146 ideas
14519 | It is a great good to show reverence for a wise man [Epicurus] |
22733 | Epicurus accepted God in his popular works, but not in his writings on nature [Epicurus, by Sext.Empiricus] |
14518 | In the study of philosophy, pleasure and knowledge arrive simultaneously [Epicurus] |
14052 | Begin philosophy when you are young, and keep going when you are old [Epicurus] |
13291 | Slavery to philosophy brings true freedom [Epicurus] |
22758 | Philosophy aims at a happy life, through argument and discussion [Epicurus] |
14523 | We should come to philosophy free from any taint of culture [Epicurus] |
22240 | The aim of medicine is removal of sickness, and philosophy similarly removes our affections [Epicurus] |
1484 | We should say nothing of the whole if our contact is with the parts [Epicurus, by Plutarch] |
14027 | If we are to use words in enquiry, we need their main, unambiguous and uncontested meanings [Epicurus] |
2670 | Epicurus despises and laughs at the whole of dialectic [Epicurus, by Cicero] |
14040 | Observation and applied thought are always true [Epicurus] |
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] |
21668 | Epicurus rejected excluded middle, because accepting it for events is fatalistic [Epicurus, by Cicero] |
21676 | Epicureans say disjunctions can be true whiile the disjuncts are not true [Epicurus, by Cicero] |
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] |
10605 | Two functions are the same if they have the same extension [Smith,P] |
10076 | The 'range' of a function is the set of elements in the output set created by the function [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] |
10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
10850 | Baby Arithmetic is complete, but not very expressive [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] |
14028 | Nothing comes to be from what doesn't exist [Epicurus] |
14029 | If disappearing things went to nothingness, nothing could return, and it would all be gone by now [Epicurus] |
14030 | The totality is complete, so there is no room for it to change, and nothing extraneous to change it [Epicurus] |
14048 | Astronomical movements are blessed, but they don't need the help of the gods [Epicurus] |
10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
14044 | The perceived accidental properties of bodies cannot be conceived of as independent natures [Epicurus] |
14045 | Accidental properties give a body its nature, but are not themselves bodies or parts of bodies [Epicurus] |
14524 | Bodies are combinations of shape, size, resistance and weight [Epicurus] |
14046 | A 'body' is a conception of an aggregate, with properties defined by application conditions [Epicurus] |
14047 | Bodies have impermanent properties, and permanent ones which define its conceived nature [Epicurus] |
14039 | Above and below us will never appear to be the same, because it is inconceivable [Epicurus] |
14050 | We aim to dissolve our fears, by understanding their causes [Epicurus] |
1823 | We can't seek for things if we have no idea of them [Epicurus, by Diog. Laertius] |
1824 | To name something, you must already have an idea of what it is [Epicurus, by Diog. Laertius] |
14037 | Atoms only have shape, weight and size, and the properties which accompany shape [Epicurus] |
5949 | Epicurus says colours are relative to the eye, not intrinsic to bodies [Epicurus, by Plutarch] |
1821 | Sensations cannot be judged, because similar sensations have equal value, and different ones have nothing in common [Epicurus, by Diog. Laertius] |
1820 | The criteria of truth are senses, preconceptions and passions [Epicurus, by Diog. Laertius] |
1822 | Reason can't judge senses, as it is based on them [Epicurus, by Diog. Laertius] |
3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
4549 | Epicurus denied knowledge in order to retain morality or hedonism as the highest values [Nietzsche on Epicurus] |
2668 | Epicurus says if one of a man's senses ever lies, none of his senses should ever be believed [Epicurus, by Cicero] |
6010 | Illusions are not false perceptions, as we accurately perceive the pattern of atoms [Epicurus, by Modrak] |
1487 | When entering a dark room it is colourless, but colour gradually appears [Epicurus] |
1483 | Bath water is too hot for some, too cold for others [Epicurus, by Plutarch] |
1482 | If two people disagree over taste, who is right? [Epicurus, by Plutarch] |
21386 | We should accept as explanations all the plausible ways in which something could come about [Epicurus] |
14526 | The rational soul is in the chest, and the non-rational soul is spread through the body [Epicurus] |
14041 | The soul is fine parts distributed through the body, resembling hot breath [Epicurus] |
6035 | Soul is made of four stuffs, giving warmth, rest, motion and perception [Epicurus, by Aetius] |
6018 | Epicurus was the first to see the free will problem, and he was a libertarian [Epicurus, by Long/Sedley] |
20922 | Epicurus showed that the swerve can give free motion in the atoms [Epicurus, by Diogenes of Oen.] |
14516 | There is no necessity to live with necessity [Epicurus] |
14521 | If everything is by necessity, then even denials of necessity are by necessity [Epicurus] |
14062 | Sooner follow mythology, than accept the 'fate' of natural philosophers [Epicurus] |
