125 ideas
| 7001 | If you begin philosophy with language, you find yourself trapped in it [Heil] |
| 7038 | A theory with few fundamental principles might still posit a lot of entities [Heil] |
| 7037 | Parsimony does not imply the world is simple, but that our theories should try to be [Heil] |
| 7004 | The view that truth making is entailment is misguided and misleading [Heil] |
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| 7035 | God does not create the world, and then add the classes [Heil] |
| 1618 | We study bound variables not to know reality, but to know what reality language asserts [Quine] |
| 8455 | Canonical notation needs quantification, variables and predicates, but not names [Quine, by Orenstein] |
| 8456 | Quine extended Russell's defining away of definite descriptions, to also define away names [Quine, by Orenstein] |
| 1611 | Names can be converted to descriptions, and Russell showed how to eliminate those [Quine] |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| 1613 | Logicists cheerfully accept reference to bound variables and all sorts of abstract entities [Quine] |
| 1616 | Formalism says maths is built of meaningless notations; these build into rules which have meaning [Quine] |
| 1615 | Intuitionism says classes are invented, and abstract entities are constructed from specified ingredients [Quine] |
| 1614 | Conceptualism holds that there are universals but they are mind-made [Quine] |
| 10241 | For Quine, there is only one way to exist [Quine, by Shapiro] |
| 4064 | The idea of a thing and the idea of existence are two sides of the same coin [Quine, by Crane] |
| 19277 | Quine rests existence on bound variables, because he thinks singular terms can be analysed away [Quine, by Hale] |
| 7017 | The reductionist programme dispenses with levels of reality [Heil] |
| 7003 | There are levels of organisation, complexity, description and explanation, but not of reality [Heil] |
| 12210 | Quine's ontology is wrong; his question is scientific, and his answer is partly philosophical [Fine,K on Quine] |
| 7045 | Realism says some of our concepts 'cut nature at the joints' [Heil] |
| 7065 | Anti-realists who reduce reality to language must explain the existence of language [Heil] |
| 8496 | What actually exists does not, of course, depend on language [Quine] |
| 1610 | To be is to be the value of a variable, which amounts to being in the range of reference of a pronoun [Quine] |
| 8459 | Fictional quantification has no ontology, so we study ontology through scientific theories [Quine, by Orenstein] |
| 8497 | An ontology is like a scientific theory; we accept the simplest scheme that fits disorderly experiences [Quine] |
| 16261 | If commitment rests on first-order logic, we obviously lose the ontology concerning predication [Maudlin on Quine] |
| 7698 | If to be is to be the value of a variable, we must already know the values available [Jacquette on Quine] |
| 7020 | Concepts don't carve up the world, which has endless overlooked or ignored divisions [Heil] |
| 7007 | I think of properties as simultaneously dispositional and qualitative [Heil] |
| 7015 | A predicate applies truly if it picks out a real property of objects [Heil] |
| 7042 | A theory of universals says similarity is identity of parts; for modes, similarity is primitive [Heil] |
| 7023 | Powers or dispositions are usually seen as caused by lower-level qualities [Heil] |
| 7025 | Are a property's dispositions built in, or contingently added? [Heil] |
| 1612 | Realism, conceptualism and nominalism in medieval universals reappear in maths as logicism, intuitionism and formalism [Quine] |
| 7034 | Universals explain one-over-many relations, and similar qualities, and similar behaviour [Heil] |
| 7039 | How could you tell if the universals were missing from a world of instances? [Heil] |
| 15402 | There is no entity called 'redness', and that some things are red is ultimate and irreducible [Quine] |
| 7009 | Similarity among modes will explain everthing universals were for [Heil] |
| 7041 | Similar objects have similar properties; properties are directly similar [Heil] |
| 4443 | Quine has argued that predicates do not have any ontological commitment [Quine, by Armstrong] |
| 7032 | Objects join sets because of properties; the property is not bestowed by set membership [Heil] |
