129 ideas
| 9408 | Science studies phenomena, but only metaphysics tells us what exists [Mumford] |
| 9429 | Many forms of reasoning, such as extrapolation and analogy, are useful but deductively invalid [Mumford] |
| 6627 | Radical pragmatists abandon the notion of truth [Stich, by Lowe] |
| 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] |
| 18191 | Axiom of Infinity: completed infinite collections can be treated mathematically [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] |
| 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] |
| 9427 | For Humeans the world is a world primarily of events [Mumford] |
| 14334 | Modest realism says there is a reality; the presumptuous view says we can accurately describe it [Mumford] |
| 14306 | Anti-realists deny truth-values to all statements, and say evidence and ontology are inseparable [Mumford] |
| 18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
| 14333 | Dispositions and categorical properties are two modes of presentation of the same thing [Mumford] |
| 14336 | Categorical predicates are those unconnected to functions [Mumford] |
| 14315 | Categorical properties and dispositions appear to explain one another [Mumford] |
| 14332 | There are four reasons for seeing categorical properties as the most fundamental [Mumford] |
| 14302 | A lead molecule is not leaden, and macroscopic properties need not be microscopically present [Mumford] |
| 14294 | Dispositions are attacked as mere regularities of events, or place-holders for unknown properties [Mumford] |
| 9446 | Properties are just natural clusters of powers [Mumford] |
| 14316 | If dispositions have several categorical realisations, that makes the two separate [Mumford] |
| 14310 | Dispositions are classifications of properties by functional role [Mumford] |
| 14317 | I say the categorical base causes the disposition manifestation [Mumford] |
| 14313 | All properties must be causal powers (since they wouldn't exist otherwise) [Mumford] |
| 14318 | Intrinsic properties are just causal powers, and identifying a property as causal is then analytic [Mumford] |
| 14293 | Dispositions are ascribed to at least objects, substances and persons [Mumford] |
| 14326 | Unlike categorical bases, dispositions necessarily occupy a particular causal role [Mumford] |
| 14298 | Dispositions can be contrasted either with occurrences, or with categorical properties [Mumford] |
| 14314 | If dispositions are powers, background conditions makes it hard to say what they do [Mumford] |
| 14325 | Maybe dispositions can replace powers in metaphysics, as what induces property change [Mumford] |
| 14312 | Orthodoxy says dispositions entail conditionals (rather than being equivalent to them) [Mumford] |
| 14291 | Dispositions are not just possibilities - they are features of actual things [Mumford] |
| 14299 | There could be dispositions that are never manifested [Mumford] |
| 14323 | If every event has a cause, it is easy to invent a power to explain each case [Mumford] |
| 14328 | Traditional powers initiate change, but are mysterious between those changes [Mumford] |
| 14331 | Categorical eliminativists say there are no dispositions, just categorical states or mechanisms [Mumford] |
| 9435 | A 'porridge' nominalist thinks we just divide reality in any way that suits us [Mumford] |
| 9447 | If properties are clusters of powers, this can explain why properties resemble in degrees [Mumford] |
| 18617 | Substances, unlike aggregates, can survive a change of parts [Mumford] |
| 14295 | Many artefacts have dispositional essences, which make them what they are [Mumford] |
| 12248 | How can we show that a universally possessed property is an essential property? [Mumford] |
| 18618 | Maybe possibilities are recombinations of the existing elements of reality [Mumford] |
| 18619 | Combinatorial possibility has to allow all elements to be combinable, which seems unlikely [Mumford] |
| 18620 | Combinatorial possibility relies on what actually exists (even over time), but there could be more [Mumford] |
| 14309 | Truth-functional conditionals can't distinguish whether they are causal or accidental [Mumford] |
| 14311 | Dispositions are not equivalent to stronger-than-material conditionals [Mumford] |
| 14319 | Nomothetic explanations cite laws, and structural explanations cite mechanisms [Mumford] |
| 14342 | General laws depend upon the capacities of particulars, not the other way around [Mumford] |
| 14322 | If fragile just means 'breaks when dropped', it won't explain a breakage [Mumford] |
| 14337 | Maybe dispositions can replace the 'laws of nature' as the basis of explanation [Mumford] |
| 14343 | To avoid a regress in explanations, ungrounded dispositions will always have to be posited [Mumford] |
| 14320 | Subatomic particles may terminate explanation, if they lack structure [Mumford] |
| 14324 | Ontology is unrelated to explanation, which concerns modes of presentation and states of knowledge [Mumford] |
| 18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
| 4765 | Stich accepts eliminativism (labelled 'pragmatism') about rationality and normativity [Stich, by Engel] |
| 14344 | Natural kinds, such as electrons, all behave the same way because we divide them by dispositions [Mumford] |
| 19068 | Causation interests us because we want to explain change [Mumford] |
| 9430 | Singular causes, and identities, might be necessary without falling under a law [Mumford] |
| 9445 | We can give up the counterfactual account if we take causal language at face value [Mumford] |
| 9443 | It is only properties which are the source of necessity in the world [Mumford] |
| 14338 | In the 'laws' view events are basic, and properties are categorical, only existing when manifested [Mumford] |
| 9444 | There are four candidates for the logical form of law statements [Mumford] |
| 14339 | Without laws, how can a dispositionalist explain general behaviour within kinds? [Mumford] |
| 14341 | Dretske and Armstrong base laws on regularities between individual properties, not between events [Mumford] |
| 9431 | Pure regularities are rare, usually only found in idealized conditions [Mumford] |
| 9441 | Regularity laws don't explain, because they have no governing role [Mumford] |
| 14340 | It is a regularity that whenever a person sneezes, someone (somewhere) promptly coughs [Mumford] |
| 9416 | Regularities are more likely with few instances, and guaranteed with no instances! [Mumford] |
| 9415 | Would it count as a regularity if the only five As were also B? [Mumford] |
| 9422 | If the best system describes a nomological system, the laws are in nature, not in the description [Mumford] |
| 9421 | The best systems theory says regularities derive from laws, rather than constituting them [Mumford] |
| 9432 | Laws of nature are necessary relations between universal properties, rather than about particulars [Mumford] |
| 9433 | If laws can be uninstantiated, this favours the view of them as connecting universals [Mumford] |
| 14345 | The necessity of an electron being an electron is conceptual, and won't ground necessary laws [Mumford] |
| 9434 | Laws of nature are just the possession of essential properties by natural kinds [Mumford] |
| 14307 | Some dispositions are so far unknown, until we learn how to manifest them [Mumford] |
| 9437 | To distinguish accidental from essential properties, we must include possible members of kinds [Mumford] |
| 9439 | The Central Dilemma is how to explain an internal or external view of laws which govern [Mumford] |
| 9412 | You only need laws if you (erroneously) think the world is otherwise inert [Mumford] |
| 9411 | There are no laws of nature in Aristotle; they became standard with Descartes and Newton [Mumford] |