111 ideas
| 12865 | Analytic philosophers may prefer formal systems because natural language is such mess [Simons] |
| 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] |
| 12815 | Classical mereology doesn't apply well to the objects around us [Simons] |
| 12819 | A 'part' has different meanings for individuals, classes, and masses [Simons] |
| 12832 | Complement: the rest of the Universe apart from some individual, written x-bar [Simons] |
| 12834 | Criticisms of mereology: parts? transitivity? sums? identity? four-dimensional? [Simons] |
| 12827 | Difference: the difference of individuals is the remainder of an overlap, written 'x - y' [Simons] |
| 12828 | General sum: the sum of objects satisfying some predicate, written σx(Fx) [Simons] |
| 12822 | Proper or improper part: x < y, 'x is (a) part of y' [Simons] |
| 12823 | Overlap: two parts overlap iff they have a part in common, expressed as 'x o y' [Simons] |
| 12824 | Disjoint: two individuals are disjoint iff they do not overlap, written 'x | y' [Simons] |
| 12825 | Product: the product of two individuals is the sum of all of their overlaps, written 'x · y' [Simons] |
| 12826 | Sum: the sum of individuals is what is overlapped if either of them are, written 'x + y' [Simons] |
| 12829 | General product: the nucleus of all objects satisfying a predicate, written πx(Fx) [Simons] |
| 12830 | Universe: the mereological sum of all objects whatever, written 'U' [Simons] |
| 12831 | Atom: an individual with no proper parts, written 'At x' [Simons] |
| 12844 | Dissective: stuff is dissective if parts of the stuff are always the stuff [Simons] |
| 12816 | Classical mereology doesn't handle temporal or modal notions very well [Simons] |
| 12821 | The part-relation is transitive and asymmetric (and thus irreflexive) [Simons] |
| 18847 | Each wheel is part of a car, but the four wheels are not a further part [Simons] |
| 12813 | Two standard formalisations of part-whole theory are the Calculus of Individuals, and Mereology [Simons] |
| 12848 | The same members may form two groups [Simons] |
| 12846 | A 'group' is a collection with a condition which constitutes their being united [Simons] |
| 12861 | 'The wolves' are the matter of 'the pack'; the latter is a group, with different identity conditions [Simons] |
| 8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
| 12876 | Philosophy is stuck on the Fregean view that an individual is anything with a proper name [Simons] |
| 12845 | Some natural languages don't distinguish between singular and plural [Simons] |
| 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] |
| 12838 | Four-dimensional ontology has no change, since that needs an object, and time to pass [Simons] |
| 12842 | There are real relational changes, as well as bogus 'Cambridge changes' [Simons] |
| 12841 | I don't believe in processes [Simons] |
| 12836 | Fans of process ontology cheat, since river-stages refer to 'rivers' [Simons] |
| 12880 | Moments are things like smiles or skids, which are founded on other things [Simons] |
| 12881 | A smiling is an event with causes, but the smile is a continuant without causes [Simons] |
| 12883 | Moving disturbances are are moments which continuously change their basis [Simons] |
| 12882 | A wave is maintained by a process, but it isn't a process [Simons] |
| 12840 | I do not think there is a general identity condition for events [Simons] |
| 12839 | Relativity has an ontology of things and events, not on space-time diagrams [Simons] |
| 12879 | Independent objects can exist apart, and maybe even entirely alone [Simons] |
| 12847 | Mass nouns admit 'much' and 'a little', and resist 'many' and 'few'. [Simons] |
| 12863 | Mass terms (unlike plurals) are used with indifference to whether they can exist in units [Simons] |
| 12862 | Gold is not its atoms, because the atoms must be all gold, but gold contains neutrons [Simons] |
| 12859 | A mixture can have different qualities from its ingredients. [Simons] |
| 12858 | Mixtures disappear if nearly all of the mixture is one ingredient [Simons] |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| 12850 | To individuate something we must pick it out, but also know its limits of variation [Simons] |
| 12860 | Sortal nouns for continuants tell you their continuance- and cessation-conditions [Simons] |
| 12886 | A whole requires some unique relation which binds together all of the parts [Simons] |
| 12835 | Does Tibbles remain the same cat when it loses its tail? [Simons] |
| 12857 | Tibbles isn't Tib-plus-tail, because Tibbles can survive its loss, but the sum can't [Simons] |
| 12820 | Without extensional mereology two objects can occupy the same position [Simons] |
| 12866 | Composition is asymmetric and transitive [Simons] |
| 12867 | A hand constitutes a fist (when clenched), but a fist is not composed of an augmented hand [Simons] |
| 12864 | We say 'b is part of a', 'b is a part of a', 'b are a part of a', or 'b are parts of a'. [Simons] |
| 12814 | Classical mereology says there are 'sums', for whose existence there is no other evidence [Simons] |
| 12817 | 'Mereological extensionality' says objects with the same parts are identical [Simons] |
| 12833 | If there are c atoms, this gives 2^c - 1 individuals, so there can't be just 2 or 12 individuals [Simons] |
| 12849 | Sums are more plausible for pluralities and masses than they are for individuals [Simons] |
| 12877 | Sums of things in different categories are found within philosophy. [Simons] |
| 12888 | The wholeness of a melody seems conventional, but of an explosion it seems natural [Simons] |
| 12871 | Objects have their essential properties because of the kind of objects they are [Simons] |
| 12870 | We must distinguish the de dicto 'must' of propositions from the de re 'must' of essence [Simons] |
| 12873 | Original parts are the best candidates for being essential to artefacts [Simons] |
| 12874 | An essential part of an essential part is an essential part of the whole [Simons] |
| 12837 | Four dimensional-objects are stranger than most people think [Simons] |
| 12856 | Intermittent objects would be respectable if they occurred in nature, as well as in artefacts [Simons] |
| 12885 | Objects like chess games, with gaps in them, are thereby less unified [Simons] |
| 12854 | An entrepreneur and a museum curator would each be happy with their ship at the end [Simons] |
| 12855 | The 'best candidate' theories mistakenly assume there is one answer to 'Which is the real ship?' [Simons] |
| 12872 | The zygote is an essential initial part, for a sexually reproduced organism [Simons] |
| 12889 | The limits of change for an individual depend on the kind of individual [Simons] |
| 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] |
| 12843 | With activities if you are doing it you've done it, with performances you must finish to have done it [Simons] |
| 12875 | One false note doesn't make it a performance of a different work [Simons] |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |