126 ideas
| 20947 | Thoughts are learnt through words, so language shows the limits and shape of our knowledge [Herder] |
| 18194 | 'Forcing' can produce new models of ZFC from old models [Maddy] |
| 18195 | A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy [Maddy] |
| 13011 | New axioms are being sought, to determine the size of the continuum [Maddy] |
| 13013 | The Axiom of Extensionality seems to be analytic [Maddy] |
| 13014 | Extensional sets are clearer, simpler, unique and expressive [Maddy] |
| 13021 | The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics [Maddy] |
| 13022 | Infinite sets are essential for giving an account of the real numbers [Maddy] |
| 18191 | Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy] |
| 13023 | The Power Set Axiom is needed for, and supported by, accounts of the continuum [Maddy] |
| 18193 | The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy] |
| 13024 | Efforts to prove the Axiom of Choice have failed [Maddy] |
| 13025 | Modern views say the Choice set exists, even if it can't be constructed [Maddy] |
| 13026 | A large array of theorems depend on the Axiom of Choice [Maddy] |
| 17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
| 18169 | Axiom of Reducibility: propositional functions are extensionally predicative [Maddy] |
| 13019 | The Iterative Conception says everything appears at a stage, derived from the preceding appearances [Maddy] |
| 13018 | Limitation of Size is a vague intuition that over-large sets may generate paradoxes [Maddy] |
| 17824 | The master science is physical objects divided into sets [Maddy] |
| 8755 | Maddy replaces pure sets with just objects and perceived sets of objects [Maddy, by Shapiro] |
| 10594 | Henkin semantics is more plausible for plural logic than for second-order logic [Maddy] |
| 17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
| 18168 | 'Propositional functions' are propositions with a variable as subject or predicate [Maddy] |
| 17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
| 17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
| 18190 | Completed infinities resulted from giving foundations to calculus [Maddy] |
| 18171 | Cantor and Dedekind brought completed infinities into mathematics [Maddy] |
| 17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
| 18172 | Infinity has degrees, and large cardinals are the heart of set theory [Maddy] |
| 18175 | For any cardinal there is always a larger one (so there is no set of all sets) [Maddy] |
| 18196 | An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy] |
| 18187 | Theorems about limits could only be proved once the real numbers were understood [Maddy] |
| 18182 | The extension of concepts is not important to me [Maddy] |
| 18177 | In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy] |
| 18164 | Frege solves the Caesar problem by explicitly defining each number [Maddy] |
| 17825 | Set theory (unlike the Peano postulates) can explain why multiplication is commutative [Maddy] |
| 17826 | Standardly, numbers are said to be sets, which is neat ontology and epistemology [Maddy] |
| 17828 | Numbers are properties of sets, just as lengths are properties of physical objects [Maddy] |
| 10718 | A natural number is a property of sets [Maddy, by Oliver] |
| 18184 | Making set theory foundational to mathematics leads to very fruitful axioms [Maddy] |
| 18185 | Unified set theory gives a final court of appeal for mathematics [Maddy] |
| 18183 | Set theory brings mathematics into one arena, where interrelations become clearer [Maddy] |
| 18186 | Identifying geometric points with real numbers revealed the power of set theory [Maddy] |
| 18188 | The line of rationals has gaps, but set theory provided an ordered continuum [Maddy] |
| 17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
| 18163 | Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy] |
| 17830 | Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy] |
| 17827 | Sets exist where their elements are, but numbers are more like universals [Maddy] |
| 17823 | If mathematical objects exist, how can we know them, and which objects are they? [Maddy] |
| 8756 | Intuition doesn't support much mathematics, and we should question its reliability [Maddy, by Shapiro] |
| 17733 | We know mind-independent mathematical truths through sets, which rest on experience [Maddy, by Jenkins] |
| 18204 | Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy] |
| 18207 | Maybe applications of continuum mathematics are all idealisations [Maddy] |
| 17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
| 17829 | Number words are unusual as adjectives; we don't say 'is five', and numbers always come first [Maddy] |
| 18167 | We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy] |
| 22355 | In the realist view, the real external world explains how it (and perceptions of it) are possible [Williams,B] |
| 18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
| 23283 | Necessity implies possibility, but in experience it matters which comes first [Williams,B] |
| 4244 | It is very confused to deduce a nonrelativist morality of universal toleration from relativism [Williams,B] |
| 4243 | Our ability to react to an alien culture shows that ethical thought extends beyond cultural boundaries [Williams,B] |
| 18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
| 3238 | 'Dead person' isn't a contradiction, so 'person' is somewhat vague [Williams,B] |
| 3239 | You can only really love a person as a token, not as a type [Williams,B] |
| 7946 | The memory criterion has a problem when one thing branches into two things [Williams,B, by Macdonald,C] |
| 2181 | It is an absurd Kantian idea that at the limit rationality and freedom coincide [Williams,B] |
| 2176 | There is only a problem of free will if you think the notion of 'voluntary' can be metaphysically deepened [Williams,B] |
| 24008 | Reference to a person's emotions is often essential to understanding their actions [Williams,B] |
| 24009 | Moral education must involve learning about various types of feeling towards things [Williams,B] |
| 20949 | Study the use of words, not their origins [Herder] |