1837 | We should not refer things to irresponsible necessity, but either to fortune or to our own will [Epicurus] |
14042 | The soul cannot be incorporeal, because then it could neither act nor be acted upon [Epicurus] |
1909 | How can pleasure or judgement occur in a heap of atoms? [Sext.Empiricus on Epicurus] |
1836 | Prudence is more valuable than philosophy, because it avoids confusions of the soul [Epicurus] |
14061 | Our own choices are autonomous, and the basis for praise and blame [Epicurus] |
7814 | It was Epicurus who made the question of the will's freedom central to ethics [Epicurus, by Grayling] |
14054 | Fearing death is absurd, because we are not present when it occurs [Epicurus] |
14053 | It is absurd to fear the pain of death when you are not even facing it [Epicurus] |
14055 | The wisdom that produces a good life also produces a good death [Epicurus] |
3562 | Fine things are worthless if they give no pleasure [Epicurus] |
14058 | Pleasure is the goal, but as lack of pain and calm mind, not as depraved or greedy pleasure [Epicurus] |
1840 | Pleasure is the chief good because it is the most natural, especially for animals [Epicurus, by Diog. Laertius] |
14057 | All pleasures are good, but it is not always right to choose them [Epicurus] |
1833 | Pleasure is the first good in life [Epicurus] |
14063 | Sooner a good decision going wrong, than a bad one turning out for the good [Epicurus] |
14522 | What happens to me if I obtain all my desires, and what if I fail? [Epicurus] |
14059 | The best life is not sensuality, but rational choice and healthy opinion [Epicurus] |
1835 | True pleasure is not debauchery, but freedom from physical and mental pain [Epicurus] |
1839 | Pains of the soul are worse than pains of the body, because it feels the past and future [Epicurus, by Diog. Laertius] |
1842 | Pleasures only differ in their duration and the part of the body affected [Epicurus] |
3557 | The end for Epicurus is static pleasure [Epicurus, by Annas] |
14056 | We only need pleasure when we have the pain of desire [Epicurus] |
3563 | Pleasure and virtue entail one another [Epicurus] |
3560 | Justice is merely a contract about not harming or being harmed [Epicurus] |
1845 | Justice has no independent existence, but arises entirely from keeping contracts [Epicurus] |
14060 | Prudence is the greatest good, and more valuable than philosophy, because it produces virtue [Epicurus] |
1841 | We choose virtue because of pleasure, not for its own sake [Epicurus, by Diog. Laertius] |
14517 | We value our own character, whatever it is, and we should respect the characters of others [Epicurus] |
14513 | Justice is a pledge of mutual protection [Epicurus] |
1829 | A wise man would be happy even under torture [Epicurus, by Diog. Laertius] |
1843 | Friendship is by far the most important ingredient of a complete and happy life [Epicurus] |
14515 | A law is not just if it is not useful in mutual associations [Epicurus] |
14520 | It is small-minded to find many good reasons for suicide [Epicurus] |
1831 | Wise men should partake of life even if they go blind [Epicurus, by Diog. Laertius] |
12044 | Only Epicurus denied purpose in nature, for the whole world, or for its parts [Epicurus, by Annas] |
14032 | Totality has no edge; an edge implies a contrast beyond the edge, and there can't be one [Epicurus] |
14033 | Bodies are unlimited as well as void, since the two necessarily go together [Epicurus] |
14034 | There exists an infinity of each shape of atom, but the number of shapes is beyond our knowledge [Epicurus] |
14035 | Atoms just have shape, size and weight; colour results from their arrangement [Epicurus] |
14038 | There cannot be unlimited division, because it would reduce things to non-existence [Epicurus] |
20907 | Democritus says atoms have size and shape, and Epicurus added weight [Epicurus, by Ps-Plutarch] |
21669 | Atoms don't swerve by being struck, because they move in parallel, so the swerve is uncaused [Cicero on Epicurus] |
21680 | What causes atomic swerves? Do they draw lots? What decides the size or number of swerves? [Cicero on Epicurus] |
14049 | We aim to know the natures which are observed in natural phenomena [Epicurus] |
14043 | The void cannot interact, but just gives the possibility of motion [Epicurus] |
14031 | Space must exist, since movement is obvious, and there must be somewhere to move in [Epicurus] |
14525 | Stoics say time is incorporeal and self-sufficient; Epicurus says it is a property of properties of things [Epicurus] |
14051 | A cosmos is a collection of stars and an earth, with some sort of boundary, movement and shape [Epicurus] |
14036 | There are endless cosmoi, some like and some unlike this one [Epicurus] |
2637 | For Epicureans gods are made of atoms, and are not eternal [Epicurus, by Cicero] |
2633 | Epicurus saw that gods must exist, because nature has imprinted them on human minds [Epicurus, by Cicero] |
1828 | God does not intervene in heavenly movements, but is beyond all action and perfectly happy [Epicurus] |
2639 | Some say Epicurus only pretended to believe in the gods, so as not to offend Athenians [Epicurus, by Cicero] |
14527 | If god answered prayers we would be destroyed, because we pray for others to suffer [Epicurus] |