83 ideas
18241 | Sufficient reason is implied by contradiction, of an insufficient possible which exists [Wolff, by Korsgaard] |
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] |
13013 | The Axiom of Extensionality seems to be analytic [Maddy] |
13014 | Extensional sets are clearer, simpler, unique and expressive [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] |
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] |
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] |
18190 | Completed infinities resulted from giving foundations to calculus [Maddy] |
18171 | Cantor and Dedekind brought completed infinities into mathematics [Maddy] |
17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
18172 | Infinity has degrees, and large cardinals are the heart of set theory [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] |
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] |
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] |
18184 | Making set theory foundational to mathematics leads to very fruitful axioms [Maddy] |
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] |
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] |
18163 | Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy] |
17830 | Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy] |
17827 | Sets exist where their elements are, but numbers are more like universals [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] |
18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
17979 | Research shows perceptual discrimination is sharper at category boundaries [Murphy] |
18690 | Induction is said to just compare properties of categories, but the type of property also matters [Murphy] |
18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
17980 | The main theories of concepts are exemplar, prototype and knowledge [Murphy] |
17973 | The theoretical and practical definitions for the classical view are very hard to find [Murphy] |
17969 | The classical definitional approach cannot distinguish typical and atypical category members [Murphy] |
17970 | Classical concepts follow classical logic, but concepts in real life don't work that way [Murphy] |
17971 | Classical concepts are transitive hierarchies, but actual categories may be intransitive [Murphy] |
17972 | The classical core is meant to be the real concept, but actually seems unimportant [Murphy] |
17975 | There is no 'ideal' bird or dog, and prototypes give no information about variability [Murphy] |
17976 | Prototypes are unified representations of the entire category (rather than of members) [Murphy] |
18691 | The prototype theory uses observed features, but can't include their construction [Murphy] |
17983 | The prototype theory handles hierarchical categories and combinations of concepts well [Murphy] |
17985 | Prototypes theory of concepts is best, as a full description with weighted typical features [Murphy] |
17986 | Learning concepts is forming prototypes with a knowledge structure [Murphy] |
17974 | The most popular theories of concepts are based on prototypes or exemplars [Murphy] |
17977 | The exemplar view of concepts says 'dogs' is the set of dogs I remember [Murphy] |
17982 | Exemplar theory struggles with hierarchical classification and with induction [Murphy] |
17981 | Children using knowing and essentialist categories doesn't fit the exemplar view [Murphy] |
17984 | Conceptual combination must be compositional, and can't be built up from exemplars [Murphy] |
17987 | The concept of birds from exemplars must also be used in inductions about birds [Murphy] |
17978 | We do not learn concepts in isolation, but as an integrated part of broader knowledge [Murphy] |
18687 | Concepts with familiar contents are easier to learn [Murphy] |
18688 | Some knowledge is involved in instant use of categories, other knowledge in explanations [Murphy] |
18689 | People categorise things consistent with their knowledge, even rejecting some good evidence [Murphy] |
18242 | Confucius shows that ethics can rest on reason, rather than on revelation [Wolff, by Korsgaard] |