55 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] |
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] |
3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
23616 | Legal excuses are duress, ignorance, and diminished responsibility [McMahan] |
23606 | Liberty Rights are permissions, and Claim Rights are freedom from intervention [McMahan] |
23594 | Wars can be unjust, despite a just cause, if they are unnecessary or excessive or of mixed cause [McMahan] |
23620 | A person or state may be attacked if they are responsible for an unjustified threat [McMahan] |
23595 | The worst unjustified wars have no aim at all [McMahan] |
23597 | Just war theory says all and only persons posing a threat are liable to attack [McMahan] |
23598 | You (e.g. a police officer) are not liable to attack just because you pose a threat [McMahan] |
23619 | A defensive war is unjust, if it is responding to a just war [McMahan] |
23600 | Proportionality in fighting can't be judged independently of the justice of each side [McMahan] |
23603 | Can an army start an unjust war, and then fight justly to defend their own civilians? [McMahan] |
23611 | Soldiers cannot freely fight in unjust wars, just because they behave well when fighting [McMahan] |
23612 | The law of war differs from criminal law; attacking just combatants is immoral, but legal [McMahan] |
23617 | If the unjust combatants are morally excused they are innocent, so how can they be killed? [McMahan] |
23596 | If all combatants are seen as morally equal, that facilitates starting unjust wars [McMahan] |
23599 | You don't become a legitimate target, just because you violently resist an unjust attack [McMahan] |
23604 | Volunteer soldiers accept the risk of attack, but they don't agree to it, or to their deaths [McMahan] |
23605 | Soldiers cannot know enough facts to evaluate the justice of their war [McMahan] |
23608 | If being part of a big collective relieves soldiers of moral responsibility, why not the leaders too? [McMahan] |
23610 | If soldiers can't refuse to fight in unjust wars, can they choose to fight in just wars? [McMahan] |
23613 | Equality is both sides have permission, or both sides are justified, or one justified the other permitted [McMahan] |
23615 | Fighting unjustly under duress does not justify it, or permit it, but it may excuse it [McMahan] |
23602 | Innocence implies not being morally responsible, rather than merely being guiltless [McMahan] |
23618 | Unconditional surrender can't be demanded, since evil losers still have legitimate conditions [McMahan] |