97 ideas
| 3240 | There is more insight in fundamental perplexity about problems than in their supposed solutions [Nagel] |
| 3269 | If your life is to be meaningful as part of some large thing, the large thing must be meaningful [Nagel] |
| 3242 | Philosophy is the childhood of the intellect, and a culture can't skip it [Nagel] |
| 3241 | It seems mad, but the aim of philosophy is to climb outside of our own minds [Nagel] |
| 1489 | Modern philosophy tends to be a theory-constructing extension of science, but there is also problem-solving [Nagel] |
| 3248 | Realism invites scepticism because it claims to be objective [Nagel] |
| 20989 | Views are objective if they don't rely on a person's character, social position or species [Nagel] |
| 22354 | Things cause perceptions, properties have other effects, hence we reach a 'view from nowhere' [Nagel, by Reiss/Sprenger] |
| 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] |
| 4242 | Pure supervenience explains nothing, and is a sign of something fundamental we don't know [Nagel] |
| 3291 | Emergent properties appear at high levels of complexity, but aren't explainable by the lower levels [Nagel] |
| 12887 | A whole must have one characteristic, an internal relation, and a structure [Rescher/Oppenheim] |
| 3249 | Modern science depends on the distinction between primary and secondary qualities [Nagel] |
| 22429 | We achieve objectivity by dropping secondary qualities, to focus on structural primary qualities [Nagel] |
| 3296 | Sense-data are a false objectification of what is essentially subjective [Nagel] |
| 3247 | Epistemology is centrally about what we should believe, not the definition of knowledge [Nagel] |
| 3271 | We can't control our own beliefs [Nagel] |
| 3270 | Justifications come to an end when we want them to [Nagel] |
| 3252 | Scepticism is based on ideas which scepticism makes impossible [Nagel] |
| 1490 | You would have to be very morally lazy to ignore criticisms of your own culture [Nagel] |
| 3251 | Observed regularities are only predictable if we assume hidden necessity [Nagel] |
| 3295 | Inner v outer brings astonishment that we are a particular person [Nagel] |
| 2957 | Brain bisection suggests unity of mind isn't all-or-nothing [Nagel, by Lockwood] |
| 3286 | An organism is conscious if and only if there is something it is like to be that organism [Nagel] |
| 3285 | We may be unable to abandon personal identity, even when split-brains have undermined it [Nagel] |
| 3293 | If you assert that we have an ego, you can still ask if that future ego will be me [Nagel] |
| 3244 | Personal identity cannot be fully known a priori [Nagel] |
| 3245 | The question of whether a future experience will be mine presupposes personal identity [Nagel] |
| 3246 | I can't even conceive of my brain being split in two [Nagel] |
| 3292 | The most difficult problem of free will is saying what the problem is [Nagel] |
| 3288 | Can we describe our experiences to zombies? [Nagel] |
| 4883 | Nagel's title creates an impenetrable mystery, by ignoring a bat's ways that may not be "like" anything [Dennett on Nagel] |
| 3287 | We can't be objective about experience [Nagel] |
| 4989 | Physicalism should explain how subjective experience is possible, but not 'what it is like' [Kirk,R on Nagel] |
| 4001 | The meaning of a word contains all its possible uses as well as its actual ones [Nagel] |
| 6479 | Noninterference requires justification as much as interference does [Nagel] |
| 6450 | Morality must be motivating, and not because of pre-moral motives [Nagel] |
| 3284 | There is no one theory of how to act (or what to believe) [Nagel] |
| 3257 | Total objectivity can't see value, but it sees many people with values [Nagel] |
| 3265 | We don't worry about the time before we were born the way we worry about death [Nagel] |
| 3263 | If our own life lacks meaning, devotion to others won't give it meaning [Nagel] |
| 3256 | Pain doesn't have a further property of badness; it gives a reason for its avoidance [Nagel] |
| 3272 | Moral luck can arise in character, preconditions, actual circumstances, and outcome [Nagel] |
| 6447 | Game theory misses out the motivation arising from the impersonal standpoint [Nagel] |
| 3261 | Something may be 'rational' either because it is required or because it is acceptable [Nagel] |
| 3258 | If cockroaches can't think about their actions, they have no duties [Nagel] |
| 6446 | In ethics we abstract from our identity, but not from our humanity [Nagel] |
| 3282 | The general form of moral reasoning is putting yourself in other people's shoes [Nagel] |
| 3294 | As far as possible we should become instruments to realise what is best from an eternal point of view [Nagel] |
| 3254 | If we can decide how to live after stepping outside of ourselves, we have the basis of a moral theory [Nagel] |
| 3264 | We should see others' viewpoints, but not lose touch with our own values [Nagel] |
| 6477 | I can only universalise a maxim if everyone else could also universalise it [Nagel] |
| 3255 | We find new motives by discovering reasons for action different from our preexisting motives [Nagel] |
| 3262 | Utilitarianism is too demanding [Nagel] |
| 3268 | If a small brief life is absurd, then so is a long and large one [Nagel] |
| 3278 | An egalitarian system must give priority to those with the worst prospects in life [Nagel] |
| 6448 | A legitimate system is one accepted as both impartial and reasonably partial [Nagel] |
| 3275 | Equality was once opposed to aristocracy, but now it opposes public utility and individual rights [Nagel] |
| 3281 | The ideal of acceptability to each individual underlies the appeal to equality [Nagel] |
| 3277 | In judging disputes, should we use one standard, or those of each individual? [Nagel] |
| 3274 | Equality can either be defended as good for society, or as good for individual rights [Nagel] |
| 3273 | Equality nowadays is seen as political, social, legal and economic [Nagel] |
| 6478 | Democracy is opposed to equality, if the poor are not a majority [Nagel] |
| 3276 | A morality of rights is very minimal, leaving a lot of human life without restrictions or duties [Nagel] |
| 3290 | Given the nature of heat and of water, it is literally impossible for water not to boil at the right heat [Nagel] |