| 4317 | We judge weakness of will by an assessment after the event is concluded [Williams,B, by Cottingham] |
| 9284 | Reasons are 'internal' if they give a person a motive to act, but 'external' otherwise [Williams,B] |
| 2174 | Responsibility involves cause, intention, state of mind, and response after the event [Williams,B] |
| 22455 | Many ethical theories neglect the power of regretting the ought not acted upon [Williams,B] |
| 4114 | Philosophers try to produce ethical theories because they falsely assume that ethics can be simple [Williams,B] |
| 22453 | Moral conflicts have a different feeling and structure from belief conflicts [Williams,B, by Foot] |
| 22454 | We tolerate inconsistency in ethics but not in other beliefs (which reflect an independent order) [Williams,B, by Foot] |
| 22450 | If moral systems can't judge other moral systems, then moral relativism is true [Williams,B, by Foot] |
| 2178 | In bad actions, guilt points towards victims, and shame to the agent [Williams,B] |
| 20168 | Blame usually has no effect if the recipient thinks it unjustified [Williams,B] |
| 20167 | Blame partly rests on the fiction that blamed agents always know their obligations [Williams,B] |
| 4128 | Intuitionism has been demolished by critics, and no longer looks interesting [Williams,B] |
| 4366 | We can't accept Aristotle's naturalism about persons, because it is normative and unscientific [Williams,B, by Hursthouse] |
| 4132 | The category of person is a weak basis for ethics, because it is not fixed but comes in degrees [Williams,B] |
| 24007 | Emotivism saw morality as expressing emotions, and influencing others' emotions [Williams,B] |
| 4134 | The weakness of prescriptivism is shown by "I simply don't like staying at good hotels" [Williams,B] |
| 4135 | Some ethical ideas, such as 'treachery' and 'promise', seem to express a union of facts and values [Williams,B] |
| 7669 | We cannot attain all the ideals of every culture, so there cannot be a perfect life [Herder, by Berlin] |
| 22410 | Maybe the unthinkable is a moral category, and considering some options is dishonourable or absurd [Williams,B] |
| 22408 | Consequentialism assumes that situations can be compared [Williams,B] |
| 22411 | For a consequentialist massacring 7 million must be better than massacring 7 million and one [Williams,B] |
| 4120 | It is an error of consequentialism to think we just aim at certain states of affairs; we also want to act [Williams,B] |
| 23282 | If all that matters in morality is motive and intention, that makes moral luck irrelevant [Williams,B] |
| 4252 | Promise keeping increases reliability, by making deliberation focus on something which would be overlooked [Williams,B] |
| 4116 | A weakness of contractual theories is the position of a person of superior ability and power [Williams,B] |
| 2169 | Greek moral progress came when 'virtue' was freed from social status [Williams,B] |
| 4112 | A crucial feature of moral thought is second-order desire - the desire to have certain desires [Williams,B] |
| 24010 | An admirable human being should have certain kinds of emotional responses [Williams,B] |
| 23279 | It is important that a person can change their character, and not just be successive 'selves' [Williams,B] |
| 23280 | Kantians have an poor account of individuals, and insist on impartiality, because they ignore character [Williams,B] |
| 3236 | Equality of opportunity without equality of respect would create a very inhuman society [Williams,B] |
| 4113 | 'Deon' in Greek means what one must do; there was no word meaning 'duty' [Williams,B] |
| 4250 | The concept of a 'duty to myself' is fraudulent [Williams,B] |
| 4110 | Obligation and duty look backwards (because of a promise or job), although the acts are in the future [Williams,B] |
| 4248 | Not all moral deliberations lead to obligations; some merely reveal what 'may' be done [Williams,B] |
| 4249 | "Ought implies can" is a famous formula in connection with moral obligation [Williams,B] |
| 2172 | The modern idea of duty is unknown in archaic Greece [Williams,B] |
| 4121 | Why should I think of myself as both the legislator and the citizen who follows the laws? [Williams,B] |
| 22409 | We don't have a duty to ensure that others do their duty [Williams,B] |
| 4122 | If the self becomes completely impartial, it no longer has enough identity to worry about its interests [Williams,B] |
| 2180 | If reason cannot lead people to good, we must hope they have an internal voice [Williams,B] |
| 24012 | Kant's love of consistency is too rigid, and it even overrides normal fairness [Williams,B] |
| 2179 | If the moral self is seen as characterless, then other people have a very limited role in our moral lives [Williams,B] |
| 22407 | Utilitarianism cannot make any serious sense of integrity [Williams,B] |
| 23278 | For utilitarians states of affairs are what have value, not matter who produced them [Williams,B] |
| 4124 | Utilitarian benevolence involves no particular attachments, and is immune to the inverse square law [Williams,B] |
| 4245 | Ethical conviction must be to some extent passive, and can't just depend on the will and decisions [Williams,B] |
| 4246 | Taking responsibility won't cure ethical uncertainty by; we are uncertain what to decide [Williams,B] |
| 7668 | Herder invented the idea of being rooted in (or cut off from) a home or a group [Herder, by Berlin] |
| 3233 | Equality implies that people are alike in potential as well as in needs [Williams,B] |
| 3234 | Equality seems to require that each person be acknowledged as having a significant point of view [Williams,B] |
| 3235 | It is a mark of extreme exploitation that the sufferers do not realise their plight [Williams,B] |
| 4247 | It is a mark of our having ethical values that we aim to reproduce them in our children [Williams,B] |
| 4131 | Most women see an early miscarriage and a late stillbirth as being very different in character [Williams,B] |
| 4133 | Speciesism isn't like racism, because the former implies a viewpoint which belongs to no one [Williams,B] |
| 2175 | There is a problem of evil only if you expect the world to be good [Williams,B] |