124 ideas
| 13466 | We are all post-Kantians, because he set the current agenda for philosophy [Hart,WD] |
| 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] |
| 13477 | The problems are the monuments of philosophy [Hart,WD] |
| 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] |
| 13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
| 224 | When questions are doubtful we should concentrate not on objects but on ideas of the intellect [Plato] |
| 232 | Opposites are as unlike as possible [Plato] |
| 8937 | Plato's 'Parmenides' is the greatest artistic achievement of the ancient dialectic [Hegel on Plato] |
| 13469 | Tarski showed how we could have a correspondence theory of truth, without using 'facts' [Hart,WD] |
| 13504 | Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do [Hart,WD] |
| 13503 | A first-order language has an infinity of T-sentences, which cannot add up to a definition of truth [Hart,WD] |
| 13500 | Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent [Hart,WD] |
| 13502 | ∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...' [Hart,WD] |
| 13456 | Set theory articulates the concept of order (through relations) [Hart,WD] |
| 13497 | Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe [Hart,WD] |
| 13443 | ∈ relates across layers, while ⊆ relates within layers [Hart,WD] |
| 13442 | Without the empty set we could not form a∩b without checking that a and b meet [Hart,WD] |
| 13493 | In the modern view, foundation is the heart of the way to do set theory [Hart,WD] |
| 13495 | Foundation Axiom: an nonempty set has a member disjoint from it [Hart,WD] |
| 13462 | With the Axiom of Choice every set can be well-ordered [Hart,WD] |
| 13461 | We can choose from finite and evident sets, but not from infinite opaque ones [Hart,WD] |
| 13516 | If we accept that V=L, it seems to settle all the open questions of set theory [Hart,WD] |
| 13441 | Naïve set theory has trouble with comprehension, the claim that every predicate has an extension [Hart,WD] |
| 13494 | The iterative conception may not be necessary, and may have fixed points or infinitely descending chains [Hart,WD] |
| 13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD] |
| 13458 | A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD] |
| 13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD] |
| 13490 | Von Neumann defines α<β as α∈β [Hart,WD] |
| 13481 | Maybe sets should be rethought in terms of the even more basic categories [Hart,WD] |
| 13506 | The universal quantifier can't really mean 'all', because there is no universal set [Hart,WD] |
| 13513 | Models are ways the world might be from a first-order point of view [Hart,WD] |
| 13505 | Model theory studies how set theory can model sets of sentences [Hart,WD] |
| 13511 | Model theory is mostly confined to first-order theories [Hart,WD] |
| 13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 [Hart,WD] |
| 13496 | First-order logic is 'compact': consequences of a set are consequences of a finite subset [Hart,WD] |
| 13986 | Plato found antinomies in ideas, Kant in space and time, and Bradley in relations [Plato, by Ryle] |
| 14150 | Plato's 'Parmenides' is perhaps the best collection of antinomies ever made [Russell on Plato] |
| 13484 | Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that [Hart,WD] |
| 13482 | The Burali-Forti paradox is a crisis for Cantor's ordinals [Hart,WD] |
| 13507 | The machinery used to solve the Liar can be rejigged to produce a new Liar [Hart,WD] |
| 13459 | The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD] |
| 13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD] |
| 13491 | The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD] |
| 13492 | Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD] |
| 13446 | 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD] |
| 13509 | We can establish truths about infinite numbers by means of induction [Hart,WD] |
| 13474 | Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD] |
| 16150 | One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato] |
| 13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
| 4168 | Matter and intellect are inseparable correlatives which only exist relatively, and for each other [Schopenhauer] |
| 229 | The one was and is and will be and was becoming and is becoming and will become [Plato] |
| 21821 | Plato's Parmenides has a three-part theory, of Primal One, a One-Many, and a One-and-Many [Plato, by Plotinus] |
| 13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
| 221 | Absolute ideas, such as the Good and the Beautiful, cannot be known by us [Plato] |
| 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] |
| 223 | If you deny that each thing always stays the same, you destroy the possibility of discussion [Plato] |
| 227 | You must always mean the same thing when you utter the same name [Plato] |
| 210 | It would be absurd to think there were abstract Forms for vile things like hair, mud and dirt [Plato] |
| 220 | The concept of a master includes the concept of a slave [Plato] |
| 228 | Greatness and smallness must exist, to be opposed to one another, and come into being in things [Plato] |
| 211 | If admirable things have Forms, maybe everything else does as well [Plato] |
| 219 | If absolute ideas existed in us, they would cease to be absolute [Plato] |
| 16151 | Plato moves from Forms to a theory of genera and principles in his later work [Plato, by Frede,M] |
| 215 | If things partake of ideas, this implies either that everything thinks, or that everything actually is thought [Plato] |
| 212 | The whole idea of each Form must be found in each thing which participates in it [Plato] |
| 216 | If things are made alike by participating in something, that thing will be the absolute idea [Plato] |
| 218 | Participation is not by means of similarity, so we are looking for some other method of participation [Plato] |
| 213 | Each idea is in all its participants at once, just as daytime is a unity but in many separate places at once [Plato] |
| 214 | If absolute greatness and great things are seen as the same, another thing appears which makes them seem great [Plato] |
| 217 | Nothing can be like an absolute idea, because a third idea intervenes to make them alike (leading to a regress) [Plato] |
| 15851 | Parts must belong to a created thing with a distinct form [Plato] |
| 15846 | In Parmenides, if composition is identity, a whole is nothing more than its parts [Plato, by Harte,V] |
| 15849 | Plato says only a one has parts, and a many does not [Plato, by Harte,V] |
| 15850 | Anything which has parts must be one thing, and parts are of a one, not of a many [Plato] |
| 13259 | It seems that the One must be composed of parts, which contradicts its being one [Plato] |
| 15847 | Two things relate either as same or different, or part of a whole, or the whole of the part [Plato] |
| 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] |
| 13480 | Fregean self-evidence is an intrinsic property of basic truths, rules and definitions [Hart,WD] |
| 13476 | The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori [Hart,WD] |
| 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] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 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] |
| 222 | Only a great person can understand the essence of things, and an even greater person can teach it [Plato] |
| 4179 | The essence of nature is the will to life itself [Schopenhauer] |
| 225 | The unlimited has no shape and is endless [Plato] |
| 231 | Everything partakes of the One in some way [Plato] |
| 2062 | The only movement possible for the One is in space or in alteration [Plato] |
| 233 | Some things do not partake of the One [Plato] |
| 234 | We couldn't discuss the non-existence of the One without knowledge of it [Plato] |
| 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] |