69 ideas
14888 | Wisdom prevents us from being ruled by the moment [Nietzsche] |
14863 | Unlike science, true wisdom involves good taste [Nietzsche] |
14890 | Suffering is the meaning of existence [Nietzsche] |
14861 | Philosophy ennobles the world, by producing an artistic conception of our knowledge [Nietzsche] |
14887 | You should only develop a philosophy if you are willing to live by it [Nietzsche] |
14885 | The first aim of a philosopher is a life, not some works [Nietzsche] |
14889 | Philosophy is pointless if it does not advocate, and live, a new way of life [Nietzsche] |
14862 | Philosophy is more valuable than much of science, because of its beauty [Nietzsche] |
14878 | It would better if there was no thought [Nietzsche] |
14881 | Why do people want philosophers? [Nietzsche] |
14876 | Philosophy is always secondary, because it cannot support a popular culture [Nietzsche] |
14860 | Kant has undermined our belief in metaphysics [Nietzsche] |
14859 | If philosophy controls science, then it has to determine its scope, and its value [Nietzsche] |
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] |
15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
15920 | Pure collections of things obey Choice, but collections defined by a rule may not [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] |
9406 | A class is natural when everybody can spot further members of it [Quinton] |
14880 | Logic is just slavery to language [Nietzsche] |
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] |
15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [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] |
14869 | If some sort of experience is at the root of matter, then human knowledge is close to its essence [Nietzsche] |
15730 | Extreme nominalists say all classification is arbitrary convention [Quinton] |
15728 | The naturalness of a class depends as much on the observers as on the objects [Quinton] |
9407 | Properties imply natural classes which can be picked out by everybody [Quinton] |
15729 | Uninstantiated properties must be defined using the instantiated ones [Quinton] |
8520 | An individual is a union of a group of qualities and a position [Quinton, by Campbell,K] |
14875 | Belief matters more than knowledge, and only begins when knowledge ceases [Nietzsche] |
14866 | It always remains possible that the world just is the way it appears [Nietzsche] |
14872 | Our knowledge is illogical, because it rests on false identities between things [Nietzsche] |
14879 | The most extreme scepticism is when you even give up logic [Nietzsche] |
14873 | If we find a hypothesis that explains many things, we conclude that it explains everything [Nietzsche] |
14868 | Our primary faculty is perception of structure, as when looking in a mirror [Nietzsche] |
14870 | We experience causation between willing and acting, and thereby explain conjunctions of changes [Nietzsche] |
14867 | It is just madness to think that the mind is supernatural (or even divine!) [Nietzsche] |
14884 | The shortest path to happiness is forgetfulness, the path of animals (but of little value) [Nietzsche] |
14886 | Education is contrary to human nature [Nietzsche] |
14883 | We should evaluate the past morally [Nietzsche] |
14882 | Protest against vivisection - living things should not become objects of scientific investigation [Nietzsche] |
14865 | We do not know the nature of one single causality [Nietzsche] |
14871 | Laws of nature are merely complex networks of relations [Nietzsche] |
14864 | The Greeks lack a normative theology: each person has their own poetic view of things [Nietzsche] |