82 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] |
| 16764 | The soul conserves the body, as we see by its dissolution when the soul leaves [Toletus] |
| 19906 | All countries are in a mutual state of nature [Locke] |
| 19882 | We are not created for solitude, but are driven into society by our needs [Locke] |
| 19864 | In nature men can dispose of possessions and their persons in any way that is possible [Locke] |
| 19865 | There is no subjection in nature, and all creatures of the same species are equal [Locke] |
| 19866 | The rational law of nature says we are all equal and independent, and should show mutual respect [Locke] |
| 19872 | The animals and fruits of the earth belong to mankind [Locke] |
| 19907 | There is a natural right to inheritance within a family [Locke] |
| 19863 | Politics is the right to make enforceable laws to protect property and the state, for the common good [Locke] |
| 5654 | The Second Treatise explores the consequences of the contractual view of the state [Locke, by Scruton] |
| 19888 | A society only begins if there is consent of all the individuals to join it [Locke] |
| 6702 | If anyone enjoys the benefits of government (even using a road) they give tacit assent to its laws [Locke] |
| 19909 | A politic society is created from a state of nature by a unanimous agreement [Locke] |
| 19910 | A single will creates the legislature, which is duty-bound to preserve that will [Locke] |
| 19893 | Anyone who enjoys the benefits of a state has given tacit consent to be part of it [Locke] |
| 19894 | You can only become an actual member of a commonwealth by an express promise [Locke] |
| 19892 | Children are not born into citizenship of a state [Locke] |
| 19885 | Absolute monarchy is inconsistent with civil society [Locke] |
| 19886 | The idea that absolute power improves mankind is confuted by history [Locke] |
| 19903 | Despotism is arbitrary power to kill, based neither on natural equality, nor any social contract [Locke] |
| 19905 | People stripped of their property are legitimately subject to despotism [Locke] |
| 19904 | Legitimate prisoners of war are subject to despotism, because that continues the state of war [Locke] |
| 19895 | Even the legislature must be preceded by a law which gives it power to make laws [Locke] |
| 19900 | The executive must not be the legislature, or they may exempt themselves from laws [Locke] |
| 19902 | Any obstruction to the operation of the legislature can be removed forcibly by the people [Locke] |
| 19908 | Rebelling against an illegitimate power is no sin [Locke] |
| 19911 | If legislators confiscate property, or enslave people, they are no longer owed obedience [Locke] |
| 19901 | The people have supreme power, to depose a legislature which has breached their trust [Locke] |
| 19887 | Unanimous consent makes a united community, which is then ruled by the majority [Locke] |
| 19913 | A master forfeits ownership of slaves he abandons [Locke] |
| 19883 | Slaves captured in a just war have no right to property, so are not part of civil society [Locke] |
| 19870 | If you try to enslave me, you have declared war on me [Locke] |
| 19871 | Freedom is not absence of laws, but living under laws arrived at by consent [Locke] |
| 19880 | All value depends on the labour involved [Locke] |
| 19884 | There is only a civil society if the members give up all of their natural executive rights [Locke] |
| 19873 | We all own our bodies, and the work we do is our own [Locke] |
| 6580 | Locke (and Marx) held that ownership of objects is a natural relation, based on the labour put into it [Locke, by Fogelin] |
| 20520 | Locke says 'mixing of labour' entitles you to land, as well as nuts and berries [Wolff,J on Locke] |
| 19875 | A man's labour gives ownership rights - as long as there are fair shares for all [Locke] |
| 19874 | If a man mixes his labour with something in Nature, he thereby comes to own it [Locke] |
| 19877 | Fountain water is everyone's, but a drawn pitcher of water has an owner [Locke] |
| 19876 | Gathering natural fruits gives ownership; the consent of other people is irrelevant [Locke] |
| 19878 | Mixing labour with a thing bestows ownership - as long as the thing is not wasted [Locke] |
| 19898 | Soldiers can be commanded to die, but not to hand over their money [Locke] |
| 19879 | A man owns land if he cultivates it, to the limits of what he needs [Locke] |
| 19881 | The aim of law is not restraint, but to make freedom possible [Locke] |
| 19868 | It is only by a law of Nature that we can justify punishing foreigners [Locke] |
| 19867 | Reparation and restraint are the only justifications for punishment [Locke] |
| 19912 | Self-defence is natural, but not the punishment of superiors by inferiors [Locke] |
| 19869 | Punishment should make crime a bad bargain, leading to repentance and deterrence [Locke] |
| 19899 | The consent of the people is essential for any tax [Locke] |