88 ideas
| 17651 | Without words or other symbols, we have no world [Goodman] |
| 17652 | Truth is irrelevant if no statements are involved [Goodman] |
| 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] |
| 15510 | Classes are a host of ethereal, platonic, pseudo entities [Goodman] |
| 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] |
| 9920 | Two objects can apparently make up quite distinct arrangements in sets [Goodman, by Burgess/Rosen] |
| 10657 | The counties of Utah, and the state, and its acres, are in no way different [Goodman] |
| 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] |
| 12394 | If the result is bad, we change the rule; if we like the rule, we reject the result [Goodman] |
| 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] |
| 17656 | Being primitive or prior always depends on a constructional system [Goodman] |
| 17661 | We don't recognise patterns - we invent them [Goodman] |
| 17659 | Reality is largely a matter of habit [Goodman] |
| 17657 | We build our world, and ignore anything that won't fit [Goodman] |
| 18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
| 17654 | A world can be full of variety or not, depending on how we sort it [Goodman] |
| 14292 | Dispositions seem more ethereal than behaviour; a non-occult account of them would be nice [Goodman] |
| 7956 | If all and only red things were round things, we would need to specify the 'respect' of the resemblance [Goodman, by Macdonald,C] |
| 7957 | Without respects of resemblance, we would collect blue book, blue pen, red pen, red clock together [Goodman, by Macdonald,C] |
| 7952 | If we apply the same word to different things, it is only because we are willing to do so [Goodman, by Macdonald,C] |
| 17653 | Things can only be judged the 'same' by citing some respect of sameness [Goodman] |
| 12191 | Counterfactuals are true if logical or natural laws imply the consequence [Goodman, by McFetridge] |
| 17660 | Discovery is often just finding a fit, like a jigsaw puzzle [Goodman] |
| 17658 | Users of digital thermometers recognise no temperatures in the gaps [Goodman] |
| 17650 | We lack frames of reference to transform physics, biology and psychology into one another [Goodman] |
| 18749 | Goodman argued that the confirmation relation can never be formalised [Goodman, by Horsten/Pettigrew] |
| 17646 | Goodman showed that every sound inductive argument has an unsound one of the same form [Goodman, by Putnam] |
| 17655 | Grue and green won't be in the same world, as that would block induction entirely [Goodman] |
| 18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
| 20440 | Art is a referential activity, hence indefinable, but it has a set of symptoms [Goodman] |
| 20439 | Artistic symbols are judged by the fruitfulness of their classifications [Goodman, by Giovannelli] |
| 8113 | Art is like understanding a natural language, and needs a grasp of a symbol system [Goodman, by Gardner] |
| 20438 | A performance is only an instance of a work if there is not a single error [Goodman] |
| 20437 | A copy only becomes an 'instance' of an artwork if there is a system of notation [Goodman] |
| 7258 | The forefather of modern intuitionism is Richard Price [Price,R, by Dancy,J] |
| 17649 | If the world is one it has many aspects, and if there are many worlds they will collect into one [Goodman] |
| 4794 | We don't use laws to make predictions, we call things laws if we make predictions with them [Goodman] |