117 ideas
18495 | The best philosophers I know are the best people I know [Heil] |
18494 | Using a technical vocabulary actually prevents discussion of the presuppositions [Heil] |
18506 | Questions of explanation should not be confused with metaphyics [Heil] |
18535 | Without abstraction we couldn't think systematically [Heil] |
4767 | Traditionally, rational beliefs are those which are justified by reasons [Psillos] |
18534 | Truth relates truthbearers to truthmakers [Heil] |
18531 | Philosophers of the past took the truthmaking idea for granted [Heil] |
18509 | Not all truths need truthmakers - mathematics and logic seem to be just true [Heil] |
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] |
4810 | Valid deduction is monotonic - that is, it remains valid if further premises are added [Psillos] |
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] |
18518 | Infinite numbers are qualitatively different - they are not just very large numbers [Heil] |
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] |
18500 | How could structures be mathematical truthmakers? Maths is just true, without truthmakers [Heil] |
8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
4768 | The 'epistemic fallacy' is inferring what does exist from what can be known to exist [Psillos] |
18539 | Our categories lack the neat arrangement needed for reduction [Heil] |
18505 | Fundamental ontology aims at the preconditions for any true theory [Heil] |
18499 | Our quantifications only reveal the truths we accept; the ontology and truthmakers are another matter [Heil] |
18512 | Ontology aims to give the fundamental categories of being [Heil] |
18508 | Most philosophers now (absurdly) believe that relations fully exist [Heil] |
18532 | If causal relations are power manifestations, that makes them internal relations [Heil] |
18510 | We need properties to explain how the world works [Heil] |
18522 | Categorical properties were introduced by philosophers as actual properties, not if-then properties [Heil] |
18513 | Emergent properties will need emergent substances to bear them [Heil] |
18540 | Predicates only match properties at the level of fundamentals [Heil] |
18533 | In Fa, F may not be a property of a, but a determinable, satisfied by some determinate [Heil] |
18511 | Properties have causal roles which sets can't possibly have [Heil] |
18523 | Are all properties powers, or are there also qualities, or do qualities have the powers? [Heil] |
18524 | Properties are both qualitative and dispositional - they are powerful qualities [Heil] |
18498 | Abstract objects wouldn't be very popular without the implicit idea of truthmakers [Heil] |
18507 | Substances bear properties, so must be simple, and not consist of further substances [Heil] |
18515 | Spatial parts are just regions, but objects depend on and are made up of substantial parts [Heil] |
18516 | A 'gunky' universe would literally have no parts at all [Heil] |
18514 | Many wholes can survive replacement of their parts [Heil] |
18517 | Dunes depend on sand grains, but line segments depend on the whole line [Heil] |
18502 | If basic physics has natures, then why not reality itself? That would then found the deepest necessities [Heil] |
18496 | If possible worlds are just fictions, they can't be truthmakers for modal judgements [Heil] |
4807 | A good barometer will predict a storm, but not explain it [Psillos] |
4808 | If we say where Mars was two months ago, we offer an explanation without a prediction [Psillos] |
4811 | Induction (unlike deduction) is non-monotonic - it can be invalidated by new premises [Psillos] |
4812 | Explanation is either showing predictability, or showing necessity, or showing causal relations [Psillos] |
4802 | Just citing a cause does not enable us to understand an event; we also need a relevant law [Psillos] |
4804 | The 'covering law model' says only laws can explain the occurrence of single events [Psillos] |
4805 | If laws explain the length of a flagpole's shadow, then the shadow also explains the length of the pole [Psillos] |
4395 | There are non-causal explanations, most typically mathematical explanations [Psillos] |
4806 | An explanation can just be a 'causal story', without laws, as when I knock over some ink [Psillos] |
4404 | Maybe explanation is entirely relative to the interests and presuppositions of the questioner [Psillos] |
4803 | An explanation is the removal of the surprise caused by the event [Psillos] |
18525 | Mental abstraction does not make what is abstracted mind-dependent [Heil] |
18504 | Only particulars exist, and generality is our mode of presentation [Heil] |
4769 | It is hard to analyse causation, if it is presupposed in our theory of the functioning of the mind [Psillos] |
18503 | You can think of tomatoes without grasping what they are [Heil] |
18537 | Linguistic thought is just as imagistic as non-linguistic thought [Heil] |
18538 | Non-conscious thought may be unlike conscious thought [Heil] |
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] |
18536 | The subject-predicate form reflects reality [Heil] |
4770 | Nothing is more usual than to apply to external bodies every internal sensation which they occasion [Psillos] |
18497 | Many reject 'moral realism' because they can't see any truthmakers for normative judgements [Heil] |
18519 | If there were infinite electrons, they could vanish without affecting total mass-energy [Heil] |
4399 | Causes clearly make a difference, are recipes for events, explain effects, and are evidence [Psillos] |
4400 | Theories of causation are based either on regularity, or on intrinsic relations of properties [Psillos] |
4403 | We can't base our account of causation on explanation, because it is the wrong way round [Psillos] |
4789 | Three divisions of causal theories: generalist/singularist, intrinsic/extrinsic, reductive/non-reductive [Psillos] |
18526 | We should focus on actual causings, rather than on laws and causal sequences [Heil] |
4790 | If causation is 'intrinsic' it depends entirely on the properties and relations of the cause and effect [Psillos] |
18527 | Probabilistic causation is not a weak type of cause; it is just a probability of there being a cause [Heil] |
4402 | Empiricists tried to reduce causation to explanation, which they reduced to logic-plus-a-law [Psillos] |
4774 | Counterfactual claims about causation imply that it is more than just regular succession [Psillos] |
4793 | "All gold cubes are smaller than one cubic mile" is a true universal generalisation, but not a law [Psillos] |
4397 | Regularity doesn't seem sufficient for causation [Psillos] |
4792 | A Humean view of causation says it is regularities, and causal facts supervene on non-causal facts [Psillos] |
4801 | The regularity of a cock's crow is used to predict dawn, even though it doesn't cause it [Psillos] |
4401 | It is not a law of nature that all the coins in my pocket are euros, though it is a regularity [Psillos] |
4796 | Laws are sets of regularities within a simple and strong coherent system of wider regularities [Psillos] |
4799 | Dispositional essentialism can't explain its key distinction between essential and non-essential properties [Psillos] |
4780 | In some counterfactuals, the counterfactual event happens later than its consequent [Psillos] |
4791 | Counterfactual theories say causes make a difference - if c hadn't occurred, then e wouldn't occur [Psillos] |
18520 | Electrons are treated as particles, but they lose their individuality in relations [Heil] |
10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
18501 | Maybe the universe is fine-tuned because it had to be, despite plans by God or Nature? [Heil] |
13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |