99 ideas
| 16841 | Good inference has mechanism, precision, scope, simplicity, fertility and background fit [Lipton] |
| 16854 | Contrary pairs entail contradictions; one member entails negation of the other [Lipton] |
| 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] |
| 7783 | Bodies, properties, relations, events, numbers, sets and propositions are 'things' if they exist [Lowe] |
| 16814 | Understanding is not mysterious - it is just more knowledge, of causes [Lipton] |
| 16825 | How do we distinguish negative from irrelevant evidence, if both match the hypothesis? [Lipton] |
| 16851 | The inference to observables and unobservables is almost the same, so why distinguish them? [Lipton] |
| 16798 | We infer from evidence by working out what would explain that evidence [Lipton] |
| 16799 | Inductive inference is not proof, but weighing evidence and probability [Lipton] |
| 16856 | It is more impressive that relativity predicted Mercury's orbit than if it had accommodated it [Lipton] |
| 16857 | Predictions are best for finding explanations, because mere accommodations can be fudged [Lipton] |
| 16827 | If we make a hypothesis about data, then a deduction, where does the hypothesis come from? [Lipton] |
| 16804 | Induction is repetition, instances, deduction, probability or causation [Lipton] |
| 16823 | Standard induction does not allow for vertical inferences, to some unobservable lower level [Lipton] |
| 16858 | We can argue to support our beliefs, so induction will support induction, for believers in induction [Lipton] |
| 16800 | An inductive inference is underdetermined, by definition [Lipton] |
| 16832 | If something in ravens makes them black, it may be essential (definitive of ravens) [Lipton] |
| 16836 | My shoes are not white because they lack some black essence of ravens [Lipton] |
| 16831 | A theory may explain the blackness of a raven, but say nothing about the whiteness of shoes [Lipton] |
| 16833 | We can't turn non-black non-ravens into ravens, to test the theory [Lipton] |
| 16834 | To pick a suitable contrast to ravens, we need a hypothesis about their genes [Lipton] |
| 16801 | A hypothesis is confirmed if an unlikely prediction comes true [Lipton] |
| 16802 | Bayes seems to rule out prior evidence, since that has a probability of one [Lipton] |
| 16803 | Bayes is too liberal, since any logical consequence of a hypothesis confirms it [Lipton] |
| 16837 | Bayes involves 'prior' probabilities, 'likelihood', 'posterior' probability, and 'conditionalising' [Lipton] |
| 16839 | Explanation may be an important part of implementing Bayes's Theorem [Lipton] |
| 16850 | Explanation may describe induction, but may not show how it justifies, or leads to truth [Lipton] |
| 16807 | An explanation gives the reason the phenomenon occurred [Lipton] |
| 16808 | An explanation is what makes the unfamiliar familiar to us [Lipton] |
| 16806 | An explanation is what is added to knowledge to yield understanding [Lipton] |
| 16822 | Seaching for explanations is a good way to discover the structure of the world [Lipton] |
| 16816 | In 'contrastive' explanation there is a fact and a foil - why that fact, rather than this foil? [Lipton] |
| 16826 | With too many causes, find a suitable 'foil' for contrast, and the field narrows right down [Lipton] |
| 16811 | An explanation unifies a phenomenon with our account of other phenomena [Lipton] |
| 16810 | Deduction explanation is too easy; any law at all will imply the facts - together with the facts! [Lipton] |
| 16829 | We reject deductive explanations if they don't explain, not if the deduction is bad [Lipton] |
| 16809 | Good explanations may involve no laws and no deductions [Lipton] |
| 16812 | An explanation shows why it was necessary that the effect occurred [Lipton] |
| 16813 | To explain is to give either the causal history, or the causal mechanism [Lipton] |
| 16815 | Mathematical and philosophical explanations are not causal [Lipton] |
| 16846 | A cause may not be an explanation [Lipton] |
| 16849 | Explanations may be easier to find than causes [Lipton] |
| 16848 | Causal inferences are clearest when we can manipulate things [Lipton] |
| 16842 | We want to know not just the cause, but how the cause operated [Lipton] |
| 16840 | To maximise probability, don't go beyond your data [Lipton] |
| 16824 | Is Inference to the Best Explanation nothing more than inferring the likeliest cause? [Lipton] |
| 16817 | Best Explanation as a guide to inference is preferable to best standard explanations [Lipton] |
| 16818 | The 'likeliest' explanation is the best supported; the 'loveliest' gives the most understanding [Lipton] |
| 16820 | Finding the 'loveliest' potential explanation links truth to understanding [Lipton] |
| 16819 | IBE is inferring that the best potential explanation is the actual explanation [Lipton] |
| 16828 | IBE is not passive treatment of data, but involves feedback between theory and data search [Lipton] |
| 16844 | A contrasting difference is the cause if it offers the best explanation [Lipton] |
| 16853 | We select possible explanations for explanatory reasons, as well as choosing among them [Lipton] |
| 16821 | Must we only have one explanation, and must all the data be made relevant? [Lipton] |
| 16838 | Bayesians say best explanations build up an incoherent overall position [Lipton] |
| 16855 | The best theory is boring: compare 'all planets move elliptically' with 'most of them do' [Lipton] |
| 16852 | Best explanation can't be a guide to truth, because the truth must precede explanation [Lipton] |
| 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] |
| 16847 | Counterfactual causation makes causes necessary but not sufficient [Lipton] |
| 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] |