108 ideas
3807 | Reason is and ought to be the slave of the passions [Hume] |
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] |
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] |
8649 | Two numbers are equal if all of their units correspond to one another [Hume] |
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] |
21291 | There is no medium state between existence and non-existence [Hume] |
11942 | Power is the possibility of action, as discovered by experience [Hume] |
11949 | There may well be powers in things, with which we are quite unacquainted [Hume] |
11950 | We have no idea of powers, because we have no impressions of them [Hume] |
11941 | The distinction between a power and its exercise is entirely frivolous [Hume] |
11098 | Momentary impressions are wrongly identified with one another on the basis of resemblance [Hume, by Quine] |
7954 | If we see a resemblance among objects, we apply the same name to them, despite their differences [Hume] |
21293 | Individuation is only seeing that a thing is stable and continuous over time [Hume] |
12048 | The only meaning we have for substance is a collection of qualities [Hume] |
13424 | Aristotelians propose accidents supported by substance, but they don't understand either of them [Hume] |
21300 | A change more obviously destroys an identity if it is quick and observed [Hume] |
21299 | Changing a part can change the whole, not absolutely, but by its proportion of the whole [Hume] |
1330 | If a republic can retain identity through many changes, so can an individual [Hume] |
1321 | If identity survives change or interruption, then resemblance, contiguity or causation must unite the parts of it [Hume] |
21302 | If a ruined church is rebuilt, its relation to its parish makes it the same church [Hume] |
21303 | We accept the identity of a river through change, because it is the river's nature [Hume] |
21301 | The purpose of the ship makes it the same one through all variations [Hume] |
21290 | Multiple objects cannot convey identity, because we see them as different [Hume] |
1207 | Both number and unity are incompatible with the relation of identity [Hume] |
21289 | 'An object is the same with itself' is meaningless; it expresses unity, not identity [Hume] |
21292 | Saying an object is the same with itself is only meaningful over a period of time [Hume] |
9428 | Nothing we clearly imagine is absolutely impossible [Hume] |
4766 | Necessity only exists in the mind, and not in objects [Hume] |
12149 | Indexicals are a problem for beliefs being just subject-proposition relations [Perry] |
6526 | Hume says objects are not a construction, but an imaginative leap [Hume, by Robinson,H] |
6489 | Associationism results from having to explain intentionality just with sense-data [Robinson,H on Hume] |
6182 | Even Hume didn't include mathematics in his empiricism [Hume, by Kant] |
12417 | Mathematicians only accept their own proofs when everyone confims them [Hume] |
5548 | Hume became a total sceptic, because he believed that reason was a deception [Hume, by Kant] |
7446 | The idea of inductive evidence, around 1660, made Hume's problem possible [Hume, by Hacking] |
21806 | Memory, senses and understanding are all founded on the imagination [Hume] |
3819 | Hume's 'bundle' won't distinguish one mind with ten experiences from ten minds [Searle on Hume] |
1317 | A person is just a fast-moving bundle of perceptions [Hume] |
1331 | The parts of a person are always linked together by causation [Hume] |
1388 | Hume gives us an interesting sketchy causal theory of personal identity [Perry on Hume] |
21297 | A person is simply a bundle of continually fluctuating perceptions [Hume] |
1316 | Introspection always discovers perceptions, and never a Self without perceptions [Hume] |
1333 | Memory only reveals personal identity, by showing cause and effect [Hume] |
1332 | We use memory to infer personal actions we have since forgotten [Hume] |
21305 | Memory not only reveals identity, but creates it, by producing resemblances [Hume] |
21307 | Who thinks that because you have forgotten an incident you are no longer that person? [Hume] |
21306 | Causation unites our perceptions, by producing, destroying and modifying each other [Hume] |
21294 | A continuous lifelong self must be justified by a single sustained impression, which we don't have [Hume] |
21295 | When I introspect I can only observe my perceptions, and never a self which has them [Hume] |
21298 | We pretend our perceptions are continuous, and imagine a self to fill the gaps [Hume] |
21304 | Identity in the mind is a fiction, like that fiction that plants and animals stay the same [Hume] |
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] |
12151 | If we replace 'I' in sentences about me, they are different beliefs and explanations of behaviour [Perry] |
18412 | Indexicals individuate certain belief states, helping in explanation and prediction [Perry] |
12150 | Indexicals reveal big problems with the traditional idea of a proposition [Perry] |
20030 | If one event causes another, the two events must be wholly distinct [Hume, by Wilson/Schpall] |
6692 | For Hume, practical reason has little force, because we can always modify our desires [Hume, by Graham] |
8257 | Reason alone can never be a motive to any action of the will [Hume] |
22374 | You can only hold people responsible for actions which arise out of their character [Hume] |
22382 | We cannot discover vice by studying a wilful murder; that only arises from our own feelings [Hume] |
4008 | Modern science has destroyed the Platonic synthesis of scientific explanation and morality [Hume, by Taylor,C] |
8067 | The problem of getting to 'ought' from 'is' would also apply in getting to 'owes' or 'needs' [Anscombe on Hume] |
4578 | You can't move from 'is' to 'ought' without giving some explanation or reason for the deduction [Hume] |
3650 | Total selfishness is not irrational [Hume] |
14301 | We have no good concept of solidity or matter, because accounts of them are all circular [Hume] |
8382 | For Hume a constant conjunction is both necessary and sufficient for causation [Hume, by Crane] |
19274 | Hume seems to presuppose necessary connections between mental events [Kripke on Hume] |
10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
15203 | Tense is essential for thought and action [Perry, by Le Poidevin] |
15204 | Actual tensed sentences cannot be tenseless, because they can cite their own context [Perry, by Le Poidevin] |
13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
21296 | If all of my perceptions were removed by death, nothing more is needed for total annihilation [Hume] |