86 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] |
10405 | In the iterative conception of sets, they form a natural hierarchy [Swoyer] |
8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
10407 | Logical Form explains differing logical behaviour of similar sentences [Swoyer] |
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] |
14592 | Some abstract things have a beginning and end, so may exist in time (though not space) [Swoyer] |
458 | Nothing could come out of nothing, and existence could never completely cease [Empedocles] |
5112 | Empedocles says things are at rest, unless love unites them, or hatred splits them [Empedocles, by Aristotle] |
10421 | Supervenience is nowadays seen as between properties, rather than linguistic [Swoyer] |
14594 | Ontologists seek existence and identity conditions, and modal and epistemic status for a thing [Swoyer] |
10410 | Anti-realists can't explain different methods to measure distance [Swoyer] |
10399 | If a property such as self-identity can only be in one thing, it can't be a universal [Swoyer] |
10416 | Can properties have parts? [Swoyer] |
14595 | Can properties exemplify other properties? [Swoyer] |
10417 | There are only first-order properties ('red'), and none of higher-order ('coloured') [Swoyer] |
10413 | The best-known candidate for an identity condition for properties is necessary coextensiveness [Swoyer] |
10402 | Various attempts are made to evade universals being wholly present in different places [Swoyer] |
10400 | Conceptualism says words like 'honesty' refer to concepts, not to properties [Swoyer] |
10403 | If properties are abstract objects, then their being abstract exemplifies being abstract [Swoyer] |
8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
14593 | Quantum field theory suggests that there are, fundamentally, no individual things [Swoyer] |
13209 | There is no coming-to-be of anything, but only mixing and separating [Empedocles, by Aristotle] |
457 | Substance is not created or destroyed in mortals, but there is only mixing and exchange [Empedocles] |
10406 | One might hope to reduce possible worlds to properties [Swoyer] |
10404 | Extreme empiricists can hardly explain anything [Swoyer] |
462 | One vision is produced by both eyes [Empedocles] |
22765 | Wisdom and thought are shared by all things [Empedocles] |
8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
1524 | For Empedocles thinking is almost identical to perception [Empedocles, by Theophrastus] |
10408 | Intensions are functions which map possible worlds to sets of things denoted by an expression [Swoyer] |
8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
10409 | Research suggests that concepts rely on typical examples [Swoyer] |
10401 | The F and G of logic cover a huge range of natural language combinations [Swoyer] |
10420 | Maybe a proposition is just a property with all its places filled [Swoyer] |
552 | Empedocles said good and evil were the basic principles [Empedocles, by Aristotle] |
589 | 'Nature' is just a word invented by people [Empedocles] |
21823 | The principle of 'Friendship' in Empedocles is the One, and is bodiless [Empedocles, by Plotinus] |
2680 | Empedocles said that there are four material elements, and two further creative elements [Empedocles, by Aristotle] |
6002 | Empedocles says bone is water, fire and earth in ratio 2:4:2 [Empedocles, by Inwood] |
13207 | Fire, Water, Air and Earth are elements, being simple as well as homoeomerous [Empedocles, by Aristotle] |
459 | All change is unity through love or division through hate [Empedocles] |
13218 | The elements combine in coming-to-be, but how do the elements themselves come-to-be? [Aristotle on Empedocles] |
13225 | Love and Strife only explain movement if their effects are distinctive [Aristotle on Empedocles] |
460 | If the one Being ever diminishes it would no longer exist, and what could ever increase it? [Empedocles] |
10412 | If laws are mere regularities, they give no grounds for future prediction [Swoyer] |
10411 | Two properties can have one power, and one property can have two powers [Swoyer] |
5090 | Maybe bodies are designed by accident, and the creatures that don't work are destroyed [Empedocles, by Aristotle] |
466 | God is pure mind permeating the universe [Empedocles] |
461 | God is a pure, solitary, and eternal sphere [Empedocles] |
1719 | In Empedocles' theory God is ignorant because, unlike humans, he doesn't know one of the elements (strife) [Aristotle on Empedocles] |
1522 | It is wretched not to want to think clearly about the gods [Empedocles] |