73 ideas
1642 | We must fight fiercely for knowledge, understanding and intelligence [Plato] |
5278 | Philosophy is no more than abstractions concerning observations of human historical development [Marx/Engels] |
1645 | The desire to split everything into its parts is unpleasant and unphilosophical [Plato] |
287 | Good analysis involves dividing things into appropriate forms without confusion [Plato] |
1644 | Dialectic should only be taught to those who already philosophise well [Plato] |
20478 | In discussion a person's opinions are shown to be in conflict, leading to calm self-criticism [Plato] |
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] |
11278 | What does 'that which is not' refer to? [Plato] |
1643 | If statements about non-existence are logically puzzling, so are statements about existence [Plato] |
7022 | To be is to have a capacity, to act on other things, or to receive actions [Plato] |
1641 | Some alarming thinkers think that only things which you can touch exist [Plato] |
5287 | Philosophical problems are resolved into empirical facts [Marx/Engels] |
10784 | Whenever there's speech it has to be about something [Plato] |
16122 | Good thinkers spot forms spread through things, or included within some larger form [Plato] |
10422 | The not-beautiful is part of the beautiful, though opposed to it, and is just as real [Plato] |
15855 | If we see everything as separate, we can then give no account of it [Plato] |
1637 | A soul without understanding is ugly [Plato] |
23872 | 'Society determines consciousness' is contradictory; society only exists in minds [Weil on Marx/Engels] |
5277 | Life is not determined by consciousness, but consciousness by life [Marx/Engels] |
5280 | Language co-exists with consciousness, and makes it social [Marx/Engels] |
5276 | The nature of an individual coincides with what they produce and how they produce it [Marx/Engels] |
5281 | Consciousness is a social product [Marx/Engels] |
1636 | Wickedness is an illness of the soul [Plato] |
5289 | When aristocracy or the bourgeoisie dominate, certain values dominate with them [Marx/Engels] |
5274 | Young Hegelians proposed changing our present consciousness for liberating critical consciousness [Marx/Engels] |
21992 | Producing their own subsistence distinguishes men from animals [Marx/Engels] |
5275 | Men distinguish themselves from animals when they begin to produce their means of subsistence [Marx/Engels] |
5292 | Individuals are mutually hostile unless they group together in competition with other groups [Marx/Engels] |
5293 | Only in community are people able to cultivate their gifts, and therefore be free [Marx/Engels] |
5273 | Young Hegelians think consciousness is chains for men, where old Hegelians think it the bond of society [Marx/Engels] |
5283 | In communist society we are not trapped in one activity, but can act freely [Marx/Engels] |
5282 | If the common interest imposes on the individual, his actions become alienated and enslaving [Marx/Engels] |
5288 | The class controlling material production also controls mental production [Marx/Engels] |
5290 | The revolutionary class is opposed to 'class', and represents all of society [Marx/Engels] |
5294 | To assert themselves as individuals, the proletarians must overthrow the State [Marx/Engels] |
5285 | Slavery cannot be abolished without the steam-engine [Marx/Engels] |
5284 | Communism abolishes private property and dissolves the powerful world market [Marx/Engels] |
5291 | The law says private property is the result of the general will [Marx/Engels] |
1638 | Didactic education is hard work and achieves little [Plato] |
5279 | Human history must always be studied in relation to industry and exchange [Marx/Engels] |
5286 | Most historians are trapped in the illusions of their own epoch [Marx/Engels] |