67 ideas
| 23770 | Reductive analysis makes a concept clearer, by giving an alternative simpler set [Williams,NE] |
| 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] |
| 23769 | Promoting an ontology by its implied good metaphysic is an 'argument-by-display' [Williams,NE] |
| 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] |
| 6007 | If you know your father, but don't recognise your father veiled, you know and don't know the same person [Eubulides, by Dancy,R] |
| 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] |
| 6006 | If you say truly that you are lying, you are lying [Eubulides, by Dancy,R] |
| 6008 | Removing one grain doesn't destroy a heap, so a heap can't be destroyed [Eubulides, by Dancy,R] |
| 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] |
| 23783 | Change exists, it is causal, and it needs an explanation [Williams,NE] |
| 23784 | Processes don't begin or end; they just change direction unexpectedly [Williams,NE] |
| 23790 | Processes are either strings of short unchanging states, or continuous and unreducible events [Williams,NE] |
| 23786 | The status quo is part of what exists, and so needs metaphysical explanation [Williams,NE] |
| 23768 | A metaphysic is a set of wider explanations derived from a basic ontology [Williams,NE] |
| 23773 | Humeans say properties are passive, possibility is vast, laws are descriptions, causation is weak [Williams,NE] |
| 23779 | We shouldn't posit the existence of anything we have a word for [Williams,NE] |
| 23780 | Every possible state of affairs is written into its originating powers [Williams,NE] |
| 23789 | Naming powers is unwise, because that it usually done by a single manifestation [Williams,NE] |
| 23775 | Powers are 'multi-track' if they can produce a variety of manifestations [Williams,NE] |
| 23771 | Fundamental physics describes everything in terms of powers [Williams,NE] |
| 23776 | Rather than pure powers or pure categoricals, I favour basics which are both at once [Williams,NE] |
| 23777 | Powers are more complicated than properties which are always on display [Williams,NE] |
| 23774 | There are basic powers, which underlie dispositions, potentialities, capacities etc [Williams,NE] |
| 23791 | Dispositions are just useful descriptions, which are explained by underlying powers [Williams,NE] |
| 23772 | If objects are property bundles, the properties need combining powers [Williams,NE] |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| 23788 | Four-Dimensional is Perdurantism (temporal parts), plus Eternalism [Williams,NE] |
| 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] |
| 23785 | Causation needs to explain stasis, as well as change [Williams,NE] |
| 23782 | Causation is the exercise of powers [Williams,NE] |
| 23787 | If causes and effects overlap, that makes changes impossible [Williams,NE] |
| 23778 | Powers contain lawlike features, pointing to possible future states [Williams,NE] |