on desires as well as beliefs [Chalmers on Ryle] |
3354 | You can't explain mind as dispositions, if they aren't real [Benardete,JA on Ryle] |
2387 | How can behaviour be the cause of behaviour? [Chalmers on Ryle] |
13980 | If you like judgments and reject propositions, what are the relata of incoherence in a judgment? [Ryle] |
13978 | Husserl and Meinong wanted objective Meanings and Propositions, as subject-matter for Logic [Ryle] |
13977 | When I utter a sentence, listeners grasp both my meaning and my state of mind [Ryle] |
13976 | 'Propositions' name what is thought, because 'thoughts' and 'judgments' are too ambiguous [Ryle] |
13981 | Several people can believe one thing, or make the same mistake, or share one delusion [Ryle] |
13987 | We may think in French, but we don't know or believe in French [Ryle] |
13989 | There are no propositions; they are just sentences, used for thinking, which link to facts in a certain way [Ryle] |
13982 | If we accept true propositions, it is hard to reject false ones, and even nonsensical ones [Ryle] |