66 ideas
21584 | A sense of timelessness is essential to wisdom [Russell] |
21572 | Philosophical disputes are mostly hopeless, because philosophers don't understand each other [Russell] |
21571 | Philosophical systems are interesting, but we now need a more objective scientific philosophy [Russell] |
21574 | Hegel's confusions over 'is' show how vast systems can be built on simple errors [Russell] |
21587 | Philosophers sometimes neglect truth and distort facts to attain a nice system [Russell] |
21582 | Physicists accept particles, points and instants, while pretending they don't do metaphysics [Russell] |
21573 | When problems are analysed properly, they are either logical, or not philosophical at all [Russell] |
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] |
21588 | Logic gives the method of research in philosophy [Russell] |
8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
21586 | The logical connectives are not objects, but are formal, and need a context [Russell] |
8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
21585 | The tortoise won't win, because infinite instants don't compose an infinitely long time [Russell] |
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] |
10580 | Mathematics is both necessary and a priori because it really consists of logical truths [Yablo] |
8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
10579 | Putting numbers in quantifiable position (rather than many quantifiers) makes expression easier [Yablo] |
8706 | Constructivism rejects too much mathematics [Friend] |
8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
21684 | Atomic facts may be inferrable from others, but never from non-atomic facts [Russell] |
10577 | Concrete objects have few essential properties, but properties of abstractions are mostly essential [Yablo] |
10578 | We are thought to know concreta a posteriori, and many abstracta a priori [Yablo] |
22316 | A positive and negative fact have the same constituents; their difference is primitive [Russell] |
21576 | With asymmetrical relations (before/after) the reduction to properties is impossible [Russell] |
21575 | When we attribute a common quality to a group, we can forget the quality and just talk of the group [Russell] |
8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
21580 | Science condemns sense-data and accepts matter, but a logical construction must link them [Russell] |
21583 | When sense-data change, there must be indistinguishable sense-data in the process [Russell] |
21577 | Empirical truths are particular, so general truths need an a priori input of generality [Russell] |
21579 | Objects are treated as real when they connect with other experiences in a normal way [Russell] |
21578 | Global scepticism is irrefutable, but can't replace our other beliefs, and just makes us hesitate [Russell] |
6416 | Other minds seem to exist, because their testimony supports realism about the world [Russell, by Grayling] |
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] |
21581 | We never experience times, but only succession of events [Russell] |