61 ideas
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry] |
17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
17800 | The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry] |
17801 | The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry] |
17803 | Limitation of size is part of the very conception of a set [Mayberry] |
17786 | The mainstream of modern logic sees it as a branch of mathematics [Mayberry] |
17788 | First-order logic only has its main theorems because it is so weak [Mayberry] |
17791 | Only second-order logic can capture mathematical structure up to isomorphism [Mayberry] |
17787 | Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry] |
17790 | No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry] |
17779 | 'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry] |
17778 | Axiomatiation relies on isomorphic structures being essentially the same [Mayberry] |
17780 | 'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry] |
17789 | No logic which can axiomatise arithmetic can be compact or complete [Mayberry] |
17784 | Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry] |
17782 | Greek quantities were concrete, and ratio and proportion were their science [Mayberry] |
17781 | Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry] |
17799 | Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry] |
17797 | Cantor extended the finite (rather than 'taming the infinite') [Mayberry] |
17775 | If proof and definition are central, then mathematics needs and possesses foundations [Mayberry] |
17776 | The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry] |
17777 | Foundations need concepts, definition rules, premises, and proof rules [Mayberry] |
17804 | Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry] |
17792 | 1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry] |
17793 | It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry] |
17794 | Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry] |
17802 | We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry] |
17805 | Set theory is not just another axiomatised part of mathematics [Mayberry] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
7518 | If folk psychology gives a network of causal laws, that fits neatly with functionalism [Churchland,PM] |
7519 | Many mental phenomena are totally unexplained by folk psychology [Churchland,PM] |
7520 | Folk psychology never makes any progress, and is marginalised by modern science [Churchland,PM] |
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] |
22823 | The rule of law is mainly to restrict governments [Charvet] |
22825 | The 1689 Bill of Rights denied the monarch new courts, or the right to sit as judge [Charvet] |
22826 | From 1701 only parliament could remove judges, whose decisions could not be discussed [Charvet] |
22827 | Justice superior to the rule of law is claimed on behalf of the workers, or the will of the nation [Charvet] |
22828 | The rule of law mainly benefits those with property and liberties [Charvet] |
22832 | Welfare is needed if citizens are to accept the obligations of a liberal state [Charvet] |