123 ideas
| 23890 | For Plato true wisdom is supernatural [Plato, by Weil] |
| 20262 | Don't use wisdom in order to become clever! [Nietzsche] |
| 3060 | Plato never mentions Democritus, and wished to burn his books [Plato, by Diog. Laertius] |
| 13466 | We are all post-Kantians, because he set the current agenda for philosophy [Hart,WD] |
| 20255 | Early 19th century German philosophers enjoyed concepts, rather than scientific explanations [Nietzsche] |
| 20260 | Carlyle spent his life vainly trying to make reason appear romantic [Nietzsche] |
| 13477 | The problems are the monuments of philosophy [Hart,WD] |
| 20256 | What we think is totally dictated by the language available to express it [Nietzsche] |
| 20265 | The desire for a complete system requires making the weak parts look equal to the rest [Nietzsche] |
| 13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
| 23891 | Two contradictories force us to find a relation which will correlate them [Plato, by Weil] |
| 20380 | Why should truth be omnipotent? It is enough that it is very powerful [Nietzsche] |
| 20235 | Like animals, we seek truth because we want safety [Nietzsche] |
| 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] |
| 13461 | We can choose from finite and evident sets, but not from infinite opaque ones [Hart,WD] |
| 13462 | With the Axiom of Choice every set can be well-ordered [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] |
| 13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD] |
| 13458 | A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD] |
| 13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [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] |
| 13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 [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] |
| 13513 | Models are ways the world might be from a first-order point of view [Hart,WD] |
| 13496 | First-order logic is 'compact': consequences of a set are consequences of a finite subset [Hart,WD] |
| 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] |
| 13491 | The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD] |
| 13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) [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] |
| 13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
| 13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
| 14502 | Plato's idea of 'structure' tends to be mathematically expressed [Plato, by Koslicki] |
| 17948 | Plato's Forms meant that the sophists only taught the appearance of wisdom and virtue [Plato, by Nehamas] |
| 20906 | Platonists argue for the indivisible triangle-in-itself [Plato, by Aristotle] |
| 3039 | When Diogenes said he could only see objects but not their forms, Plato said it was because he had eyes but no intellect [Plato, by Diog. Laertius] |
| 556 | If there is one Form for both the Form and its participants, they must have something in common [Aristotle on Plato] |
| 563 | If gods are like men, they are just eternal men; similarly, Forms must differ from particulars [Aristotle on Plato] |
| 565 | The Forms cannot be changeless if they are in changing things [Aristotle on Plato] |
| 557 | A Form is a cause of things only in the way that white mixed with white is a cause [Aristotle on Plato] |
| 9607 | The greatest discovery in human thought is Plato's discovery of abstract objects [Brown,JR on Plato] |
| 13263 | We can grasp whole things in science, because they have a mathematics and a teleology [Plato, by Koslicki] |
| 13261 | Plato sees an object's structure as expressible in mathematics [Plato, by Koslicki] |
| 13265 | Plato was less concerned than Aristotle with the source of unity in a complex object [Plato, by Koslicki] |
| 593 | Plato's holds that there are three substances: Forms, mathematical entities, and perceptible bodies [Plato, by Aristotle] |
| 13260 | Plato says wholes are either containers, or they're atomic, or they don't exist [Plato, by Koslicki] |
| 11237 | Only universals have essence [Plato, by Politis] |
| 11238 | Plato and Aristotle take essence to make a thing what it is [Plato, by Politis] |
| 20258 | Most people treat knowledge as a private possession [Nietzsche] |
| 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] |
| 20250 | We may be unable to remember, but we may never actually forget [Nietzsche] |
| 20270 | There is no one scientific method; we must try many approaches, and many emotions [Nietzsche] |
| 17085 | A good explanation totally rules out the opposite explanation (so Forms are required) [Plato, by Ruben] |
| 20131 | We can cultivate our drives, of anger, pity, curiosity, vanity, like a gardener, with good or bad taste [Nietzsche] |
