105 ideas
6851 | We overvalue whether arguments are valid, and undervalue whether they are interesting [Monk] |
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] |
4098 | The theory of descriptions supports internalism, since they are thinkable when the object is non-existent [Crane] |
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] |
4077 | Aesthetic properties of thing supervene on their physical properties [Crane] |
4078 | Constitution (as in a statue constituted by its marble) is supervenience without identity [Crane] |
8386 | Events are picked out by descriptions, and facts by whole sentences [Crane] |
18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
4082 | The distinction between 'resultant' properties (weight) and 'emergent' properties is a bit vague [Crane] |
4083 | If mental properties are emergent they add a new type of causation, and physics is not complete [Crane] |
4079 | Properties are causes [Crane] |
4068 | Traditional substance is separate from properties and capable of independent existence [Crane] |
4096 | Maybe beliefs don't need to be conscious, if you are not conscious of the beliefs guiding your actions [Crane] |
4097 | Maybe there are two kinds of belief - 'de re' beliefs and 'de dicto' beliefs [Crane] |
4093 | Many cases of knowing how can be expressed in propositional terms (like how to get somewhere) [Crane] |
4108 | Phenol-thio-urea tastes bitter to three-quarters of people, but to the rest it is tasteless, so which is it? [Crane] |
4105 | The traditional supports for the sense datum theory were seeing double and specks before one's eyes [Crane] |
4104 | One can taste that the wine is sour, and one can also taste the sourness of the wine [Crane] |
4101 | If we smell something we are aware of the smell separately, but we don't perceive a 'look' when we see [Crane] |
4102 | The problems of perception disappear if it is a relation to an intentional state, not to an object or sense datum [Crane] |
4109 | If perception is much richer than our powers of description, this suggests that it is non-conceptual [Crane] |
4103 | The adverbial theory of perceptions says it is the experiences which have properties, not the objects [Crane] |
4065 | Is knowledge just a state of mind, or does it also involve the existence of external things? [Crane] |
4092 | The core of the consciousness problem is the case of Mary, zombies, and the Hard Question [Crane] |
4087 | Intentionalism does not require that all mental states be propositional attitudes [Crane] |
4095 | Object-directed attitudes like love are just as significant as propositional attitudes [Crane] |
4106 | If someone removes their glasses the content of experience remains, but the quality changes [Crane] |
4089 | Pains have a region of the body as their intentional content, not some pain object [Crane] |
4090 | Weak intentionalism says qualia are extra properties; strong intentionalism says they are intentional [Crane] |
4107 | With inverted qualia a person's experiences would change, but their beliefs remain the same [Crane] |
18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
4069 | Descartes did not think of minds as made of a substance, because they are not divisible [Crane] |
4074 | Functionalism defines mental states by their causal properties, which rules out epiphenomenalism [Crane] |
4091 | The problems of misrepresentation and error have dogged physicalist reductions of intentionality [Crane] |
4070 | Properties dualism says mental properties are distinct from physical, despite a single underlying substance [Crane] |
4084 | Non-reductive physicalism seeks an explanation of supervenience, but emergentists accept it as basic [Crane] |
4080 | If mental supervenes on the physical, then every physical cause will be accompanied by a mental one [Crane] |
4075 | Identity theory is either of particular events, or of properties, depending on your theory of causation [Crane] |
4085 | Physicalism may be the source of the mind-body problem, rather than its solution [Crane] |
4073 | Overdetermination occurs if two events cause an effect, when each would have caused it alone [Crane] |
4072 | The completeness of physics must be an essential component of any physicalist view of mind [Crane] |
4094 | Experience teaches us propositions, because we can reason about our phenomenal experience [Crane] |
7535 | If all beliefs are propositional, then belief and judgement are the same thing [Monk] |
4100 | The Twin Earth argument depends on reference being determined by content, which may be false. [Crane] |
4067 | Broad content entails the existence of the object of the thought [Crane] |
4063 | In intensional contexts, truth depends on how extensions are conceived. [Crane] |
6850 | Wittgenstein pared his life down in his search for decency [Monk] |
4071 | Causation can be seen in counterfactual terms, or as increased probability, or as energy flow [Crane] |
8387 | A cause has its effects in virtue of its properties [Crane] |
4076 | Causes are properties, not events, because properties are what make a difference in a situation [Crane] |
8384 | The regularity theory explains a causal event by other items than the two that are involved [Crane] |
4066 | It seems that 'exists' could sometimes be a predicate [Crane] |