201 ideas
| 23889 | Among the Greeks Aristotle is the only philosopher in the modern style [Weil] |
| 23881 | All thought about values is philosophical, and thought about anything else is not philosophy [Weil] |
| 23885 | Philosophy aims to change the soul, not to accumulate knowledge [Weil] |
| 23886 | Systems are not unique to each philosopher. The platonist tradition is old and continuous [Weil] |
| 24215 | We call experience 'objective' when it seems necessary [Weil] |
| 23884 | Truth is a value of thought [Weil] |
| 23877 | Most people won't question an idea's truth if they depend on it [Weil] |
| 23755 | Genius and love of truth are always accompanied by great humility [Weil] |
| 23825 | We seek truth only because it is good [Weil] |
| 23853 | Truth is not a object we love - it is the radiant manifestation of reality [Weil] |
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| 10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
| 10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
| 10605 | Two functions are the same if they have the same extension [Smith,P] |
| 10075 | A 'partial function' maps only some elements to another set [Smith,P] |
| 10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
| 10074 | A 'total function' maps every element to one element in another set [Smith,P] |
| 10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
| 10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
| 10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
| 10613 | No nice theory can define truth for its own language [Smith,P] |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| 10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
| 10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
| 10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
| 10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P] |
| 10598 | A theory is 'negation complete' if it proves all sentences or their negation [Smith,P] |
| 10597 | 'Complete' applies both to whole logics, and to theories within them [Smith,P] |
| 10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
| 10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P] |
| 10087 | A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P] |
| 10088 | Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P] |
| 10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
| 10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
| 10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
| 10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |
| 10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
| 10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
| 10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
| 10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
| 10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
| 10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
| 10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
| 10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
| 10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
| 10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
| 10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
| 10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
| 10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| 24189 | The criterion of the real is contradictions [Weil] |
| 23855 | Creation produced a network or web of determinations [Weil] |
| 24208 | Bodies classify things prior to thought (such as chicks knowing what hits of the egg to peck) [Weil] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 23900 | Chance is compatible with necessity, and the two occur together [Weil] |
| 24194 | Wanting new discoveries blocks good thinking about what has been discovered [Weil] |
| 24195 | Don't reject opinions; arrange them all in a hierarchy [Weil] |
| 23888 | Knowledge is beyond question, as an unavoidable component of thinking [Weil] |
| 24209 | Senses are unaware of each other, and give isolated information [Weil] |
| 24211 | Associations are not lawlike, because we make arbitrary choice of which representation matters [Weil] |
| 24214 | Pragmatists are right that science is action on nature - but it must be methodical [Weil] |
| 24213 | Explanations always concern how one thing changes into another [Weil] |
| 24210 | Abstraction is just the character of generalisation [Weil] |
| 24212 | We don't infer the straight from the twisted, because judging the twisted needs the straight [Weil] |
| 23747 | What is sacred is not a person, but the whole physical human being [Weil] |
| 24207 | Observing oneself in the present is impossible, and oneself in the past may be wrong [Weil] |
| 24182 | We must be obedient, and love necessity [Weil] |
| 23756 | The mind is imprisoned and limited by language, restricting our awareness of wider thoughts [Weil] |
| 24177 | Higher emotions have less energy, and actions may need the lower emotions [Weil] |
| 23878 | Weakness of will is the inadequacy of the original impetus to carry through the action [Weil] |
| 24184 | What matters about an action is not its aim, but the origin of its compulsion [Weil] |
| 23899 | The secret of art is that beauty is a just blend of unity and its opposite [Weil] |
| 23832 | We both desire what is beautiful, and want it to remain as it is [Weil] |
| 23848 | The aesthete's treatment of beauty as amusement is sacreligious; beauty should nourish [Weil] |