| 20242 | Things are the boundaries of humanity, so all things must be known, for self-knowledge [Nietzsche] |
| 20249 | Our knowledge of the many drives that constitute us is hopelessly incomplete [Nietzsche] |
| 20231 | People used to think that outcomes were from God, rather than consequences of acts [Nietzsche] |
| 1651 | Plato wanted to somehow control and purify the passions [Vlastos on Plato] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 3324 | Plato's whole philosophy may be based on being duped by reification - a figure of speech [Benardete,JA on Plato] |
| 20266 | It is essential that wise people learn to express their wisdom, possibly even as foolishness [Nietzsche] |
| 20251 | Actions done for a purpose are least understood, because we complacently think it's obvious [Nietzsche] |
| 20271 | Beauty in art is the imitation of happiness [Nietzsche] |
| 20230 | The very idea of a critique of morality is regarded as immoral! [Nietzsche] |
| 20234 | Morality prevents us from developing better customs [Nietzsche] |
| 7503 | Plato never refers to examining the conscience [Plato, by Foucault] |
| 20237 | Moral feelings are entirely different from the moral concepts used to judge actions [Nietzsche] |
| 20238 | Treating morality as feelings is just obeying your ancestors [Nietzsche] |
| 2173 | As religion and convention collapsed, Plato sought morals not just in knowledge, but in the soul [Williams,B on Plato] |
| 20243 | Human beings are not majestic, either through divine origins, or through grand aims [Nietzsche] |
| 20268 | Most dying people have probably lost more important things than what they are about to lose [Nietzsche] |
| 20252 | Marriage is too serious to be permitted for people in love! [Nietzsche] |
| 20263 | Fear reveals the natures of other people much more clearly than love does [Nietzsche] |
| 20236 | Marriage upholds the idea that love, though a passion, can endure [Nietzsche] |
| 9274 | Plato's legacy to European thought was the Good, the Beautiful and the True [Plato, by Gray] |
| 94 | Pleasure is better with the addition of intelligence, so pleasure is not the good [Plato, by Aristotle] |
| 20233 | Punishment has distorted the pure innocence of the contingency of outcomes [Nietzsche] |
| 17947 | Plato decided that the virtuous and happy life was the philosophical life [Plato, by Nehamas] |
| 20248 | People do nothing for their real ego, but only for a phantom ego created by other people [Nietzsche] |
| 20246 | If you feel to others as they feel to themselves, you must hate a self-hater [Nietzsche] |
| 6015 | Plato, unusually, said that theoretical and practical wisdom are inseparable [Plato, by Kraut] |
| 20272 | Honesty is a new young virtue, and we can promote it, or not [Nietzsche] |
| 20240 | The Jews treated great anger as holy, and were in awe of those who expressed it [Nietzsche] |
| 20244 | Christianity replaces rational philosophical virtues with great passions focused on God [Nietzsche] |
| 20274 | The cardinal virtues want us to be honest, brave, magnanimous and polite [Nietzsche] |
| 20257 | Cool courage and feverish bravery have one name, but are two very different virtues [Nietzsche] |
| 20259 | Teach youth to respect people who differ with them, not people who agree with them [Nietzsche] |
| 20267 | Seeing duty as a burden makes it a bit cruel, and it can thus never become a habit [Nietzsche] |
| 2912 | Plato is boring [Nietzsche on Plato] |
| 20275 | Most people think they are already complete, but we can cultivate ourselves [Nietzsche] |
| 20229 | No authority ever willingly accepts criticism [Nietzsche] |
| 20254 | People govern for the pleasure of it, or just to avoid being governed [Nietzsche] |
| 20273 | The French Revolution gave trusting Europe the false delusion of instant recovery [Nietzsche] |
| 20232 | Get rid of the idea of punishment! It is a noxious weed! [Nietzsche] |
| 20253 | Modern wars arise from the study of history [Nietzsche] |
| 20261 | History does not concern what really happened, but supposed events, which have all the influence [Nietzsche] |
| 1526 | Almost everyone except Plato thinks that time could not have been generated [Plato, by Aristotle] |
| 20241 | Enquirers think finding our origin is salvation, but it turns out to be dull [Nietzsche] |
| 20245 | Christianity hoped for a short cut to perfection, that skipped the hard labour of morality [Nietzsche] |
| 20247 | Christianity was successful because of its heathen rituals [Nietzsche] |
| 20269 | 'I believe because it is absurd' - but how about 'I believe because I am absurd' [Nietzsche] |
| 20264 | The easy and graceful aspects of a person are called 'soul', and inner awkwardness is called 'soulless' [Nietzsche] |