95 ideas
| 18559 | Philosophy is empty if it does not in some way depend on matters of fact [Machery] |
| 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] |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| 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] |
| 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] |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| 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] |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| 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] |
| 18564 | Do categories store causal knowledge, or typical properties, or knowledge of individuals? [Machery] |
| 18604 | Are quick and slow categorisation the same process, or quite different? [Machery] |
| 18573 | For each category of objects (such as 'dog') an individual seems to have several concepts [Machery] |
| 18602 | A thing is classified if its features are likely to be generated by that category's causal laws [Machery] |
| 18565 | There may be ad hoc categories, such as the things to pack in your suitcase for a trip [Machery] |
| 18570 | There may be several ways to individuate things like concepts [Machery] |
| 14082 | No sortal could ever exactly pin down which set of particles count as this 'cup' [Schaffer,J] |
| 14081 | Identities can be true despite indeterminate reference, if true under all interpretations [Schaffer,J] |
| 18615 | Horizontal arguments say eliminate a term if it fails to pick out a natural kind [Machery] |
| 18616 | If a term doesn't pick out a kind, keeping it may block improvements in classification [Machery] |
| 18614 | Vertical arguments say eliminate a term if it picks out different natural kinds in different theories [Machery] |
| 18609 | Psychologists use 'induction' as generalising a property from one category to another [Machery] |
| 18610 | 'Ampliative' induction infers that all members of a category have a feature found in some of them [Machery] |
| 18562 | Connectionists cannot distinguish concept-memories from their background, or the processes [Machery] |
| 18561 | We can identify a set of cognitive capacities which are 'higher order' [Machery] |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| 18574 | Concepts for categorisation and for induction may be quite different [Machery] |
| 18588 | Concept theories aim at their knowledge, processes, format, acquisition, and location [Machery] |
| 18611 | We should abandon 'concept', and just use 'prototype', 'exemplar' and 'theory' [Machery] |
| 18567 | In the philosophy of psychology, concepts are usually introduced as constituents of thoughts [Machery] |
| 18569 | In philosophy theories of concepts explain how our propositional attitudes have content [Machery] |
| 18563 | By 'concept' psychologists mean various sorts of representation or structure [Machery] |
| 18557 | Psychologists treat concepts as long-term knowledge bodies which lead to judgements [Machery] |
| 18560 | Psychologist treat concepts as categories [Machery] |
| 18558 | Concept theorists examine their knowledge, format, processes, acquisition and location [Machery] |
| 18592 | The concepts OBJECT or AGENT may be innate [Machery] |
| 18566 | Concepts should contain working memory, not long-term, because they control behaviour [Machery] |
| 18584 | One hybrid theory combines a core definition with a prototype for identification [Machery] |
| 18585 | Heterogeneous concepts might have conflicting judgements, where hybrid theories will not [Machery] |
| 18578 | Concepts as definitions was rejected, and concepts as prototypes, exemplars or theories proposed [Machery] |
| 18575 | The concepts for a class typically include prototypes, and exemplars, and theories [Machery] |
| 18583 | Many categories don't seem to have a definition [Machery] |
| 18590 | Classical theory implies variety in processing times, but this does not generally occur [Machery] |
| 18591 | Classical theory can't explain facts like typical examples being categorised quicker [Machery] |
| 18594 | Knowing typical properties of things is especially useful in induction [Machery] |
| 18593 | The term 'prototype' is used for both typical category members, and the representation [Machery] |
| 18606 | The prototype view predicts that typical members are easier to categorise [Machery] |
| 18595 | Prototype theories are based on computation of similarities with the prototype [Machery] |
| 18596 | Prototype theorists don't tell us how we select the appropriate prototype [Machery] |
| 18603 | Maybe concepts are not the typical properties, but the ideal properties [Machery] |
| 18605 | It is more efficient to remember the prototype, than repeatedly create it from exemplars [Machery] |
| 18597 | Concepts as exemplars are based on the knowledge of properties of each particular [Machery] |
| 18598 | Exemplar theories need to explain how the relevant properties are selected from a multitude of them [Machery] |
| 18599 | In practice, known examples take priority over the rest of the set of exemplars [Machery] |
| 18587 | The theory account is sometimes labelled as 'knowledge' or 'explanation' in approach [Machery] |
| 18600 | Theory Theory says category concepts are knowledge stores explaining membership [Machery] |
| 18601 | Theory Theory says concepts are explanatory knowledge, and concepts form domains [Machery] |
| 18607 | Theory theorists rely on best explanation, rather than on similarities [Machery] |
| 18608 | If categorisation is not by similarity, it seems to rely on what properties things might have [Machery] |
| 18577 | The word 'grandmother' may be two concepts, with a prototype and a definition [Machery] |
| 18589 | For behaviourists concepts are dispositions to link category members to names [Machery] |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| 18612 | Americans are more inclined to refer causally than the Chinese are [Machery] |
| 18613 | Artifacts can be natural kinds, when they are the object of historical enquiry [Machery] |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |