190 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] |
6779 | Instrumentalists say distinctions between observation and theory vanish with ostensive definition [Bird] |
5434 | Philosophy is similar to science, and has no special source of wisdom [Russell] |
5396 | Three Laws of Thought: identity, contradiction, and excluded middle [Russell] |
5405 | The law of contradiction is not a 'law of thought', but a belief about things [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] |
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] |
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] |
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] |
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] |
6780 | Anti-realism is more plausible about laws than about entities and theories [Bird] |
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] |
4427 | Every sentence contains at least one word denoting a universal, so we need universals to know truth [Russell] |
4030 | Russell claims that universals are needed to explain a priori knowledge (as their relations) [Russell, by Mellor/Oliver] |
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] |
6796 | Subjective probability measures personal beliefs; objective probability measures the chance of an event happening [Bird] |
6797 | Objective probability of tails measures the bias of the coin, not our beliefs about it [Bird] |
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] |
6800 | Many philosophers rate justification as a more important concept than knowledge [Bird] |
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] |
6786 | As science investigates more phenomena, the theories it needs decreases [Bird] |
5365 | Dreams can be explained fairly scientifically if we assume a physical world [Russell] |
6792 | If theories need observation, and observations need theories, how do we start? [Bird] |
6757 | Explanation predicts after the event; prediction explains before the event [Bird] |
6777 | Realists say their theories involve truth and the existence of their phenomena [Bird] |
6804 | There is no agreement on scientific method - because there is no such thing [Bird] |
6805 | Relativity ousted Newtonian mechanics despite a loss of simplicity [Bird] |
5391 | Science aims to find uniformities to which (within the limits of experience) there are no exceptions [Russell] |
6778 | Instrumentalists regard theories as tools for prediction, with truth being irrelevant [Bird] |
6775 | Induction is inference to the best explanation, where the explanation is a law [Bird] |
5394 | We can't prove induction from experience without begging the question [Russell] |
5390 | Chickens are not very good at induction, and are surprised when their feeder wrings their neck [Russell] |
5392 | It doesn't follow that because the future has always resembled the past, that it always will [Russell] |
6791 | If Hume is right about induction, there is no scientific knowledge [Bird] |
6790 | Anything justifying inferences from observed to unobserved must itself do that [Bird] |
6738 | Any conclusion can be drawn from an induction, if we use grue-like predicates [Bird] |
6739 | Several months of observing beech trees supports the deciduous and evergreen hypotheses [Bird] |
6799 | We normally learn natural kinds from laws, but Goodman shows laws require prior natural kinds [Bird] |
6798 | Bayesianism claims to find rationality and truth in induction, and show how science works [Bird] |
6752 | The objective component of explanations is the things that must exist for the explanation [Bird] |
6754 | We talk both of 'people' explaining things, and of 'facts' explaining things [Bird] |
6750 | Explanations are causal, nomic, psychological, psychoanalytic, Darwinian or functional [Bird] |
6761 | Contrastive explanations say why one thing happened but not another [Bird] |
6758 | 'Covering law' explanations only work if no other explanations are to be found [Bird] |
6759 | Livers always accompany hearts, but they don't explain hearts [Bird] |
6756 | Probabilistic-statistical explanations don't entail the explanandum, but makes it more likely [Bird] |
6760 | An operation might reduce the probability of death, yet explain a death [Bird] |
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] |
6785 | Inference to the Best Explanation is done with facts, so it has to be realist [Bird] |
6788 | Maybe bad explanations are the true ones, in this messy world [Bird] |
6787 | Which explanation is 'best' is bound to be subjective, and no guide to truth [Bird] |
6751 | Maybe explanation is so subjective that it cannot be a part of science [Bird] |
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] |
8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
5384 | A universal of which we are aware is called a 'concept' [Russell] |
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] |
6776 | Natural kinds are those that we use in induction [Bird] |
6767 | Rubies and sapphires are both corundum, with traces of metals varying their colours [Bird] |
6768 | Tin is not one natural kind, but appears to be 21, depending on isotope [Bird] |
6770 | Membership of a purely random collection cannot be used as an explanation [Bird] |
6771 | Natural kinds may overlap, or be sub-kinds of one another [Bird] |
6773 | If F is a universal appearing in a natural law, then Fs form a natural kind [Bird] |
6769 | In the Kripke-Putnam view only nuclear physicists can know natural kinds [Bird] |
6774 | Darwinism suggests that we should have a native ability to detect natural kinds [Bird] |
6764 | Nominal essence of a natural kind is the features that make it fit its name [Bird] |
6766 | Jadeite and nephrite are superficially identical, but have different composition [Bird] |
6808 | Reference to scientific terms is by explanatory role, not by descriptions [Bird] |
6753 | Laws are more fundamental in science than causes, and laws will explain causes [Bird] |
6762 | Newton's laws cannot be confirmed individually, but only in combinations [Bird] |
6763 | Parapsychology is mere speculation, because it offers no mechanisms for its working [Bird] |
6772 | Existence requires laws, as inertia or gravity are needed for mass or matter [Bird] |
6740 | 'All uranium lumps are small' is a law, but 'all gold lumps are small' is not [Bird] |
6741 | There can be remarkable uniformities in nature that are purely coincidental [Bird] |
6742 | A law might have no instances, if it was about things that only exist momentarily [Bird] |
6743 | If laws are just instances, the law should either have gaps, or join the instances arbitrarily [Bird] |
6744 | Where is the regularity in a law predicting nuclear decay? [Bird] |
6747 | Laws cannot explain instances if they are regularities, as something can't explain itself [Bird] |
6748 | Similar appearance of siblings is a regularity, but shared parents is what links them [Bird] |
6749 | We can only infer a true regularity if something binds the instances together [Bird] |
6803 | If we only infer laws from regularities among observations, we can't infer unobservable entities. [Bird] |
6801 | Accidental regularities are not laws, and an apparent regularity may not be actual [Bird] |
6746 | There may be many laws, each with only a few instances [Bird] |
6745 | A regularity is only a law if it is part of a complete system which is simple and strong [Bird] |
6802 | With strange enough predicates, anything could be made out to be a regularity [Bird] |
6789 | If flame colour is characteristic of a metal, that is an empirical claim needing justification [Bird] |
5393 | We can't know that our laws are exceptionless, or even that there are any laws [Russell] |
6807 | In Newton mass is conserved, but in Einstein it can convert into energy [Bird] |
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] |