130 ideas
| 5361 | Philosophers must get used to absurdities [Russell] |
| 5368 | Philosophy verifies that our hierarchy of instinctive beliefs is harmonious and consistent [Russell] |
| 5432 | Metaphysics cannot give knowledge of the universe as a whole [Russell] |
| 5434 | Philosophy is similar to science, and has no special source of wisdom [Russell] |
| 5405 | The law of contradiction is not a 'law of thought', but a belief about things [Russell] |
| 5396 | Three Laws of Thought: identity, contradiction, and excluded middle [Russell] |
| 5420 | Truth is a property of a belief, but dependent on its external relations, not its internal qualities [Russell] |
| 5419 | Truth and falsehood are properties of beliefs and statements [Russell] |
| 5417 | A good theory of truth must make falsehood possible [Russell] |
| 7395 | Truth as congruence may work for complex beliefs, but not for simple beliefs about existence [Joslin on Russell] |
| 5428 | Beliefs are true if they have corresponding facts, and false if they don't [Russell] |
| 5421 | The coherence theory says falsehood is failure to cohere, and truth is fitting into a complete system of Truth [Russell] |
| 5422 | More than one coherent body of beliefs seems possible [Russell] |
| 5423 | If we suspend the law of contradiction, nothing will appear to be incoherent [Russell] |
| 5424 | Coherence is not the meaning of truth, but an important test for truth [Russell] |
| 5401 | The mortality of Socrates is more certain from induction than it is from deduction [Russell] |
| 16456 | For modality Lewis rejected boxes and diamonds, preferring worlds, and an index for the actual one [Lewis, by Stalnaker] |
| 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] |
| 5395 | Demonstration always relies on the rule that anything implied by a truth is true [Russell] |
| 5386 | Proper names are really descriptions, and can be replaced by a description in a person's mind [Russell] |
| 5385 | The phrase 'a so-and-so' is an 'ambiguous' description'; 'the so-and-so' (singular) is a 'definite' description [Russell] |
| 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] |
| 5399 | Maths is not known by induction, because further instances are not needed to support it [Russell] |
| 5370 | Space is neutral between touch and sight, so it cannot really be either of them [Russell] |
| 5418 | In a world of mere matter there might be 'facts', but no truths [Russell] |
| 5371 | Because we depend on correspondence, we know relations better than we know the items that relate [Russell] |
| 5407 | That Edinburgh is north of London is a non-mental fact, so relations are independent universals [Russell] |
| 5383 | Every complete sentence must contain at least one word (a verb) which stands for a universal [Russell] |
| 4428 | Propositions express relations (prepositions and verbs) as well as properties (nouns and adjectives) [Russell] |
| 5406 | Confused views of reality result from thinking that only nouns and adjectives represent universals [Russell] |
| 4479 | All universals are like the relation "is north of", in having no physical location at all [Russell, by Loux] |
| 4030 | Russell claims that universals are needed to explain a priori knowledge (as their relations) [Russell, by Mellor/Oliver] |
| 4427 | Every sentence contains at least one word denoting a universal, so we need universals to know truth [Russell] |
| 5409 | Normal existence is in time, so we must say that universals 'subsist' [Russell] |
| 5408 | If we identify whiteness with a thought, we can never think of it twice; whiteness is the object of a thought [Russell] |
| 4441 | 'Resemblance Nominalism' won't work, because the theory treats resemblance itself as a universal [Russell] |
| 4429 | If we consider whiteness to be merely a mental 'idea', we rob it of its universality [Russell] |
| 5400 | In any possible world we feel that two and two would be four [Russell] |
| 5431 | Knowledge cannot be precisely defined, as it merges into 'probable opinion' [Russell] |
| 5426 | Belief relates a mind to several things other than itself [Russell] |
| 5366 | We have an 'instinctive' belief in the external world, prior to all reflection [Russell] |
| 5359 | Descartes showed that subjective things are the most certain [Russell] |
| 5377 | 'Acquaintance' is direct awareness, without inferences or judgements [Russell] |
| 6510 | Russell (1912) said phenomena only resemble reality in abstract structure [Russell, by Robinson,H] |
| 5372 | There is no reason to think that objects have colours [Russell] |
| 5373 | 'Idealism' says that everything which exists is in some sense mental [Russell] |
