71 ideas
22338 | An unexamined life can be virtuous [Murdoch] |
22337 | Philosophy must keep returning to the beginning [Murdoch] |
23563 | Philosophy moves continually between elaborate theories and the obvious facts [Murdoch] |
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] |
8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
8663 | Raising omega to successive powers of omega reveal an infinity of infinities [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] |
8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [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] |
22591 | We know perfection when we see what is imperfect [Murdoch] |
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] |
7357 | People who control others with fluent language often end up being hated [Kongzi (Confucius)] |
22709 | We should first decide what are the great works of art, with aesthetic theory following from that [Murdoch] |
22341 | Literature is the most important aspect of culture, because it teaches understanding of living [Murdoch] |
22715 | Great art proves the absurdity of art for art's sake [Murdoch] |
22714 | Because art is love, it improves us morally [Murdoch] |
22712 | Art and morals are essentially the same, and are both identical with love [Murdoch] |
22347 | Appreciating beauty in art or nature opens up the good life, by restricting selfishness [Murdoch] |
7358 | All men prefer outward appearance to true excellence [Kongzi (Confucius)] |
7362 | Humans are similar, but social conventions drive us apart (sages and idiots being the exceptions) [Kongzi (Confucius)] |
22713 | Love is realising something other than oneself is real [Murdoch] |
22339 | Love is a central concept in morals [Murdoch] |
22348 | Ordinary human love is good evidence of transcendent goodness [Murdoch] |
7360 | Do not do to others what you would not desire yourself [Kongzi (Confucius)] |
22343 | If I attend properly I will have no choices [Murdoch] |
22349 | Art trains us in the love of virtue [Murdoch] |
22340 | It is hard to learn goodness from others, because their virtues are part of their personal history [Murdoch] |
7359 | Excess and deficiency are equally at fault [Kongzi (Confucius)] |
22346 | Moral reflection and experience gradually reveals unity in the moral world [Murdoch] |
22350 | Only trivial virtues can be possessed on their own [Murdoch] |
7363 | The virtues of the best people are humility, maganimity, sincerity, diligence, and graciousness [Kongzi (Confucius)] |
20765 | Man is a brave naked will, separate from a background of values and realities [Murdoch] |
22342 | Kantian existentialists care greatly for reasons for action, whereas Surrealists care nothing [Murdoch] |
22351 | Only a philosopher might think choices create values [Murdoch] |
7361 | Men of the highest calibre avoid political life completely [Kongzi (Confucius)] |
23393 | Confucianism assumes that all good developments have happened, and there is only one Way [Norden on Kongzi (Confucius)] |
22345 | Moral philosophy needs a central concept with all the traditional attributes of God [Murdoch] |