121 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] |
17892 | For clear questions posed by reason, reason can also find clear answers [Gödel] |
10041 | Impredicative Definitions refer to the totality to which the object itself belongs [Gödel] |
21752 | Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine] |
17751 | Gödel proved the completeness of first order predicate logic in 1930 [Gödel, by Walicki] |
8679 | We perceive the objects of set theory, just as we perceive with our senses [Gödel] |
17835 | Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M] |
9942 | Gödel proved the classical relative consistency of the axiom V = L [Gödel, by Putnam] |
21716 | In simple type theory the axiom of Separation is better than Reducibility [Gödel, by Linsky,B] |
9188 | Gödel proved that first-order logic is complete, and second-order logic incomplete [Gödel, by Dummett] |
10035 | Mathematical Logic is a non-numerical branch of mathematics, and the supreme science [Gödel] |
10042 | Reference to a totality need not refer to a conjunction of all its elements [Gödel] |
10620 | Originally truth was viewed with total suspicion, and only demonstrability was accepted [Gödel] |
17886 | The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner] |
10071 | Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P] |
19123 | If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh] |
17883 | Gödel's Theorems did not refute the claim that all good mathematical questions have answers [Gödel, by Koellner] |
10621 | Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel] |
17888 | The undecidable sentence can be decided at a 'higher' level in the system [Gödel] |
10038 | A logical system needs a syntactical survey of all possible expressions [Gödel] |
18062 | Set-theory paradoxes are no worse than sense deception in physics [Gödel] |
10132 | There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman] |
10046 | The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers [Gödel] |
10868 | The Continuum Hypothesis is not inconsistent with the axioms of set theory [Gödel, by Clegg] |
13517 | If set theory is consistent, we cannot refute or prove the Continuum Hypothesis [Gödel, by Hart,WD] |
17885 | Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable [Gödel, by Koellner] |
10614 | The real reason for Incompleteness in arithmetic is inability to define truth in a language [Gödel] |
3198 | Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey] |
10072 | First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P] |
9590 | Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman] |
11069 | Gödel's Second says that semantic consequence outruns provability [Gödel, by Hanna] |
10118 | First Incompleteness: a decent consistent system is syntactically incomplete [Gödel, by George/Velleman] |
10122 | Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by George/Velleman] |
10611 | There is a sentence which a theory can show is true iff it is unprovable [Gödel, by Smith,P] |
10867 | 'This system can't prove this statement' makes it unprovable either way [Gödel, by Clegg] |
10039 | Some arithmetical problems require assumptions which transcend arithmetic [Gödel] |
10043 | Mathematical objects are as essential as physical objects are for perception [Gödel] |
10271 | Basic mathematics is related to abstract elements of our empirical ideas [Gödel] |
10045 | Impredicative definitions are admitted into ordinary mathematics [Gödel] |
8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro] |
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] |
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] |
9446 | Properties are just natural clusters of powers [Mumford] |
14294 | Dispositions are attacked as mere regularities of events, or place-holders for unknown properties [Mumford] |
14310 | Dispositions are classifications of properties by functional role [Mumford] |
14317 | I say the categorical base causes the disposition manifestation [Mumford] |
14316 | If dispositions have several categorical realisations, that makes the two separate [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] |
17472 | Thick mechanisms map whole reactions, and thin mechanism chart the steps [Weisberg/Needham/Hendry] |
17471 | Using mechanisms as explanatory schemes began in chemistry [Weisberg/Needham/Hendry] |
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] |
3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
17465 | Lavoisier's elements included four types of earth [Weisberg/Needham/Hendry] |
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] |
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] |
9431 | Pure regularities are rare, usually only found in idealized conditions [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] |
17468 | Over 100,000,000 compounds have been discovered or synthesised [Weisberg/Needham/Hendry] |
17470 | Water molecules dissociate, and form large polymers, explaining its properties [Weisberg/Needham/Hendry] |
17473 | It is unlikely that chemistry will ever be reduced to physics [Weisberg/Needham/Hendry] |
17474 | Quantum theory won't tell us which structure a set of atoms will form [Weisberg/Needham/Hendry] |
17475 | For temperature to be mean kinetic energy, a state of equilibrium is also required [Weisberg/Needham/Hendry] |
17469 | 'H2O' just gives the element proportions, not the microstructure [Weisberg/Needham/Hendry] |
17467 | Isotopes (such as those of hydrogen) can vary in their rates of chemical reaction [Weisberg/Needham/Hendry] |
17466 | Mendeleev systematised the elements, and also gave an account of their nature [Weisberg/Needham/Hendry] |