73 ideas
| 8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
| 8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
| 3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
| 8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
| 8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
| 8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
| 8694 | Free logic was developed for fictional or non-existent objects [Friend] |
| 8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
| 8672 | A 'powerset' is all the subsets of a set [Friend] |
| 8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
| 8666 | Infinite sets correspond one-to-one with a subset [Friend] |
| 8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
| 8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
| 8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
| 8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
| 8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
| 8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
| 8668 | The 'rational' numbers are those representable as fractions [Friend] |
| 8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
| 8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
| 8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
| 8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
| 8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
| 8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
| 8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
| 8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
| 8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
| 8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
| 8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
| 8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
| 8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
| 8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
| 8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
| 8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
| 8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
| 8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
| 8706 | Constructivism rejects too much mathematics [Friend] |
| 8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
| 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
| 22849 | Rawls's theory cannot justify liberalism, since it presupposes free and equal participants [Charvet] |
| 22848 | People with strong prior beliefs would have nothing to do with a veil of ignorance [Charvet] |
| 22838 | Societies need shared values, so conservatism is right if rational discussion of values is impossible [Charvet] |
| 22846 | The universalism of utilitarianism implies a world state [Charvet] |
| 22835 | Liberals value freedom and equality, but the society itself must decide on its values [Charvet] |
| 22831 | Modern libertarian societies still provide education and some housing [Charvet] |
| 22839 | Liberalism needs people to either have equal autonomy, or everyone to have enough autonomy [Charvet] |
| 22847 | Kant places a higher value on the universal rational will than on the people asserting it [Charvet] |
| 22821 | Liberalism asserts maximum freedom, but that must be equal for all participants [Charvet] |
| 22834 | Egalitarian liberals prefer equality (either of input or outcome) to liberty [Charvet] |
| 22822 | Liberals promote community and well-being - because all good societies need them [Charvet] |
| 22841 | Identity multiculturalism emerges from communitarianism, preferring community to humanity [Charvet] |
| 22842 | For communitarians it seems that you must accept the culture you are born into [Charvet] |
| 22830 | Give by ability and receive by need, rather than a free labour market [Charvet] |
| 22829 | Allowing defamatory speech is against society's interests, by blurring which people are trustworthy [Charvet] |
| 22836 | 'Freedom from' is an empty idea, if the freedom is not from impediments to my desires [Charvet] |
| 22837 | Positive freedom can lead to coercion, if you are forced to do what you chose to do [Charvet] |
| 22844 | First level autonomy is application of personal values; second level is criticising them [Charvet] |
| 22840 | Mere equality, as in two trees being the same height, has no value at all [Charvet] |
| 22843 | Inequalities are worse if they seem to be your fault, rather than social facts [Charvet] |
| 22845 | Money allows unlimited inequalities, and we obviously all agree to money [Charvet] |
| 22828 | The rule of law mainly benefits those with property and liberties [Charvet] |
| 22825 | The 1689 Bill of Rights denied the monarch new courts, or the right to sit as judge [Charvet] |
| 22823 | The rule of law is mainly to restrict governments [Charvet] |
| 22827 | Justice superior to the rule of law is claimed on behalf of the workers, or the will of the nation [Charvet] |
| 22826 | From 1701 only parliament could remove judges, whose decisions could not be discussed [Charvet] |
| 22832 | Welfare is needed if citizens are to accept the obligations of a liberal state [Charvet] |
| 21277 | Even an imperfect machine can exhibit obvious design [Paley] |
| 21275 | Unlike a stone, the parts of a watch are obviously assembled in order to show the time [Paley] |
| 21276 | From the obvious purpose and structure of a watch we must infer that it was designed [Paley] |
| 21357 | No organ shows purpose more obviously than the eyelid [Paley] |
| 21278 | All the signs of design found in a watch are also found in nature [Paley] |