66 ideas
| 22864 | Philosophy is the study and criticsm of cultural beliefs, to achieve new possibilities [Dewey] |
| 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] |
| 22873 | Liberalism should improve the system, and not just ameliorate it [Dewey] |
| 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] |
| 22869 | Knowledge is either the product of competent enquiry, or it is meaningless [Dewey] |
| 22868 | The value and truth of knowledge are measured by success in activity [Dewey] |
| 21516 | We want certainty in order achieve secure results for action [Dewey] |
| 22867 | The quest for certainty aims for peace, and avoidance of the stress of action [Dewey] |
| 22870 | No belief can be so settled that it is not subject to further inquiry [Dewey] |
| 411 | If we succeed in speaking the truth, we cannot know we have done it [Xenophanes] |
| 412 | If God had not created honey, men would say figs are sweeter [Xenophanes] |
| 22866 | Mind is never isolated, but only exists in its interactions [Dewey] |
| 22865 | Habits constitute the self [Dewey] |
| 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] |
| 8478 | Dewey argued long before Wittgenstein that there could not seriously be a private language [Dewey, by Orenstein] |
| 22871 | The good people are those who improve; the bad are those who deteriorate [Dewey] |
| 22876 | Democracy is the development of human nature when it shares in the running of communal activities [Dewey] |
| 22875 | Democracy is not just a form of government; it is a mode of shared living [Dewey] |
| 22872 | Liberals aim to allow individuals to realise their capacities [Dewey] |
| 22874 | Individuality is only developed within groups [Dewey] |
| 22880 | The things in civilisation we prize are the products of other members of our community [Dewey] |
| 1640 | The basic Eleatic belief was that all things are one [Xenophanes, by Plato] |
| 3055 | Xenophanes said the essence of God was spherical and utterly inhuman [Xenophanes, by Diog. Laertius] |
| 22879 | 'God' is an imaginative unity of ideal values [Dewey] |
| 408 | Ethiopian gods have black hair, and Thracian gods have red hair [Xenophanes] |
| 407 | Mortals believe gods are born, and have voices and clothes just like mortals [Xenophanes] |
| 22877 | We should try attaching the intensity of religious devotion to intelligent social action [Dewey] |
| 22878 | Religions are so shockingly diverse that they have no common element [Dewey] |