96 ideas
| 2797 | As coherence expands its interrelations become steadily tighter, culminating only in necessary truth [Dancy,J] |
| 2768 | The correspondence theory also has the problem that two sets of propositions might fit the facts equally well [Dancy,J] |
| 2765 | Rescher says that if coherence requires mutual entailment, this leads to massive logical redundancy [Dancy,J] |
| 2769 | If one theory is held to be true, all the other theories appear false, because they can't be added to the true one [Dancy,J] |
| 2766 | Even with a tight account of coherence, there is always the possibility of more than one set of coherent propositions [Dancy,J] |
| 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] |
| 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] |
| 2781 | Realism says that most perceived objects exist, and have some of their perceived properties [Dancy,J] |
| 14617 | Predicates can't apply to what doesn't exist [Stalnaker] |
| 2745 | A pupil who lacks confidence may clearly know something but not be certain of it [Dancy,J] |
| 2755 | If senses are fallible, then being open to correction is an epistemological virtue [Dancy,J] |
| 5677 | Naïve direct realists hold that objects retain all of their properties when unperceived [Dancy,J] |
| 5678 | Scientific direct realism says we know some properties of objects directly [Dancy,J] |
| 5681 | Maybe we are forced from direct into indirect realism by the need to explain perceptual error [Dancy,J] |
| 5682 | Internal realism holds that we perceive physical objects via mental objects [Dancy,J] |
| 5683 | Indirect realism depends on introspection, the time-lag, illusions, and neuroscience [Dancy,J, by PG] |
| 2778 | Phenomenalism includes possible experiences, but idealism only refers to actual experiences [Dancy,J] |
| 5684 | Eliminative idealists say there are no objects; reductive idealists say objects exist as complex experiences [Dancy,J] |
| 2777 | Extreme solipsism only concerns current experience, but it might include past and future [Dancy,J] |
| 2794 | Knowing that a cow is not a horse seems to be a synthetic a priori truth [Dancy,J] |
| 2780 | Perception is either direct realism, indirect realism, or phenomenalism [Dancy,J] |
| 5679 | We can't grasp the separation of quality types, or what a primary-quality world would be like [Dancy,J] |
| 5680 | For direct realists the secondary and primary qualities seem equally direct [Dancy,J] |
| 2782 | We can be looking at distant stars which no longer actually exist [Dancy,J] |
| 2775 | It is not clear from the nature of sense data whether we should accept them as facts [Dancy,J] |
| 2784 | Appearances don't guarantee reality, unless the appearance is actually caused by the reality [Dancy,J] |
| 2785 | Perceptual beliefs may be directly caused, but generalisations can't be [Dancy,J] |
| 2788 | If perception and memory are indirect, then two things stand between mind and reality [Dancy,J] |
| 2787 | Memories aren't directly about the past, because time-lags and illusions suggest representation [Dancy,J] |
| 2791 | Phenomenalism about memory denies the past, or reduces it to present experience [Dancy,J] |
| 2790 | I can remember plans about the future, and images aren't essential (2+3=5) [Dancy,J] |
| 2754 | Foundations are justified by non-beliefs, or circularly, or they need no justification [Dancy,J] |
| 2749 | For internalists we must actually know that the fact caused the belief [Dancy,J] |
| 2770 | Internalists tend to favour coherent justification, but not the coherence theory of truth [Dancy,J] |
| 2752 | Foundationalism requires inferential and non-inferential justification [Dancy,J] |
| 2771 | Foundationalists must accept not only the basic beliefs, but also rules of inference for further progress [Dancy,J] |
| 2756 | If basic beliefs can be false, falsehood in non-basic beliefs might by a symptom [Dancy,J] |
| 2753 | Beliefs can only be infallible by having almost no content [Dancy,J] |
| 2773 | Coherentism gives a possible justification of induction, and opposes scepticism [Dancy,J] |
| 2779 | Idealists must be coherentists, but coherentists needn't be idealists [Dancy,J] |
| 2786 | For coherentists justification and truth are not radically different things [Dancy,J] |
| 2767 | If it is empirical propositions which have to be coherent, this eliminates coherent fiction [Dancy,J] |
| 2776 | Externalism could even make belief unnecessary (e.g. in animals) [Dancy,J] |
| 2746 | How can a causal theory of justification show that all men die? [Dancy,J] |
| 2747 | Causal theories don't allow for errors in justification [Dancy,J] |
| 2772 | Coherentism moves us towards a more social, shared view of knowledge [Dancy,J] |
| 2743 | What is the point of arguing against knowledge, if being right undermines your own argument? [Dancy,J] |
| 2751 | Probabilities can only be assessed relative to some evidence [Dancy,J] |
| 2757 | The argument from analogy rests on one instance alone [Dancy,J] |
| 2758 | You can't separate mind and behaviour, as the analogy argument attempts [Dancy,J] |
| 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] |
| 2744 | Verificationism (the 'verification principle') is an earlier form of anti-realism [Dancy,J] |
| 2760 | Logical positivism implies foundationalism, by dividing weak from strong verifications [Dancy,J] |
| 2761 | If the meanings of sentences depend on other sentences, how did we learn language? [Dancy,J] |
| 14616 | A 'Russellian proposition' is an ordered sequence of individual, properties and relations [Stalnaker] |
| 2763 | There is an indeterminacy in juggling apparent meanings against probable beliefs [Dancy,J] |
| 2762 | Charity makes native beliefs largely true, and Humanity makes them similar to ours [Dancy,J] |
| 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] |