64 ideas
| 21422 | Moral self-knowledge is the beginning of all human wisdom [Kant] |
| 21408 | For any subject, its system of non-experiential concepts needs a metaphysics [Kant] |
| 21416 | Philosophers should not offer multiple proofs - suggesting the weakness of each of them [Kant] |
| 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] |
| 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
| 15947 | The infinite is extrapolation from the experience of indefinitely large size [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] |
| 21410 | That a concept is not self-contradictory does not make what it represents possible [Kant] |
| 21421 | Within nature man is unimportant, but as moral person he is above any price [Kant] |
| 21415 | Duty is impossible without prior moral feeling, conscience, love and self-respect [Kant] |
| 21409 | Moral principles do not involve feelings [Kant] |
| 21431 | The love of man is required in order to present the world as a beautiful and perfect moral whole [Kant] |
| 21437 | All morality directs the will to love of others' ends, and respect for others' rights [Kant] |
| 21429 | The duty of love is to makes the ends of others one's own [Kant] |
| 21411 | A duty of virtue is a duty which is also an end [Kant] |
| 21413 | Virtue is strong maxims for duty [Kant] |
| 21414 | The supreme principle of virtue is to find universal laws for ends [Kant] |
| 21436 | We are obliged to show the social virtues, but at least they make a virtuous disposition fashionable [Kant] |
| 21419 | If virtue becomes a habit, that is a loss of the freedom needed for adopting maxims [Kant] |
| 21417 | How do we distinguish a mean? The extremes can involve quite different maxims [Kant] |
| 21420 | If virtue is the mean between vices, then virtue is just the vanishing of vice [Kant] |
| 21418 | There is one principle of virtues; the virtues are distinguished by their objects [Kant] |
| 21425 | We can love without respect, and show respect without love [Kant] |
| 21427 | Respect is limiting our self-esteem by attending to the human dignity of other persons [Kant] |
| 21430 | Disrespect is using a person as a mere means to my own ends [Kant] |
| 21428 | Respect is purely negative (of not exalting oneself over others), and is thus a duty of Right [Kant] |
| 21426 | Love urges us to get closer to people, but respect to keep our distance [Kant] |
| 21434 | We must respect the humanity even in a vicious criminal [Kant] |
| 21412 | Humans are distinguished from animals by their capacity to set themselves any sort of end [Kant] |
| 21435 | Man is both social, and unsociable [Kant] |
| 21433 | Violation of rights deserves punishment, which is vengeance, rather than restitution [Kant] |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| 21423 | Men can only have duties to those who qualify as persons [Kant] |
| 21424 | Cruelty to animals is bad because it dulls our empathy for pain in humans [Kant] |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |