125 ideas
| 3426 | If one theory is reduced to another, we make fewer independent assumptions about the world [Kim] |
| 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] |
| 3431 | Supervenience suggest dependence without reduction (e.g. beauty) [Kim] |
| 3437 | 'Physical facts determine all the facts' is the physicalists' slogan [Kim] |
| 3430 | Resemblance or similarity is the core of our concept of a property [Kim] |
| 3432 | Is weight a 'resultant' property of water, but transparency an 'emergent' property? [Kim] |
| 3434 | Emergent properties are 'brute facts' (inexplicable), but still cause things [Kim] |
| 8780 | Attributes are functions, not objects; this distinguishes 'square of 2' from 'double of 2' [Geach] |
| 3436 | Should properties be individuated by their causal powers? [Kim] |
| 11910 | Being 'the same' is meaningless, unless we specify 'the same X' [Geach] |
| 3406 | Counterfactuals are either based on laws, or on nearby possible worlds [Kim, by PG] |
| 3368 | Mind is basically qualities and intentionality, but how do they connect? [Kim] |
| 3392 | Mind is only interesting if it has causal powers [Kim] |
| 3396 | Experiment requires mental causation [Kim] |
| 3397 | Beliefs cause other beliefs [Kim] |
| 3367 | Both thought and language have intentionality [Kim] |
| 3365 | Intentionality involves both reference and content [Kim] |
| 3360 | Are pains pure qualia, or do they motivate? [Kim] |
| 3366 | Pain has no reference or content [Kim] |
| 3389 | Inverted qualia and zombies suggest experience isn't just functional [Kim] |
| 3391 | Crosswiring would show that pain and its function are separate [Kim, by PG] |
| 8775 | A big flea is a small animal, so 'big' and 'small' cannot be acquired by abstraction [Geach] |
| 8776 | We cannot learn relations by abstraction, because their converse must be learned too [Geach] |
| 3422 | Externalism about content makes introspection depend on external evidence [Kim] |
| 3412 | How do we distinguish our anger from embarrassment? [Kim] |
| 3363 | We often can't decide what emotion, or even sensation, we are experiencing [Kim] |
| 3409 | Mental substance causation makes physics incomplete [Kim] |
| 3399 | If epiphenomenalism were true, we couldn't report consciousness [Kim] |
| 3390 | Are inverted or absent qualia coherent ideas? [Kim] |
| 3414 | What could demonstrate that zombies and inversion are impossible? [Kim] |
| 3359 | Cartesian dualism fails because it can't explain mental causation [Kim] |
| 3369 | Logical behaviourism translates mental language to behavioural [Kim] |
| 3428 | Behaviourism reduces mind to behaviour via bridging principles [Kim] |
| 2567 | You can't define real mental states in terms of behaviour that never happens [Geach] |
| 3380 | Are dispositions real, or just a type of explanation? [Kim] |
| 2568 | Beliefs aren't tied to particular behaviours [Geach] |
| 3370 | What behaviour goes with mathematical beliefs? [Kim] |
| 3371 | Behaviour depends on lots of mental states together [Kim] |
| 3372 | Behaviour is determined by society as well as mental states [Kim] |
| 3373 | Snakes have different pain behaviour from us [Kim] |
| 3388 | Machine functionalism requires a Turing machine, causal-theoretical version doesn't [Kim] |
| 3379 | Neurons seem to be very similar and interchangeable [Kim] |
| 3384 | The person couldn't run Searle's Chinese Room without understanding Chinese [Kim] |
| 3393 | How do functional states give rise to mental causation? [Kim] |
| 3439 | Reductionism gets stuck with qualia [Kim] |
| 3427 | Reductionism is impossible if there aren't any 'bridge laws' between mental and physical [Kim] |
| 3376 | We can't assess evidence about mind without acknowledging phenomenal properties [Kim] |
| 3424 | Most modern physicalists are non-reductive property dualists [Kim] |
| 3362 | Supervenience says all souls are identical, being physically indiscernible [Kim] |
| 3413 | Zombies and inversion suggest non-reducible supervenience [Kim] |
| 3374 | Token physicalism isn't reductive; it just says all mental events have some physical properties [Kim] |
| 3433 | The core of the puzzle is the bridge laws between mind and brain [Kim] |
| 3377 | Elimination can either be by translation or by causal explanation [Kim] |
| 3438 | Reductionists deny new causal powers at the higher level [Kim] |
| 3440 | Without reductionism, mental causation is baffling [Kim] |
| 3375 | If an orange image is a brain state, are some parts of the brain orange? [Kim] |
| 3411 | How do we distinguish our attitudes from one another? [Kim] |
| 3387 | A culture without our folk psychology would be quite baffling [Kim] |
| 3386 | Folk psychology has been remarkably durable [Kim] |
| 3410 | Folk psychology has adapted to Freudianism [Kim] |
| 3394 | Maybe folk psychology is a simulation, not a theory [Kim] |
| 3382 | A machine with a mind might still fail the Turing Test [Kim] |
| 3383 | The Turing Test is too specifically human in its requirements [Kim] |
| 3408 | Two identical brain states could have different contents in different worlds [Kim] |
| 3420 | Two types of water are irrelevant to accounts of behaviour [Kim] |
| 3418 | 'Arthritis in my thigh' requires a social context for its content to be meaningful [Kim] |
| 3416 | Content may match several things in the environment [Kim] |
| 3421 | Content is best thought of as truth conditions [Kim] |
| 3417 | Content depends on other content as well as the facts [Kim] |
| 3419 | Pain, our own existence, and negative existentials, are not external [Kim] |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| 8781 | The mind does not lift concepts from experience; it creates them, and then applies them [Geach] |
| 8769 | If someone has aphasia but can still play chess, they clearly have concepts [Geach] |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| 8770 | 'Abstractionism' is acquiring a concept by picking out one experience amongst a group [Geach] |
| 8771 | 'Or' and 'not' are not to be found in the sensible world, or even in the world of inner experience [Geach] |
| 8772 | We can't acquire number-concepts by extracting the number from the things being counted [Geach] |
| 8773 | Abstractionists can't explain counting, because it must precede experience of objects [Geach] |
| 8774 | The numbers don't exist in nature, so they cannot have been abstracted from there into our languages [Geach] |
| 8778 | Blind people can use colour words like 'red' perfectly intelligently [Geach] |
| 8777 | If 'black' and 'cat' can be used in the absence of such objects, how can such usage be abstracted? [Geach] |
| 8779 | We can form two different abstract concepts that apply to a single unified experience [Geach] |
| 3403 | We assume people believe the obvious logical consequences of their known beliefs [Kim] |
| 3402 | If someone says "I do and don't like x", we don't assume a contradiction [Kim] |
| 3401 | A common view is that causal connections must be instances of a law [Kim] |
| 3407 | Laws are either 'strict', or they involve a 'ceteris paribus' clause [Kim] |
| 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] |