85 ideas
19767 | Reason leads to prudent selfishness, which overrules natural compassion [Rousseau] |
8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
8694 | Free logic was developed for fictional or non-existent objects [Friend] |
8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
8672 | A 'powerset' is all the subsets of a set [Friend] |
8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
8666 | Infinite sets correspond one-to-one with a subset [Friend] |
8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
8668 | The 'rational' numbers are those representable as fractions [Friend] |
8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
8706 | Constructivism rejects too much mathematics [Friend] |
8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
19757 | No one would bother to reason, and try to know things, without a desire for enjoyment [Rousseau] |
19760 | General ideas are purely intellectual; imagining them is immediately particular [Rousseau] |
19759 | Only words can introduce general ideas into the mind [Rousseau] |
8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
19758 | Language may aid thinking, but powerful thought was needed to produce language [Rousseau] |
19773 | Without love, what use is beauty? [Rousseau] |
468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
19769 | Rational morality is OK for brainy people, but ordinary life can't rely on that [Rousseau] |
19752 | If we should not mistreat humans, it is mainly because of sentience, not rationality [Rousseau] |
19768 | The better Golden Rule is 'do good for yourself without harming others' [Rousseau] |
19766 | The fact that we weep (e.g. in theatres) shows that we are naturally compassionate [Rousseau] |
19756 | Humans are less distinguished from other animals by understanding, than by being free agents [Rousseau] |
19755 | Most human ills are self-inflicted; the simple, solitary, regular natural life is good [Rousseau] |
19762 | Is language a pre-requisite for society, or might it emerge afterwards? [Rousseau] |
19763 | I doubt whether a savage person ever complains of life, or considers suicide [Rousseau] |
19765 | Savages avoid evil because they are calm, and never think of it (not because they know goodness) [Rousseau] |
19779 | Primitive man was very gentle [Rousseau] |
19771 | Savage men quietly pursue desires, without the havoc of modern frenzied imagination [Rousseau] |
19778 | Leisure led to envy, inequality, vice and revenge, which we now see in savages [Rousseau] |
19751 | Our two starting principles are concern for self-interest, and compassion for others [Rousseau] |
19774 | A savage can steal fruit or a home, but there is no means of achieving obedience [Rousseau] |
19772 | In a state of nature people are much more equal; it is society which increases inequalities [Rousseau] |
19789 | It is against nature for children to rule old men, fools to rule the wise, and the rich to hog resources [Rousseau] |
19787 | People accept the right to be commanded, because they themselves wish to command [Rousseau] |
19780 | We seem to have made individual progress since savagery, but actually the species has decayed [Rousseau] |
19747 | Revolutionaries usually confuse liberty with total freedom, and end up with heavier chains [Rousseau] |
19748 | Plebiscites are bad, because they exclude the leaders from crucial decisions [Rousseau] |
19749 | In a direct democracy, only the leaders should be able to propose new laws [Rousseau] |
19775 | People must be made dependent before they can be enslaved [Rousseau] |
19784 | Enslaved peoples often boast of their condition, calling it a state of 'peace' [Rousseau] |
19785 | If the child of a slave woman is born a slave, then a man is not born a man [Rousseau] |
19746 | Like rich food, liberty can ruin people who are too weak to cope with it [Rousseau] |
19786 | Three stages of the state produce inequalities of wealth, power, and enslavement [Rousseau] |
19788 | The pleasure of wealth and power is largely seeing others deprived of them [Rousseau] |
19777 | Persuading other people that some land was 'owned' was the beginning of society [Rousseau] |
19782 | What else could property arise from, but the labour people add to it? [Rousseau] |
19781 | Land cultivation led to a general right of ownership, administered justly [Rousseau] |
19754 | If we have a natural right to property, what exactly does 'belonging to' mean? [Rousseau] |
19750 | Writers just propose natural law as the likely useful agreements among people [Rousseau] |
19770 | Primitive people simply redressed the evil caused by violence, without thought of punishing [Rousseau] |
19783 | A state of war remains after a conquest, if the losers don't accept the winners [Rousseau] |
19753 | Both men and animals are sentient, which should give the latter the right not to be mistreated [Rousseau] |
19761 | Men started with too few particular names, but later had too few natural kind names [Rousseau] |
19776 | Small uninterrupted causes can have big effects [Rousseau] |