68 ideas
| 343 | The unexamined life is not worth living for men [Socrates] |
| 125 | Is a gifted philosopher unmanly if he avoids the strife of the communal world? [Plato] |
| 1654 | In "Gorgias" Socrates is confident that his 'elenchus' will decide moral truth [Vlastos on Plato] |
| 4321 | We should test one another, by asking and answering questions [Plato] |
| 15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
| 15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
| 15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
| 15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
| 15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
| 15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
| 15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
| 15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
| 15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
| 15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
| 15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
| 15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
| 15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
| 15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
| 15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
| 15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
| 15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
| 15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
| 15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
| 15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
| 18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
| 15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
| 15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
| 15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
| 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
| 15940 | The intuitionist endorses only the potential infinite [Lavine] |
| 15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
| 15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
| 15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
| 15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
| 15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
| 15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
| 15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
| 116 | Rhetoric is irrational about its means and its ends [Plato] |
| 114 | Rhetoric can produce conviction, but not educate people about right and wrong [Plato] |
| 135 | All activity aims at the good [Plato] |
| 122 | Moral rules are made by the weak members of humanity [Plato] |
| 339 | Men fear death as a great evil when it may be a great blessing [Socrates] |
| 344 | If death is like a night of dreamless sleep, such nights are very pleasant [Socrates] |
| 139 | A good person is bound to act well, and this brings happiness [Plato] |
| 128 | Is it natural to simply indulge our selfish desires? [Plato] |
| 4322 | In slaking our thirst the goodness of the action and the pleasure are clearly separate [Plato] |
| 136 | Good should be the aim of pleasant activity, not the other way round [Plato] |
| 134 | Good and bad people seem to experience equal amounts of pleasure and pain [Plato] |
| 4319 | In a fool's mind desire is like a leaky jar, insatiable in its desires, and order and contentment are better [Plato] |
| 132 | If happiness is the satisfaction of desires, then a life of scratching itches should be happiness [Plato] |
| 130 | Is the happiest state one of sensual, self-indulgent freedom? [Plato] |
| 2 | We should not even harm someone who harms us [Socrates] |
| 345 | A good man cannot be harmed, either in life or in death [Socrates] |
| 120 | Should we avoid evil because it will bring us bad consequences? [Plato] |
| 118 | I would rather be a victim of crime than a criminal [Plato] |
| 140 | Self-indulgent desire makes friendship impossible, because it makes a person incapable of co-operation [Plato] |
| 131 | If absence of desire is happiness, then nothing is happier than a stone or a corpse [Plato] |
| 119 | A criminal is worse off if he avoids punishment [Plato] |
| 346 | One ought not to return a wrong or injury to any person, whatever the provocation [Socrates] |
| 129 | Do most people praise self-discipline and justice because they are too timid to gain their own pleasure? [Plato] |
| 4320 | The popular view is that health is first, good looks second, and honest wealth third [Plato] |
| 341 | Wealth is good if it is accompanied by virtue [Socrates] |
| 137 | As with other things, a good state is organised and orderly [Plato] |
| 141 | A good citizen won't be passive, but will redirect the needs of the state [Plato] |
| 123 | Do most people like equality because they are second-rate? [Plato] |
| 124 | Does nature imply that it is right for better people to have greater benefits? [Plato] |
| 347 | Will I stand up against the law, simply because I have been unjustly judged? [Socrates] |
| 338 | Socrates is accused of denying the gods, saying sun is stone and moon is earth [Socrates, by Plato] |