143 ideas
| 15209 | Like disastrous small errors in navigation, small misunderstandings can wreck intellectual life [Harré/Madden] |
| 15215 | Philosophy devises and assesses conceptual schemes in the service of worldviews [Harré/Madden] |
| 15212 | Analysis of concepts based neither on formalism nor psychology can arise from examining what we know [Harré/Madden] |
| 15210 | Humeans see analysis in terms of formal logic, because necessities are fundamentally logical relations [Harré/Madden] |
| 15236 | Positivism says science only refers to immediate experiences [Harré/Madden] |
| 17892 | For clear questions posed by reason, reason can also find clear answers [Gödel] |
| 15227 | Logically, definitions have a subject, and a set of necessary predicates [Harré/Madden] |
| 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] |
| 15273 | Points can be 'dense' by unending division, but must meet a tougher criterion to be 'continuous' [Harré/Madden] |
| 15274 | Points are 'continuous' if any 'cut' point participates in both halves of the cut [Harré/Madden] |
| 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] |
| 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] |
| 3198 | Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey] |
| 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] |
| 15211 | There is not an exclusive dichotomy between the formal and the logical [Harré/Madden] |
| 15261 | Humeans can only explain change with continuity as successive replacement [Harré/Madden] |
| 15268 | Humeans construct their objects from events, but we construct events from objects [Harré/Madden] |
| 15257 | The induction problem fades if you work with things, rather than with events [Harré/Madden] |
| 15300 | Fundamental particulars can't change [Harré/Madden] |
| 15319 | Hard individual blocks don't fix what 'things' are; fluids are no less material things [Harré/Madden] |
| 15320 | Magnetic and gravity fields can occupy the same place without merging [Harré/Madden] |
| 15318 | Gravitational and electrical fields are, for a materialist, distressingly empty of material [Harré/Madden] |
| 15267 | Events are changes in states of affairs (which consist of structured particulars, with powers and relations) [Harré/Madden] |
| 15281 | Humeans see predicates as independent, but science says they are connected [Harré/Madden] |
| 15279 | Energy was introduced to physics to refer to the 'store of potency' of a moving ball [Harré/Madden] |
| 15276 | Some powers need a stimulus, but others are just released [Harré/Madden] |
| 15305 | Some powers are variable, others cannot change (without destroying an identity) [Harré/Madden] |
| 15218 | Scientists define copper almost entirely (bar atomic number) in terms of its dispositions [Harré/Madden] |
| 15302 | We explain powers by the natures of things, but explanations end in inexplicable powers [Harré/Madden] |
| 15303 | Maybe a physical field qualifies as ultimate, if its nature is identical with its powers [Harré/Madden] |
| 15258 | Powers are not qualities; they just point to directions of empirical investigation [Harré/Madden] |
| 15315 | What is a field of potentials, if it only consists of possible events? [Harré/Madden] |
| 15272 | The good criticism of substance by Humeans also loses them the vital concept of a thing [Harré/Madden] |
| 15304 | We can escape substance and its properties, if we take fields of pure powers as ultimate [Harré/Madden] |
| 15309 | The assumption that shape and solidity are fundamental implies dubious 'substance' in bodies [Harré/Madden] |
| 15264 | The notorious substratum results from substance-with-qualities; individuals-with-powers solves this [Harré/Madden] |
| 15262 | In logic the nature of a kind, substance or individual is the essence which is inseparable from what it is [Harré/Madden] |
| 15297 | We can infer a new property of a thing from its other properties, via its essential nature [Harré/Madden] |
| 15266 | We say the essence of particles is energy, but only so we can tell a story about the nature of things [Harré/Madden] |
| 15220 | To say something remains the same but lacks its capacities and powers seems a contradiction [Harré/Madden] |
| 15222 | Some individuals can gain or lose capacities or powers, without losing their identity [Harré/Madden] |
| 15296 | A particular might change all of its characteristics, retaining mere numerical identity [Harré/Madden] |
| 15275 | 'Dense' time raises doubts about continuous objects, so they need 'continuous' time [Harré/Madden] |
| 15271 | If things are successive instantaneous events, nothing requires those events to resemble one another [Harré/Madden] |
| 15256 | Humeans cannot step in the same river twice, because they cannot strictly form the concept of 'river' [Harré/Madden] |
| 15290 | What reduces the field of the possible is a step towards necessity [Harré/Madden] |
| 15291 | There is 'absolute' necessity (implied by all propositions) and 'relative' necessity (from what is given) [Harré/Madden] |
| 15230 | Logical necessity is grounded in the logical form of a statement [Harré/Madden] |
| 15221 | The relation between what a thing is and what it can do or undergo relate by natural necessity [Harré/Madden] |
| 15214 | Natural necessity is not logical necessity or empirical contingency in disguise [Harré/Madden] |
| 15224 | A necessity corresponds to the nature of the actual [Harré/Madden] |
| 15232 | Natural necessity is when powerful particulars must produce certain results in a situation [Harré/Madden] |
| 15288 | People doubt science because if it isn't logically necessary it seems to be absolutely contingent [Harré/Madden] |
| 15289 | Property or event relations are naturally necessary if generated by essential mechanisms [Harré/Madden] |
| 15231 | Transcendental necessity is conditions of a world required for a rational being to know its nature [Harré/Madden] |
| 15234 | There is a transcendental necessity for each logical necessity, but the transcendental extends further [Harré/Madden] |