| 8498 | Treating scattered sensations as single objects simplifies our understanding of experience [Quine] |
| 7008 | Trope theorists usually see objects as 'bundles' of tropes [Heil] |
| 7018 | Objects are substances, which are objects considered as the bearer of properties [Heil] |
| 7019 | Maybe there is only one substance, space-time or a quantum field [Heil] |
| 7046 | Rather than 'substance' I use 'objects', which have properties [Heil] |
| 7047 | Statues and bronze lumps have discernible differences, so can't be identical [Heil] |
| 7048 | Do we reduce statues to bronze, or eliminate statues, or allow statues and bronze? [Heil] |
| 8856 | Quine's indispensability argument said arguments for abstracta were a posteriori [Quine, by Yablo] |
| 12443 | Can an unactualized possible have self-identity, and be distinct from other possibles? [Quine] |
| 18209 | We can never translate our whole language of objects into phenomenalism [Quine] |
| 7030 | Properties don't possess ways they are, because that just is the property [Heil] |
| 7028 | If properties were qualities without dispositions, they would be undetectable [Heil] |
| 7029 | Can we distinguish the way a property is from the property? [Heil] |
| 7051 | Objects only have secondary qualities because they have primary qualities [Heil] |
| 7044 | Secondary qualities are just primary qualities considered in the light of their effect on us [Heil] |
| 7052 | Colours aren't surface properties, because of radiant sources and the colour of the sky [Heil] |
| 7053 | Treating colour as light radiation has the implausible result that tomatoes are not red [Heil] |
| 7066 | If the world is just texts or social constructs, what are texts and social constructs? [Heil] |
| 7021 | If the world is theory-dependent, the theories themselves can't be theory-dependent [Heil] |
| 7026 | Science is sometimes said to classify powers, neglecting qualities [Heil] |
| 7060 | One form of explanation is by decomposition [Heil] |
| 7010 | Dispositionality provides the grounding for intentionality [Heil] |
| 7054 | Intentionality now has internalist (intrinsic to thinkers) and externalist (environment or community) views [Heil] |
| 7011 | Qualia are not extra appendages, but intrinsic ingredients of material states and processes [Heil] |
| 7061 | Philosophers' zombies aim to show consciousness is over and above the physical world [Heil] |
| 7063 | Zombies are based on the idea that consciousness relates contingently to the physical [Heil] |
| 7064 | Functionalists deny zombies, since identity of functional state means identity of mental state [Heil] |
| 7027 | Functionalists say objects can be the same in disposition but differ in quality [Heil] |
| 7062 | Functionalism cannot explain consciousness just by functional organisation [Heil] |
| 7059 | The 'explanatory gap' is used to say consciousness is inexplicable, at least with current concepts [Heil] |
| 7012 | If a car is a higher-level entity, distinct from its parts, how could it ever do anything? [Heil] |
| 7043 | Multiple realisability is actually one predicate applying to a diverse range of properties [Heil] |
| 7058 | Externalism is causal-historical, or social, or biological [Heil] |
| 7057 | Intentionality is based in dispositions, which are intrinsic to agents, suggesting internalism [Heil] |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| 7013 | The Picture Theory claims we can read reality from our ways of speaking about it [Heil] |
| 1619 | There is an attempt to give a verificationist account of meaning, without the error of reducing everything to sensations [Dennett on Quine] |
| 1609 | I do not believe there is some abstract entity called a 'meaning' which we can 'have' [Quine] |
| 1617 | The word 'meaning' is only useful when talking about significance or about synonymy [Quine] |
| 19159 | Quine relates predicates to their objects, by being 'true of' them [Quine, by Davidson] |
| 7002 | If propositions are states of affairs or sets of possible worlds, these lack truth values [Heil] |
| 7016 | The standard view is that causal sequences are backed by laws, and between particular events [Heil] |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| 7036 | The real natural properties are sparse, but there are many complex properties [Heil] |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |