73 ideas
20981 | What justifies reliance on reason? Is it just a tool? Why is it better than blind belief? [Sen] |
20982 | In politics and ethics, scrutiny from different perspectives is essential for objectivity [Sen] |
18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
18122 | Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock] |
18114 | There is no single agreed structure for set theory [Bostock] |
18107 | A 'proper class' cannot be a member of anything [Bostock] |
18115 | We could add axioms to make sets either as small or as large as possible [Bostock] |
18139 | The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock] |
18105 | Replacement enforces a 'limitation of size' test for the existence of sets [Bostock] |
18109 | The completeness of first-order logic implies its compactness [Bostock] |
18108 | First-order logic is not decidable: there is no test of whether any formula is valid [Bostock] |
18123 | Substitutional quantification is just standard if all objects in the domain have a name [Bostock] |
18120 | The Deduction Theorem is what licenses a system of natural deduction [Bostock] |
18125 | Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock] |
18100 | ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock] |
18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock] |
18102 | A cardinal is the earliest ordinal that has that number of predecessors [Bostock] |
18106 | Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock] |
18095 | Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock] |
18099 | The number of reals is the number of subsets of the natural numbers [Bostock] |
18093 | For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock] |
18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock] |
18156 | Modern axioms of geometry do not need the real numbers [Bostock] |
18097 | The Peano Axioms describe a unique structure [Bostock] |
18149 | There are many criteria for the identity of numbers [Bostock] |
18148 | Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock] |
18145 | Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock] |
18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock] |
18116 | Numbers can't be positions, if nothing decides what position a given number has [Bostock] |
18117 | Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock] |
18141 | Nominalism about mathematics is either reductionist, or fictionalist [Bostock] |
18157 | Nominalism as based on application of numbers is no good, because there are too many applications [Bostock] |
18150 | Actual measurement could never require the precision of the real numbers [Bostock] |
18158 | Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock] |
18127 | Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock] |
18144 | Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock] |
18147 | Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock] |
18111 | Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock] |
18129 | Many crucial logicist definitions are in fact impredicative [Bostock] |
18146 | If Hume's Principle is the whole story, that implies structuralism [Bostock] |
18159 | Higher cardinalities in sets are just fairy stories [Bostock] |
18155 | A fairy tale may give predictions, but only a true theory can give explanations [Bostock] |
18140 | The best version of conceptualism is predicativism [Bostock] |
18138 | Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock] |
18133 | The usual definitions of identity and of natural numbers are impredicative [Bostock] |
18131 | If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock] |
18134 | Predicativism makes theories of huge cardinals impossible [Bostock] |
18135 | If mathematics rests on science, predicativism may be the best approach [Bostock] |
18136 | If we can only think of what we can describe, predicativism may be implied [Bostock] |
18132 | The predicativity restriction makes a difference with the real numbers [Bostock] |
5959 | Some philosophers say the soul is light [Plutarch] |
20990 | Rationality is conformity to reasons that can be sustained even after scrutiny [Sen] |
18121 | In logic a proposition means the same when it is and when it is not asserted [Bostock] |
21005 | A human right is not plausible if public scrutiny might reject it [Sen] |
20983 | The original position insures that the agreements reached are fair [Sen] |
20987 | The veil of ignorance encourages neutral interests, but not a wider view of values [Sen] |
20984 | A social contract limits the pursuit of justice to members of a single society [Sen] |
20986 | A person's voice may count because of their interests, or because of their good sense [Sen] |
21001 | Famines tend to be caused by authoritarian rule [Sen] |
21002 | Effective democracy needs tolerant values [Sen] |
20979 | Democracy needs more than some institutions; diverse sections of the people must be heard [Sen] |
20999 | Democracy as 'government by discussion' now has wide support [Sen] |
20993 | Eradicating smallpox does not impoverish nature [Sen] |
20998 | Freedom can involve capabilities, independence and non-interference [Sen] |
20995 | Capabilities are part of freedom, involving real opportunities [Sen] |
20997 | The need for equality among people arises from impartiality and objectivity [Sen] |
20996 | All modern theories of justice demand equality of something [Sen] |
20988 | Freedom from torture or terrorist attacks is independent of citizenship [Sen] |
21000 | If justice needs public reasoning, which needs democracy, then justice and democracy are linked [Sen] |
20980 | You don't need a complete theory of justice to see that slavery is wrong [Sen] |
20978 | Practical justice concerns not only ideals, but ways to achieve them [Sen] |
20985 | Our institutions should promote justice, rather than embodying it [Sen] |
20994 | We must focus on removing manifest injustice, not just try to design a perfect society [Sen] |