| 15260 | Counterfactuals are just right for analysing statements about the powers which things have [Harré/Madden] |
| 15233 | If natural necessity is used to include or exclude some predicate, the predicate is conceptually necessary [Harré/Madden] |
| 15242 | Having a child is contingent for a 'man', necessary for a 'father'; the latter reflects a necessity of nature [Harré/Madden] |
| 15216 | Is conceptual necessity just conventional, or does it mirror something about nature? [Harré/Madden] |
| 15235 | There is a conceptual necessity when properties become a standard part of a nominal essence [Harré/Madden] |
| 15228 | Necessity and contingency are separate from the a priori and the a posteriori [Harré/Madden] |
| 15252 | If Goldbach's Conjecture is true (and logically necessary), we may be able to conceive its opposite [Harré/Madden] |
| 15245 | It is silly to say that direct experience must be justified, either by reason, or by more experience [Harré/Madden] |
| 15244 | We experience qualities as of objects, not on their own [Harré/Madden] |
| 15248 | Inference in perception is unconvincingly defended as non-conscious and almost instantaneous [Harré/Madden] |
| 15269 | Humean impressions are too instantaneous and simple to have structure or relations [Harré/Madden] |
| 15286 | Clavius's Paradox: purely syntactic entailment theories won't explain, because they are too profuse [Harré/Madden] |
| 15283 | Simplicity can sort theories out, but still leaves an infinity of possibilities [Harré/Madden] |
| 15316 | The powers/natures approach has been so successful (for electricity, magnetism, gravity) it may be universal [Harré/Madden] |
| 15298 | We prefer the theory which explains and predicts the powers and capacities of particulars [Harré/Madden] |
| 15225 | Science investigates the nature and constitution of things or substances [Harré/Madden] |
| 15255 | Conjunctions explain nothing, and so do not give a reason for confidence in inductions [Harré/Madden] |
| 15270 | Hume's atomic events makes properties independent, and leads to problems with induction [Harré/Madden] |
| 15284 | Contraposition may be equivalent in truth, but not true in nature, because of irrelevant predicates [Harré/Madden] |
| 15285 | The items put forward by the contraposition belong within different natural clusters [Harré/Madden] |
| 15287 | The possibility that all ravens are black is a law depends on a mechanism producing the blackness [Harré/Madden] |
| 15306 | Only changes require explanation [Harré/Madden] |
| 15293 | If explanation is by entailment, that lacks a causal direction, unlike natural necessity [Harré/Madden] |
| 15294 | Powers can explain the direction of causality, and make it a natural necessity [Harré/Madden] |
| 15254 | If the nature of particulars explains their powers, it also explains their relations and behaviour [Harré/Madden] |
| 15317 | Powers and natures lead us to hypothesise underlying mechanisms, which may be real [Harré/Madden] |
| 15310 | Solidity comes from the power of repulsion, and shape from the power of attraction [Harré/Madden] |
| 15219 | Essence explains passive capacities as well as active powers [Harré/Madden] |
| 15301 | The very concepts of a particular power or nature imply the possibility of being generalised [Harré/Madden] |
| 3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
| 15226 | What properties a thing must have to be a type of substance can be laid down a priori [Harré/Madden] |
| 15229 | We say there is 'no alternative' in all sorts of contexts, and there are many different grounds for it [Harré/Madden] |
| 15292 | We can base the idea of a natural kind on the mechanisms that produce natural necessity [Harré/Madden] |
| 15299 | Species do not have enough constancy to be natural kinds [Harré/Madden] |
| 15253 | If the concept of a cause includes its usual effects, we call it a 'power' [Harré/Madden] |
| 15278 | Humean accounts of causal direction by time fail, because cause and effect can occur together [Harré/Madden] |
| 15246 | Active causal power is just objects at work, not something existing in itself [Harré/Madden] |
| 15213 | Causation always involves particular productive things [Harré/Madden] |
| 15217 | Efficient causes combine stimulus to individuals, absence of contraints on activity [Harré/Madden] |
| 15277 | The cause (or part of it) is what stimulates or releases the powerful particular thing involved [Harré/Madden] |
| 15237 | Originally Humeans based lawlike statements on pure qualities, without particulars [Harré/Madden] |
| 15238 | Being lawlike seems to resist formal analysis, because there are always counter-examples [Harré/Madden] |
| 15223 | Necessary effects will follow from some general theory specifying powers and structure of a world [Harré/Madden] |
| 15241 | Humeans say there is no necessity in causation, because denying an effect is never self-contradictory [Harré/Madden] |
| 15240 | In lawful universal statements (unlike accidental ones) we see why the regularity holds [Harré/Madden] |
| 15239 | We could call any generalisation a law, if it had reasonable support and no counter-evidence [Harré/Madden] |
| 15243 | We perceive motion, and not just successive occupations of different positions [Harré/Madden] |
| 15280 | 'Kinetic energy' is used to explain the effects of moving things when they are stopped [Harré/Madden] |
| 15265 | 'Energy' is a quasi-substance invented as the bearer of change during interactions [Harré/Madden] |
| 15321 | Space can't be an individual (in space), but it is present in all places [Harré/Madden] |
| 15263 | Chemistry is not purely structural; CO2 is not the same as SO2 [Harré/Madden] |
| 15259 | Chemical atoms have two powers: to enter certain combinations, and to emit a particular spectrum [Harré/Madden] |
| 15295 | Theism is supposed to make the world more intelligible - and should offer results [Harré/Madden] |
| 3029 | Stilpo said if Athena is a daughter of Zeus, then a statue is only the child of a sculptor, and so is not a god [Stilpo, by Diog. Laertius] |