66 ideas
| 17774 | Definitions make our intuitions mathematically useful [Mayberry] |
| 17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
| 17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
| 17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry] |
| 17800 | The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry] |
| 15717 | Using Choice, you can cut up a small ball and make an enormous one from the pieces [Kaplan/Kaplan] |
| 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] |
| 15712 | 1 and 0, then add for naturals, subtract for negatives, divide for rationals, take roots for irrationals [Kaplan/Kaplan] |
| 17784 | Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry] |
| 15711 | The rationals are everywhere - the irrationals are everywhere else [Kaplan/Kaplan] |
| 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] |
| 15714 | 'Commutative' laws say order makes no difference; 'associative' laws say groupings make no difference [Kaplan/Kaplan] |
| 15715 | 'Distributive' laws say if you add then multiply, or multiply then add, you get the same result [Kaplan/Kaplan] |
| 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] |
| 15713 | The first million numbers confirm that no number is greater than a million [Kaplan/Kaplan] |
| 18658 | The 'Kantian' self steps back from commitment to its social situation [Kymlicka] |
| 18650 | Teleological theories give the good priority over concern for people [Kymlicka] |
| 18664 | Maybe the particularist moral thought of women is better than the impartial public thinking of men [Kymlicka] |
| 18624 | Utilitarianism is not a decision-procedure; choice of the best procedure is an open question [Kymlicka] |
| 18626 | One view says start with equality, and infer equal weight to interests, and hence maximum utility [Kymlicka] |
| 18627 | A second view says start with maximising the good, implying aggregation, and hence equality [Kymlicka] |
| 18625 | To maximise utility should we double the population, even if life somewhat deteriorates? [Kymlicka] |
| 18638 | The difference principles says we must subsidise the costs of other people's choices [Kymlicka] |
| 18635 | Social contract theories are usually rejected because there never was such a contract [Kymlicka] |
| 18630 | Utilitarianism is no longer a distinctive political position [Kymlicka] |
| 18623 | The quest of the general good is partly undermined by people's past entitlements [Kymlicka] |
| 18628 | We shouldn't endorse preferences which reject equality, and show prejudice and selfishness [Kymlicka] |
| 18629 | Using utilitarian principles to make decisions encourages cold detachment from people [Kymlicka] |
| 18637 | Utilitarianism is irrational if it tells you to trade in your rights and resources just for benefits [Kymlicka] |
| 18663 | Modern liberalism has added personal privacy to our personal social lives [Kymlicka] |
| 18632 | Liberalism tends to give priority to basic liberties [Kymlicka] |
| 18656 | Marxists say liberalism is unjust, because it allows exploitation in the sale of labour [Kymlicka] |
| 18659 | The 'Kantian' view of the self misses the way it is embedded or situated in society [Kymlicka] |
| 18660 | Communitarians say we should pay more attention to our history [Kymlicka] |
| 18657 | Communitarian states only encourage fairly orthodox ideas of the good life [Kymlicka] |
| 18649 | If everyone owned himself, that would prevent slavery [Kymlicka] |
| 18640 | Libertarians like the free market, but they also think that the free market is just [Kymlicka] |
| 18651 | The most valuable liberties to us need not be the ones with the most freedom [Kymlicka] |
| 18661 | Ancient freedom was free participation in politics, not private independence of life [Kymlicka] |
| 18633 | Equal opportunities seems fair, because your fate is from your choices, not your circumstances [Kymlicka] |
| 18634 | Equal opportunity arbitrarily worries about social circumstances, but ignores talents [Kymlicka] |
| 18654 | Marxists say justice is unneeded in the truly good community [Kymlicka] |
| 18652 | The Lockean view of freedom depends on whether you had a right to what is restricted [Kymlicka] |
| 18655 | Justice corrects social faults, but also expresses respect to individuals as ends [Kymlicka] |