76 ideas
| 15879 | The Square of Opposition has two contradictory pairs, one contrary pair, and one sub-contrary pair [Harré] |
| 15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
| 15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
| 15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
| 15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
| 15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
| 15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
| 15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
| 15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
| 15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
| 15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
| 15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
| 15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
| 15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
| 15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
| 15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
| 15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
| 15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
| 15891 | Traditional quantifiers combine ordinary language generality and ontology assumptions [Harré] |
| 15878 | Some quantifiers, such as 'any', rule out any notion of order within their range [Harré] |
| 15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
| 15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
| 15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
| 18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
| 15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
| 15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
| 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
| 15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
| 15940 | The intuitionist endorses only the potential infinite [Lavine] |
| 15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
| 15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
| 15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
| 15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
| 15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
| 15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
| 15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
| 15874 | Scientific properties are not observed qualities, but the dispositions which create them [Harré] |
| 20339 | Classes rarely share properties with their members - unlike universals and types [Wollheim] |
| 15884 | Laws of nature remain the same through any conditions, if the underlying mechanisms are unchanged [Harré] |
| 15880 | In physical sciences particular observations are ordered, but in biology only the classes are ordered [Harré] |
| 15869 | Reports of experiments eliminate the experimenter, and present results as the behaviour of nature [Harré] |
| 15881 | We can save laws from counter-instances by treating the latter as analytic definitions [Harré] |
| 15882 | Since there are three different dimensions for generalising laws, no one system of logic can cover them [Harré] |
| 15887 | 'Grue' introduces a new causal hypothesis - that emeralds can change colour [Harré] |
| 15888 | The grue problem shows that natural kinds are central to science [Harré] |
| 15889 | It is because ravens are birds that their species and their colour might be connected [Harré] |
| 15890 | Non-black non-ravens just aren't part of the presuppositions of 'all ravens are black' [Harré] |
| 15885 | The necessity of Newton's First Law derives from the nature of material things, not from a mechanism [Harré] |
| 20338 | We often treat a type as if it were a sort of token [Wollheim] |
| 15868 | Idealisation idealises all of a thing's properties, but abstraction leaves some of them out [Harré] |
| 20343 | A love of nature must precede a love of art [Wollheim] |
| 20342 | Interpretation is performance for some arts, and critical for all arts [Wollheim] |
| 20348 | A criterion of identity for works of art would be easier than a definition [Wollheim] |
| 20347 | If beauty needs organisation, then totally simple things can't be beautiful [Wollheim] |
| 20345 | Some say art must have verbalisable expression, and others say the opposite! [Wollheim] |
| 20331 | It is claimed that the expressive properties of artworks are non-physical [Wollheim] |
| 20336 | Style can't be seen directly within a work, but appreciation needs a grasp of style [Wollheim] |
| 20337 | The traditional view is that knowledge of its genre to essential to appreciating literature [Wollheim] |
| 20333 | If artworks are not physical objects, they are either ideal entities, or collections of phenomena [Wollheim] |
| 20334 | The ideal theory says art is an intuition, shaped by a particular process, and presented in public [Wollheim] |
| 20335 | The ideal theory of art neglects both the audience and the medium employed [Wollheim] |
| 20340 | A musical performance has virtually the same features as the piece of music [Wollheim] |
| 20341 | An interpretation adds further properties to the generic piece of music [Wollheim] |
| 20332 | A drawing only represents Napoleon if the artist intended it to [Wollheim] |
| 15886 | Science rests on the principle that nature is a hierarchy of natural kinds [Harré] |
| 15864 | Classification is just as important as laws in natural science [Harré] |
| 15865 | Newton's First Law cannot be demonstrated experimentally, as that needs absence of external forces [Harré] |
| 15862 | Laws can come from data, from theory, from imagination and concepts, or from procedures [Harré] |
| 15870 | Are laws of nature about events, or types and universals, or dispositions, or all three? [Harré] |
| 15871 | Are laws about what has or might happen, or do they also cover all the possibilities? [Harré] |
| 15876 | Maybe laws of nature are just relations between properties? [Harré] |
| 15860 | We take it that only necessary happenings could be laws [Harré] |
| 15872 | Must laws of nature be universal, or could they be local? [Harré] |
| 15867 | Laws describe abstract idealisations, not the actual mess of nature [Harré] |
| 15892 | Laws of nature state necessary connections of things, events and properties, based on models of mechanisms [Harré] |
| 15875 | In counterfactuals we keep substances constant, and imagine new situations for them [Harré] |