78 ideas
| 17925 | Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan] |
| 17926 | Rejecting double negation elimination undermines reductio proofs [Colyvan] |
| 17924 | Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan] |
| 17929 | Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan] |
| 17930 | Axioms are 'categorical' if all of their models are isomorphic [Colyvan] |
| 17928 | Ordinal numbers represent order relations [Colyvan] |
| 17923 | Intuitionists only accept a few safe infinities [Colyvan] |
| 17941 | Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan] |
| 17922 | Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan] |
| 17936 | Transfinite induction moves from all cases, up to the limit ordinal [Colyvan] |
| 17940 | Most mathematical proofs are using set theory, but without saying so [Colyvan] |
| 17931 | Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan] |
| 17932 | If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan] |
| 3016 | Even the gods cannot strive against necessity [Pittacus, by Diog. Laertius] |
| 17943 | Probability supports Bayesianism better as degrees of belief than as ratios of frequencies [Colyvan] |
| 17939 | Mathematics can reveal structural similarities in diverse systems [Colyvan] |
| 17938 | Mathematics can show why some surprising events have to occur [Colyvan] |
| 17934 | Proof by cases (by 'exhaustion') is said to be unexplanatory [Colyvan] |
| 17933 | Reductio proofs do not seem to be very explanatory [Colyvan] |
| 17935 | If inductive proofs hold because of the structure of natural numbers, they may explain theorems [Colyvan] |
| 17942 | Can a proof that no one understands (of the four-colour theorem) really be a proof? [Colyvan] |
| 17937 | Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan] |
| 19956 | True goodness is political, and consists of love of and submission to the laws [Montesquieu] |
| 19962 | Men do not desire to subjugate one another; domination is a complex and advanced idea [Montesquieu] |
| 19961 | Primitive people would be too vulnerable and timid to attack anyone, so peace would reign [Montesquieu] |
| 19963 | People are drawn into society by needs, shared fears, pleasure, and knowledge [Montesquieu] |
| 20008 | People are guided by a multitude of influences, from which the spirit of a nation emerges [Montesquieu] |
| 19993 | In small republics citizens identify with the public good, and abuses are fewer [Montesquieu] |
| 19992 | In a large republic there is too much wealth for individuals to manage it [Montesquieu] |
| 20005 | The rich would never submit to a lottery deciding which part of their society should be slaves [Montesquieu] |
| 19995 | All states aim at preservation, and then have distinctive individual purposes [Montesquieu] |
| 19964 | The natural power of a father suggests rule by one person, but that authority can be spread [Montesquieu] |
| 19972 | The nobility are an indispensable part of a monarchy [Montesquieu] |
| 19974 | Monarchs must not just have links to the people; they need a body which maintains the laws [Montesquieu] |
| 19976 | Ambition is good in a monarchy, because the monarch can always restrain it [Montesquieu] |
| 19978 | In monarchies, men's actions are judged by their grand appearance, not their virtues [Montesquieu] |
| 19985 | In a monarchy, the nobility must be hereditary, to bind them together [Montesquieu] |
| 19986 | Monarchies can act more quickly, because one person is in charge [Montesquieu] |
| 19988 | A despot's agents must be given power, so they inevitably become corrupt [Montesquieu] |
| 19975 | Despots are always lazy and ignorant, so they always delegate their power to a vizier [Montesquieu] |
| 19977 | Despotism and honour are incompatible, because honour scorns his power, and lives by rules [Montesquieu] |
| 20007 | Tyranny is either real violence, or the imposition of unpopular legislation [Montesquieu] |
| 19989 | The will of a despot is an enigma, so magistrates can only follow their own will [Montesquieu] |
| 19970 | If the nobility is numerous, the senate is the artistocracy, and the nobles are a democracy [Montesquieu] |
| 19971 | Aristocracy is democratic if they resemble the people, but not if they resemble the monarch [Montesquieu] |
| 19984 | Great inequality between aristocrats and the rest is bad - and also among aristocrats themselves [Montesquieu] |
| 19980 | If a government is to be preserved, it must first be loved [Montesquieu] |
| 19996 | A government has a legislature, an international executive, and a domestic executive [Montesquieu] |
| 19997 | The judiciary must be separate from the legislature, to avoid arbitrary power [Montesquieu] |
| 19965 | The fundamental laws of a democracy decide who can vote [Montesquieu] |
| 19968 | It is basic to a democracy that the people themselves must name their ministers [Montesquieu] |
| 19969 | Voting should be public, so the lower classes can be influenced by the example of notable people [Montesquieu] |
| 19999 | All citizens (apart from the very humble poor) should choose their representatives [Montesquieu] |
| 19967 | In a democracy the people should manage themselves, and only delegate what they can't do [Montesquieu] |
| 19966 | A democratic assembly must have a fixed number, to see whether everyone has spoken [Montesquieu] |
| 19998 | If deputies represent people, they are accountable, but less so if they represent places [Montesquieu] |
| 20000 | Slavery is entirely bad; the master abandons the virtues, and they are pointless in the slave [Montesquieu] |
| 20003 | Slaves are not members of the society, so no law can forbid them to run away [Montesquieu] |
| 20006 | The demand for slavery is just the masters' demand for luxury [Montesquieu] |
| 20009 | Freedom of speech and writing, within the law, is essential to preserve liberty [Montesquieu] |
| 19994 | Freedom in society is ability to do what is right, and not having to do what is wrong [Montesquieu] |
| 19981 | No one even thinks of equality in monarchies and despotism; they all want superiority [Montesquieu] |
| 19991 | Equality is not command by everyone or no one, but command and obedience among equals [Montesquieu] |
| 19990 | Democracy is corrupted by lack of equality, or by extreme equality (between rulers and ruled) [Montesquieu] |
| 19982 | Some equality can be achieved by social categories, combined with taxes and poor relief [Montesquieu] |
| 19983 | Democracies may sometimes need to restrict equality [Montesquieu] |
| 19959 | Prior to positive laws there is natural equity, of obedience, gratitude, dependence and merit [Montesquieu] |
| 19960 | Sensation gives animals natural laws, but knowledge can make them break them [Montesquieu] |
| 20002 | The death penalty is permissible, because its victims enjoyed the protection of that law [Montesquieu] |
| 20010 | If religion teaches determinism, penalties must be severe; if free will, then that is different [Montesquieu] |
| 20001 | The only right victors have over captives is the protection of the former [Montesquieu] |
| 19973 | The clergy are essential to a monarchy, but dangerous in a republic [Montesquieu] |
| 20011 | Religion can support the state when the law fails to do so [Montesquieu] |
| 19987 | Religion has the most influence in despotic states, and reinforces veneration for the ruler [Montesquieu] |
| 20004 | French slavery was accepted because it was the best method of religious conversion [Montesquieu] |
| 19979 | In monarchies education ennobles people, and in despotisms it debases them [Montesquieu] |
| 19957 | Teaching is the best practice of the general virtue that leads us to love everyone [Montesquieu] |
| 19958 | Laws are the necessary relations that derive from the nature of things [Montesquieu] |