| 23758 | Beauty is an attractive mystery, leaving nothing to be desired [Weil] |
| 23887 | Art (like philosophy) establishes a relation between world and self, and between oneself and others [Weil] |
| 23903 | When we admire a work, we see ourselves as its creator [Weil] |
| 24198 | Perfect works of art seem to be essentially anonymous [Weil] |
| 23898 | Those who say immorality is not an aesthetic criterion must show that all criteria are aesthetic [Weil] |
| 24216 | Everyone is devoted to morality, if they don't have to implement it [Weil] |
| 23854 | Beauty is the proof of what is good [Weil] |
| 23826 | Beauty, goodness and truth are only achieved by applying full attention [Weil] |
| 23814 | Every human yearns for an unattainable transcendent good [Weil] |
| 23824 | Where human needs are satisfied we find happiness, friendship and beauty [Weil] |
| 23879 | In a violent moral disagreement, it can't be that both sides are just following social morality [Weil] |
| 24191 | We want our values to be eternal [Weil] |
| 23882 | Ends, unlike means, cannot be defined, which is why people tend to pursue means [Weil] |
| 24197 | Power and money are supreme means, thus blinding people to ends [Weil] |
| 23760 | All we need are the unity of justice, truth and beauty [Weil] |
| 23883 | Minds essentially and always strive towards value [Weil] |
| 23748 | The sacred in every human is their expectation of good rather than evil [Weil] |
| 24181 | We need love to have a good death [Weil] |
| 23759 | Everything which originates in love is beautiful [Weil] |
| 24179 | We should never desire the immortality of the people we love [Weil] |
| 23762 | Evil is transmitted by comforts and pleasures, but mostly by doing harm to people [Weil] |
| 24193 | If we focus on the good, our whole soul is drawn towards it [Weil] |
| 23808 | There are two goods - the absolute good we want, and the reachable opposite of evil [Weil] |
| 23833 | The good is a nothingness, and yet real [Weil] |
| 23865 | Morality would improve if people could pursue private interests [Weil] |
| 24196 | Loving others as ourselves implies varied love, and varied suffering [Weil] |
| 23896 | We see our character as a restricting limit, but also as an unshakable support [Weil] |
| 23893 | We don't see character in a single moment, but only over a period of time [Weil] |
| 23894 | The concept of character is at the centre of morality [Weil] |
| 23895 | We modify our character by placing ourselves in situations, or by attending to what seems trivial [Weil] |
| 24183 | We should only perform the good actions which we can't help doing [Weil] |
| 23837 | Respect is our only obligation, which can only be expressed through deeds, not words [Weil] |
| 23815 | We cannot equally respect what is unequal, so equal respect needs a shared ground [Weil] |
| 24185 | Friendship is a virtue, not a state we should dream of [Weil] |
| 23834 | Friendship is partly universal - the love of a person is like the ideal of loving everyone [Weil] |
| 24219 | My neighbour's pleasure can't be an end for me [Weil] |
| 24188 | It is absurd to say that evil proves life is worthless. If it were, why would evil matter? [Weil] |
| 24205 | Monotony is beautiful as a reflection of eternity, or atrocious as unvarying perpetuity [Weil] |
| 23823 | Life needs risks to avoid sickly boredom [Weil] |
| 23844 | The most important human need is to have multiple roots [Weil] |
| 23838 | The need for order stands above all others, and is understood via the other needs [Weil] |
| 23836 | Obligations only bind individuals, not collectives [Weil] |
| 24202 | Obedience to an illegitimate ruler is a nightmare [Weil] |
| 24218 | People can't be citizens in public life if they are oppressed in economic life [Weil] |
| 23840 | A citizen should be able to understand the whole of society [Weil] |
| 23843 | Even the poorest should feel collective ownership, and participation in grand display [Weil] |
| 23822 | We all need to partipate in public tasks, and take some initiative [Weil] |
| 24200 | A citizen is defined by their subjection to the laws [Weil] |
| 23846 | Culture is an instrument for creating an ongoing succession of teachers [Weil] |
| 23866 | In oppressive societies the scope of actual control is extended by a religion of power [Weil] |
| 23812 | Force is what turns man into a thing, and ultimately into a corpse [Weil] |
| 24201 | Social order is equilibrium of forces, which must be corrected when imbalanced [Weil] |
| 24199 | There is no oppression, or oppressive class; there is only an oppressive society [Weil] |
| 23831 | The essence of power is illusory prestige [Weil] |
| 23857 | People in power always try to increase their power [Weil] |
| 23839 | A lifelong head of society should only be a symbol, not a ruler [Weil] |
| 23871 | No central authority can initiate decentralisation [Weil] |
| 23856 | Spontaneous movements are powerless against organised repression [Weil] |
| 23867 | After a bloody revolution the group which already had the power comes to the fore [Weil] |
| 24203 | Atheistic materialism must be revolutionary, because its good is in the future [Weil] |
| 23830 | A group is only dangerous if it endorses an abstract entity [Weil] |
| 23870 | Decentralisation is only possible by co-operation between strong and weak - which is absurd [Weil] |
| 23809 | Our only social duty is to try to limit evil [Weil] |
