158 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] |
17892 | For clear questions posed by reason, reason can also find clear answers [Gödel] |
10041 | Impredicative Definitions refer to the totality to which the object itself belongs [Gödel] |
23884 | Truth is a value of thought [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] |
23877 | Most people won't question an idea's truth if they depend on it [Weil] |
21752 | Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine] |
13913 | The four 'perfect syllogisms' are called Barbara, Celarent, Darii and Ferio [Engelbretsen/Sayward] |
13914 | Syllogistic logic has one rule: what is affirmed/denied of wholes is affirmed/denied of their parts [Engelbretsen/Sayward] |
13915 | Syllogistic can't handle sentences with singular terms, or relational terms, or compound sentences [Engelbretsen/Sayward] |
13916 | Term logic uses expression letters and brackets, and '-' for negative terms, and '+' for compound terms [Engelbretsen/Sayward] |
17751 | Gödel proved the completeness of first order predicate logic in 1930 [Gödel, by Walicki] |
8679 | We perceive the objects of set theory, just as we perceive with our senses [Gödel] |
17835 | Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M] |
9942 | Gödel proved the classical relative consistency of the axiom V = L [Gödel, by Putnam] |
21716 | In simple type theory the axiom of Separation is better than Reducibility [Gödel, by Linsky,B] |
13850 | In modern logic all formal validity can be characterised syntactically [Engelbretsen/Sayward] |
13849 | Classical logic rests on truth and models, where constructivist logic rests on defence and refutation [Engelbretsen/Sayward] |
9188 | Gödel proved that first-order logic is complete, and second-order logic incomplete [Gödel, by Dummett] |
10035 | Mathematical Logic is a non-numerical branch of mathematics, and the supreme science [Gödel] |
13851 | Unlike most other signs, = cannot be eliminated [Engelbretsen/Sayward] |
10042 | Reference to a totality need not refer to a conjunction of all its elements [Gödel] |
10620 | Originally truth was viewed with total suspicion, and only demonstrability was accepted [Gödel] |
17886 | The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner] |
10071 | Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P] |
19123 | If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh] |
13852 | Axioms are ω-incomplete if the instances are all derivable, but the universal quantification isn't [Engelbretsen/Sayward] |
17883 | Gödel's Theorems did not refute the claim that all good mathematical questions have answers [Gödel, by Koellner] |
10621 | Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel] |
17888 | The undecidable sentence can be decided at a 'higher' level in the system [Gödel] |
10038 | A logical system needs a syntactical survey of all possible expressions [Gödel] |
18062 | Set-theory paradoxes are no worse than sense deception in physics [Gödel] |
10132 | There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman] |
10046 | The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers [Gödel] |
10868 | The Continuum Hypothesis is not inconsistent with the axioms of set theory [Gödel, by Clegg] |
13517 | If set theory is consistent, we cannot refute or prove the Continuum Hypothesis [Gödel, by Hart,WD] |
17885 | Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable [Gödel, by Koellner] |
10614 | The real reason for Incompleteness in arithmetic is inability to define truth in a language [Gödel] |
10072 | First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P] |
9590 | Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman] |
3198 | Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey] |
11069 | Gödel's Second says that semantic consequence outruns provability [Gödel, by Hanna] |
10118 | First Incompleteness: a decent consistent system is syntactically incomplete [Gödel, by George/Velleman] |
10122 | Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by George/Velleman] |
10611 | There is a sentence which a theory can show is true iff it is unprovable [Gödel, by Smith,P] |
10867 | 'This system can't prove this statement' makes it unprovable either way [Gödel, by Clegg] |
10039 | Some arithmetical problems require assumptions which transcend arithmetic [Gödel] |
10043 | Mathematical objects are as essential as physical objects are for perception [Gödel] |
10271 | Basic mathematics is related to abstract elements of our empirical ideas [Gödel] |
10045 | Impredicative definitions are admitted into ordinary mathematics [Gödel] |
8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro] |
23855 | Creation produced a network or web of determinations [Weil] |
23900 | Chance is compatible with necessity, and the two occur together [Weil] |
23888 | Knowledge is beyond question, as an unavoidable component of thinking [Weil] |
23747 | What is sacred is not a person, but the whole physical human being [Weil] |
3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
23756 | The mind is imprisoned and limited by language, restricting our awareness of wider thoughts [Weil] |
23878 | Weakness of will is the inadequacy of the original impetus to carry through the action [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] |
23899 | The secret of art is that beauty is a just blend of unity and its opposite [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] |
23898 | Those who say immorality is not an aesthetic criterion must show that all criteria are aesthetic [Weil] |
23854 | Beauty is the proof of what is good [Weil] |
23814 | Every human yearns for an unattainable transcendent good [Weil] |
23826 | Beauty, goodness and truth are only achieved by applying full attention [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] |
23882 | Ends, unlike means, cannot be defined, which is why people tend to pursue means [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] |
23759 | Everything which originates in love is beautiful [Weil] |
23762 | Evil is transmitted by comforts and pleasures, but mostly by doing harm to people [Weil] |
23833 | The good is a nothingness, and yet real [Weil] |
23808 | There are two goods - the absolute good we want, and the reachable opposite of evil [Weil] |
23865 | Morality would improve if people could pursue private interests [Weil] |
23894 | The concept of character is at the centre of morality [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] |
23895 | We modify our character by placing ourselves in situations, or by attending to what seems trivial [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] |
23834 | Friendship is partly universal - the love of a person is like the ideal of loving everyone [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] |
23840 | A citizen should be able to understand the whole of society [Weil] |
23822 | We all need to partipate in public tasks, and take some initiative [Weil] |
23843 | Even the poorest should feel collective ownership, and participation in grand display [Weil] |
23846 | Culture is an instrument for creating an ongoing succession of teachers [Weil] |
23831 | The essence of power is illusory prestige [Weil] |
23857 | People in power always try to increase their power [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] |
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] |
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] |
23829 | National leaders want to preserve necessary order - but always the existing order [Weil] |
23817 | We need both equality (to attend to human needs) and hierarchy (as a scale of responsibilities) [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] |
23750 | It is not more money which the wretched members of society need [Weil] |
23847 | Socialism tends to make a proletariat of the whole population [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] |
23868 | The pleasure of completing tasks motivates just as well as the whip of slavery [Weil] |
23811 | If effort is from necessity rather than for a good, it is 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] |
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] |
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] |
23764 | The only thing in society worse than crime is repressive justice [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] |
23860 | Even if a drowning man is doomed, he should keep swimming to the last [Weil] |
23816 | Attention to a transcendent reality motivates a duty to foster the good of humanity [Weil] |
23754 | The only choice is between supernatural good, or evil [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] |
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] |
23765 | The soul is the intrinsic value of a human [Weil] |