68 ideas
23966 | The personal view can still be objective, so I call sciences 'impersonal', rather than objective [Goldie] |
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] |
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] |
21982 | I only wish I had such eyes as to see Nobody! It's as much as I can do to see real people. [Carroll,L] |
24005 | We know other's emotions by explanation, contagion, empathy, imagination, or sympathy [Goldie] |
24006 | Empathy and imagining don't ensure sympathy, and sympathy doesn't need them [Goldie] |
23978 | 'Having an emotion' differs from 'being emotional' [Goldie] |
23973 | Unlike moods, emotions have specific objects, though the difference is a matter of degree [Goldie] |
23974 | Emotional intentionality as belief and desire misses out the necessity of feelings [Goldie] |
23972 | A long lasting and evolving emotion is still seen as a single emotion, such as love [Goldie] |
23992 | Some Aborigines have fifteen different words for types of fear [Goldie] |
23979 | Emotional responses can reveal to us our values, which might otherwise remain hidden [Goldie] |
23976 | If we have a 'feeling towards' an object, that gives the recognition a different content [Goldie] |
23977 | When actions are performed 'out of' emotion, they appear to be quite different [Goldie] |
23980 | It is best to see emotions holistically, as embedded in a person's life narrative [Goldie] |
23982 | If emotions are 'towards' things, they can't be bodily feelings, which lack aboutness [Goldie] |
24001 | Moods can focus as emotions, and emotions can blur into moods [Goldie] |
23968 | If reasons are seen impersonally (as just causal), then feelings are an irrelevant extra [Goldie] |
23969 | We have feelings of which we are hardly aware towards things in the world [Goldie] |
23984 | An emotion needs episodes of feeling, but not continuously [Goldie] |
23970 | Emotions are not avocado pears, with a rigid core and changeable surface [Goldie] |
23985 | A basic emotion is the foundation of a hierarchy, such as anger for types of annoyance [Goldie] |
23986 | Early Chinese basic emotions: joy, anger, sadness, fear, love, disliking, and liking [Goldie] |
23991 | Cross-cultural studies of facial expressions suggests seven basic emotions [Goldie] |
23967 | Some emotions are direct responses, and neither rational nor irrational [Goldie] |
23971 | Emotional thought is not rational, but it can be intelligible [Goldie] |
23975 | Learning an evaluative property like 'dangerous' is also learning an emotion [Goldie] |
23983 | We call emotions 'passions' because they are not as controlled as we would like [Goldie] |
23999 | Emotional control is hard, but we are responsible for our emotions over long time periods [Goldie] |
23994 | Emotions are not easily changed, as new knowledge makes little difference, and akrasia is possible [Goldie] |
23998 | Emotional control is less concerned with emotional incidents, and more with emotional tendencies [Goldie] |
23995 | Akrasia can be either overruling our deliberation, or failing to deliberate [Goldie] |
24000 | Justifying reasons say you were right; excusing reasons say your act was explicable [Goldie] |
24003 | Character traits are both possession of and lack of dispositions [Goldie] |
24002 | We over-estimate the role of character traits when explaining behaviour [Goldie] |
24004 | Psychologists suggest we are muddled about traits, and maybe they should be abandoned [Goldie] |
23993 | Our capabilities did not all evolve during the hunter gathering period [Goldie] |