| 24190 | Anarchists thought (hopelessly) that empowering the oppressed would end evil [Weil] |
| 23817 | We need both equality (to attend to human needs) and hierarchy (as a scale of responsibilities) [Weil] |
| 23829 | National leaders want to preserve necessary order - but always the existing order [Weil] |
| 23842 | Party politics in a democracy can't avoid an anti-democratic party [Weil] |
| 23859 | True democracy is the subordination of society to the individual [Weil] |
| 23863 | Only individual people of good will can achieve social progress [Weil] |
| 23869 | In the least evil societies people can think, control community life, and be autonomous [Weil] |
| 23847 | Socialism tends to make a proletariat of the whole population [Weil] |
| 23750 | It is not more money which the wretched members of society need [Weil] |
| 23749 | The problem of the collective is not suppression of persons, but persons erasing themselves [Weil] |
| 23807 | The collective is the one and only object of false idolatry [Weil] |
| 23861 | Marx showed that capitalist oppression, because of competition, is unstoppable [Weil] |
| 23897 | Once money is the main aim, society needs everyone to think wealth is possible [Weil] |
| 23845 | The capitalists neglect the people and the nation, and even their own interests [Weil] |
| 23828 | National prestige consists of behaving as if you could beat the others in a war [Weil] |
| 23810 | Charity is the only love, and you can feel that for a country (a place with traditions), but not a nation [Weil] |
| 23811 | If effort is from necessity rather than for a good, it is slavery [Weil] |
| 23868 | The pleasure of completing tasks motivates just as well as the whip of slavery [Weil] |
| 23819 | Deliberate public lying should be punished [Weil] |
| 23818 | We have liberty in the space between nature and accepted authority [Weil] |
| 23901 | Relationships depend on equality, so unequal treatment kills them [Weil] |
| 23753 | People absurdly claim an equal share of things which are essentially privileged [Weil] |
| 24221 | Equality is the result of unlimited freedom [Weil] |
| 23841 | By making money the sole human measure, inequality has become universal [Weil] |
| 23864 | Inequality could easily be mitigated, if it were not for the struggle for power [Weil] |
| 23751 | Rights are asserted contentiously, and need the backing of force [Weil] |
| 23752 | Giving centrality to rights stifles all impulses of charity [Weil] |
| 23835 | People have duties, and only have rights because of the obligations of others to them [Weil] |
| 23820 | People need personal and collective property, and a social class lacking property is shameful [Weil] |
| 23813 | Only people who understand force, and don't respect it, are capable of justice [Weil] |
| 23757 | The spirit of justice needs the full attention of truth, and that attention is love [Weil] |
| 23761 | Justice (concerning harm) is distinct from rights (concerning inequality) [Weil] |
| 23852 | To punish people we must ourselves be innocent - but that undermines the desire to punish [Weil] |
| 23764 | The only thing in society worse than crime is repressive justice [Weil] |
| 23821 | Crime should be punished, to bring the perpetrator freely back to morality [Weil] |
| 23763 | Punishment aims at the good for men who don't desire it [Weil] |
| 23827 | Modern wars are fought in the name of empty words which are given capital letters [Weil] |
| 23880 | When war was a profession, customary morality justified any act of war [Weil] |
| 23850 | The soldier-civilian distinction should be abolished; every citizen is committed to a war [Weil] |
| 23858 | War is perpetuated by its continual preparations [Weil] |
| 23851 | Education is essentially motivation [Weil] |
| 23873 | Dividing history books into separate chapters is disastrous [Weil] |
| 24217 | History is scientific when it relies on accurate documents [Weil] |
| 23860 | Even if a drowning man is doomed, he should keep swimming to the last [Weil] |
| 6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
| 24204 | The past is known to us but unreachable - a perfect image of eternal, supernatural reality [Weil] |
| 24222 | If we ignore all our thoughts of the past and the future, there is nothing left of the present [Weil] |
| 24220 | As the highest value, God cannot be proved [Weil] |
| 23754 | The only choice is between supernatural good, or evil [Weil] |
| 23816 | Attention to a transcendent reality motivates a duty to foster the good of humanity [Weil] |
| 24192 | My love makes me believe in God; the inconceivability of this God makes me disbelieve [Weil] |
| 23892 | The only legitimate proof of God by order derives from beauty [Weil] |
| 23904 | The cruelty of the Old Testament put me off Christianity [Weil] |
| 24178 | We must leave on one side the ordinary 'consolations' of religion [Weil] |
| 24206 | Revolution (not religion) is the opium of the people [Weil] |
| 23849 | Religion should quietly suffuse all human life with its light [Weil] |
| 23902 | I attach little importance to immortality, which is an undecidable fact, and irrelevant to us [Weil] |
| 24180 | We just see immortality as prolongation of life, making death meaningless [Weil] |
| 23765 | The soul is the intrinsic value of a human [Weil] |
| 24186 | If the world lacked evil, then the evil would be in our desires, which would be worse [Weil] |
| 24187 | Without worldly affliction, we'd think this is paradise [Weil] |