50 ideas
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] |
13331 | Part and whole contribute asymmetrically to one another, so must differ [Fine,K] |
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] |
13332 | Hierarchical set membership models objects better than the subset or aggregate relations do [Fine,K] |
13333 | The matter is a relatively unstructured version of the object, like a set without membership structure [Fine,K] |
13326 | A 'temporary' part is a part at one time, but may not be at another, like a carburetor [Fine,K] |
13327 | A 'timeless' part just is a part, not a part at some time; some atoms are timeless parts of a water molecule [Fine,K] |
13329 | An 'aggregative' sum is spread in time, and exists whenever a component exists [Fine,K] |
13330 | An 'compound' sum is not spread in time, and only exists when all the components exists [Fine,K] |
13328 | Two sorts of whole have 'rigid embodiment' (timeless parts) or 'variable embodiment' (temporary parts) [Fine,K] |
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] |
468 | Musical performance can reveal a range of virtues [Damon of Ath.] |