171 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] |
| 8797 | The negation of all my beliefs about my current headache would be fully coherent [Sosa] |
| 8877 | We can't attain a coherent system by lopping off any beliefs that won't fit [Sosa] |
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 23884 | Truth is a value of thought [Weil] |
| 23755 | Genius and love of truth are always accompanied by great humility [Weil] |
| 23877 | Most people won't question an idea's truth if they depend on it [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] |
| 10098 | The 'power set' of A is all the subsets of A [George/Velleman] |
| 10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman] |
| 10101 |
Cartesian Product A x B: the set of all ordered pairs |
| 10103 | Grouping by property is common in mathematics, usually using equivalence [George/Velleman] |
| 10104 | 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman] |
| 10096 | Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman] |
| 10097 | Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman] |
| 10100 | Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman] |
| 17900 | The Axiom of Reducibility made impredicative definitions possible [George/Velleman] |
| 10109 | ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman] |
| 10108 | As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman] |
| 10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
| 10129 | A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman] |
| 10105 | Differences between isomorphic structures seem unimportant [George/Velleman] |
| 10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman] |
| 10126 | A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman] |
| 10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman] |
| 10127 | A 'complete' theory contains either any sentence or its negation [George/Velleman] |
| 10106 | Rational numbers give answers to division problems with integers [George/Velleman] |
| 10102 | The integers are answers to subtraction problems involving natural numbers [George/Velleman] |
| 10107 | Real numbers provide answers to square root problems [George/Velleman] |
| 9946 | Logicists say mathematics is applicable because it is totally general [George/Velleman] |
| 10125 | The classical mathematician believes the real numbers form an actual set [George/Velleman] |
| 17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman] |
| 10128 | The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman] |
| 17902 | A successor is the union of a set with its singleton [George/Velleman] |
| 10133 | Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman] |
| 10130 | Set theory can prove the Peano Postulates [George/Velleman] |
| 10089 | Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman] |
| 8884 | The phenomenal concept of an eleven-dot pattern does not include the concept of eleven [Sosa] |
| 10131 | If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman] |
| 10092 | In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman] |
| 10094 | The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman] |
| 10095 | Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman] |
| 17901 | Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman] |
| 10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman] |
| 10134 | Much infinite mathematics can still be justified finitely [George/Velleman] |
| 10123 | The intuitionists are the idealists of mathematics [George/Velleman] |
| 10124 | Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman] |
| 23855 | Creation produced a network or web of determinations [Weil] |
| 8443 | Mereological essentialism says an entity must have exactly those parts [Sosa] |
| 23900 | Chance is compatible with necessity, and the two occur together [Weil] |
| 8878 | It is acceptable to say a supermarket door 'knows' someone is approaching [Sosa] |
| 23888 | Knowledge is beyond question, as an unavoidable component of thinking [Weil] |
| 8880 | In reducing arithmetic to self-evident logic, logicism is in sympathy with rationalism [Sosa] |
| 8881 | Most of our knowledge has insufficient sensory support [Sosa] |
| 8794 | There are very few really obvious truths, and not much can be proved from them [Sosa] |
| 8882 | Perception may involve thin indexical concepts, or thicker perceptual concepts [Sosa] |
| 8883 | Do beliefs only become foundationally justified if we fully attend to features of our experience? [Sosa] |
| 8885 | Some features of a thought are known directly, but others must be inferred [Sosa] |
| 8876 | Much propositional knowledge cannot be formulated, as in recognising a face [Sosa] |
| 8796 | A single belief can trail two regresses, one terminating and one not [Sosa] |
| 8799 | If mental states are not propositional, they are logically dumb, and cannot be foundations [Sosa] |
| 8795 | Mental states cannot be foundational if they are not immune to error [Sosa] |
| 8879 | Fully comprehensive beliefs may not be knowledge [Sosa] |
| 8798 | Vision causes and justifies beliefs; but to some extent the cause is the justification [Sosa] |
| 23747 | What is sacred is not a person, but the whole physical human being [Weil] |
| 23756 | The mind is imprisoned and limited by language, restricting our awareness of wider thoughts [Weil] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 23878 | Weakness of will is the inadequacy of the original impetus to carry through the action [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] |
| 23898 | Those who say immorality is not an aesthetic criterion must show that all criteria are aesthetic [Weil] |
| 23814 | Every human yearns for an unattainable transcendent good [Weil] |
| 23826 | Beauty, goodness and truth are only achieved by applying full attention [Weil] |
| 23854 | Beauty is the proof of what is 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] |
| 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] |
| 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] |
| 23896 | We see our character as a restricting limit, but also as an unshakable support [Weil] |
| 23894 | The concept of character is at the centre of morality [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] |
| 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] |
| 23831 | The essence of power is illusory prestige [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] |
| 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] |
| 23807 | The collective is the one and only object of false idolatry [Weil] |
| 23749 | The problem of the collective is not suppression of persons, but persons erasing themselves [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] |
| 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] |
| 23835 | People have duties, and only have rights because of the obligations of others to them [Weil] |
| 23751 | Rights are asserted contentiously, and need the backing of force [Weil] |
| 23752 | Giving centrality to rights stifles all impulses of charity [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] |
| 23860 | Even if a drowning man is doomed, he should keep swimming to the last [Weil] |
| 8442 | What law would explain causation in the case of causing a table to come into existence? [Sosa] |
| 8445 | The necessitated is not always a result or consequence of the necessitator [Sosa] |
| 8444 | Where is the necessary causation in the three people being tall making everybody tall? [Sosa] |
| 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] |
| 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] |