| 5362 | It is not illogical to think that only myself and my mental events exist [Russell] |
| 5412 | Some propositions are self-evident, but their implications may also be self-evident [Russell] |
| 5413 | Particular instances are more clearly self-evident than any general principles [Russell] |
| 5415 | As shown by memory, self-evidence comes in degrees [Russell] |
| 5416 | If self-evidence has degrees, we should accept the more self-evident as correct [Russell] |
| 5397 | The rationalists were right, because we know logical principles without experience [Russell] |
| 4430 | All a priori knowledge deals with the relations of universals [Russell] |
| 5411 | We can know some general propositions by universals, when no instance can be given [Russell] |
| 6514 | Russell's representationalism says primary qualities only show the structure of reality [Russell, by Robinson,H] |
| 6415 | After 1912, Russell said sense-data are last in analysis, not first in experience [Russell, by Grayling] |
| 5358 | 'Sense-data' are what are immediately known in sensation, such as colours or roughnesses [Russell] |
| 7290 | If Russell rejects innate ideas and direct a priori knowledge, he is left with a tabula rasa [Russell, by Thompson] |
| 5357 | It is natural to begin from experience, and presumably that is the basis of knowledge [Russell] |
| 5382 | We are acquainted with outer and inner sensation, memory, Self, and universals [Russell, by PG] |
| 5389 | Knowledge by descriptions enables us to transcend private experience [Russell] |
| 5376 | I can know the existence of something with which nobody is acquainted [Russell] |
| 5414 | Images are not memory, because they are present, and memories are of the past [Russell] |
| 5430 | A true belief is not knowledge if it is reached by bad reasoning [Russell] |
| 5429 | True belief is not knowledge when it is deduced from false belief [Russell] |
| 5378 | All knowledge (of things and of truths) rests on the foundations of acquaintance [Russell] |
| 5365 | Dreams can be explained fairly scientifically if we assume a physical world [Russell] |
| 5391 | Science aims to find uniformities to which (within the limits of experience) there are no exceptions [Russell] |
| 5390 | Chickens are not very good at induction, and are surprised when their feeder wrings their neck [Russell] |
| 5394 | We can't prove induction from experience without begging the question [Russell] |
| 5392 | It doesn't follow that because the future has always resembled the past, that it always will [Russell] |
| 5363 | If the cat reappears in a new position, presumably it has passed through the intermediate positions [Russell] |
| 5367 | Belief in real objects makes our account of experience simpler and more systematic [Russell] |
| 5364 | It is hard not to believe that speaking humans are expressing thoughts, just as we do ourselves [Russell] |
| 5379 | If we didn't know our own minds by introspection, we couldn't know that other people have minds [Russell] |
| 5410 | I learn the universal 'resemblance' by seeing two shades of green, and their contrast with red [Russell] |
| 5381 | In seeing the sun, we are acquainted with our self, but not as a permanent person [Russell] |
| 5380 | In perceiving the sun, I am aware of sun sense-data, and of the perceiver of the data [Russell] |
| 5369 | It is rational to believe in reality, despite the lack of demonstrative reasons for it [Russell] |
| 5375 | Knowledge of truths applies to judgements; knowledge by acquaintance applies to sensations and things [Russell] |
| 21711 | Russell's 'multiple relations' theory says beliefs attach to ingredients, not to propositions [Russell, by Linsky,B] |
| 5427 | Truth is when a mental state corresponds to a complex unity of external constituents [Russell] |
| 5425 | In order to explain falsehood, a belief must involve several terms, not two [Russell] |
| 5384 | A universal of which we are aware is called a 'concept' [Russell] |
| 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] |
| 13468 | Russell started philosophy of language, by declaring some plausible sentences to be meaningless [Russell, by Hart,WD] |
| 5388 | Every understood proposition is composed of constituents with which we are acquainted [Russell] |
| 5387 | It is pure chance which descriptions in a person's mind make a name apply to an individual [Russell] |
| 21724 | The main aim of the multiple relations theory of judgement was to dispense with propositions [Russell, by Linsky,B] |
| 5398 | Judgements of usefulness depend on judgements of value [Russell] |
| 5393 | We can't know that our laws are exceptionless, or even that there are any laws [Russell] |
| 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] |