87 ideas
2945 | Most philosophers start with reality and then examine knowledge; Descartes put the study of knowledge first [Lehrer] |
4171 | Philosophy considers only the universal, in nature as everywhere else [Schopenhauer] |
4186 | Everyone is conscious of all philosophical truths, but philosophers bring them to conceptual awareness [Schopenhauer] |
12171 | Absurdity is incongruity between correct and false points of view [Schopenhauer] |
21366 | Metaphysics must understand the world thoroughly, as a principal source of knowledge [Schopenhauer] |
2946 | You cannot demand an analysis of a concept without knowing the purpose of the analysis [Lehrer] |
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] |
10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
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] |
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] |
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] |
17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
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] |
15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
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] |
8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
4168 | Matter and intellect are inseparable correlatives which only exist relatively, and for each other [Schopenhauer] |
21926 | Schopenhauer, unlike other idealists, says reality is irrational [Schopenhauer, by Lewis,PB] |
4167 | The knowing subject and the crude matter of the world are both in themselves unknowable [Schopenhauer] |
4165 | Descartes found the true beginning of philosophy with the Cogito, in the consciousness of the individual [Schopenhauer] |
21923 | Schopenhauer can't use force/energy instead of 'will', because he is not a materialist [Lewis,PB on Schopenhauer] |
4162 | The world only exists in relation to something else, as an idea of the one who conceives it [Schopenhauer] |
21922 | We know reality because we know our own bodies and actions [Schopenhauer] |
21913 | Kant rightly separates appearance and thing-in-itself [Schopenhauer] |
4164 | Direct feeling of the senses are merely data; perception of the world comes with understanding causes [Schopenhauer] |
4163 | All perception is intellectual [Schopenhauer] |
4166 | A consciousness without an object is no consciousness [Schopenhauer] |
21369 | We have hidden and unadmitted desires and fears, suppressed because of vanity [Schopenhauer] |
21367 | I know both aspects of my body, as representation, and as will [Schopenhauer] |
4175 | It is as perverse to resent our individuality being replaced by others, as to resent the body renewing itself [Schopenhauer] |
4176 | We all regard ourselves a priori as free, but see from experience that character and motive compel us [Schopenhauer] |
4170 | Man's actions are not free, because they follow strictly from impact of motive on character [Schopenhauer] |
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] |
4169 | Every true act of will is also at once and without exception a movement of the body [Schopenhauer] |
7187 | Schopenhauer was caught in Christian ideals, because he didn't deify his 'will' [Nietzsche on Schopenhauer] |
21365 | Only the will is thing-in-itself, seen both in blind nature and in human action [Schopenhauer] |
4173 | If we were essentially intellect rather than will, our moral worth would depend on imagined motives [Schopenhauer] |
21370 | Schopenhauer is a chief proponent of aesthetic experience as 'disinterested' [Schopenhauer, by Janaway] |
4182 | A principal pleasure of the beautiful is that it momentarily silences the will [Schopenhauer] |
21928 | The Sublime fights for will-less knowing, when faced with a beautiful threat to humanity [Schopenhauer, by Lewis,PB] |
21927 | Schopenhauer emphasises Ideas in art, unlike most romantics [Schopenhauer, by Lewis,PB] |
8116 | The will-less contemplation of art brings a liberation from selfhood [Schopenhauer, by Gardner] |
4174 | Man is more beautiful than anything else, and the loftiest purpose of art is to reveal his nature [Schopenhauer] |
21380 | The only aim of our existence is to grasp that non-existence would be better [Schopenhauer] |
21374 | We should no more expect ethical theory to produce good people than aesthetics to produce artists [Schopenhauer] |
4181 | Every good is essentially relative, for it has its essential nature only in its relation to a desiring will [Schopenhauer] |
5649 | Will casts aside each of its temporary fulfilments, so human life has no ultimate aim [Schopenhauer, by Scruton] |
4177 | Most people would probably choose non-existence at the end of their life, rather than relive the whole thing [Schopenhauer] |
4185 | Altruistic people make less distinction than usual between themselves and others [Schopenhauer] |
4183 | Only self-love can motivate morality, but that also makes it worthless [Schopenhauer] |
4172 | Happiness is the swift movement from desire to satisfaction, and then again on to desire [Schopenhauer] |
21371 | We can never attain happiness while our will is pursuing desires [Schopenhauer] |
4184 | Virtue must spring from an intuitive recognition that other people are essentially like us [Schopenhauer] |
4179 | The essence of nature is the will to life itself [Schopenhauer] |
10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
4178 | Christianity is a pessimistic religion, in which the world is equated with evil [Schopenhauer] |
4180 | Religion is the mythical clothing of the truth which is inaccessible to the crude human intellect [Schopenhauer] |