78 ideas
14456 | 'Socrates is human' expresses predication, and 'Socrates is a man' expresses identity [Russell] |
14426 | A definition by 'extension' enumerates items, and one by 'intension' gives a defining property [Russell] |
22289 | Dedekind proved definition by recursion, and thus proved the basic laws of arithmetic [Dedekind, by Potter] |
8468 | The sentence 'procrastination drinks quadruplicity' is meaningless, rather than false [Russell, by Orenstein] |
14454 | An argument 'satisfies' a function φx if φa is true [Russell] |
14453 | The Darapti syllogism is fallacious: All M is S, all M is P, so some S is P' - but if there is no M? [Russell] |
10183 | An infinite set maps into its own proper subset [Dedekind, by Reck/Price] |
14427 | We can enumerate finite classes, but an intensional definition is needed for infinite classes [Russell] |
14428 | Members define a unique class, whereas defining characteristics are numerous [Russell] |
22288 | We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available [Dedekind, by Potter] |
14440 | We may assume that there are infinite collections, as there is no logical reason against them [Russell] |
14447 | Infinity says 'for any inductive cardinal, there is a class having that many terms' [Russell] |
14443 | The British parliament has one representative selected from each constituency [Russell] |
14445 | Choice shows that if any two cardinals are not equal, one must be the greater [Russell] |
14444 | Choice is equivalent to the proposition that every class is well-ordered [Russell] |
14446 | We can pick all the right or left boots, but socks need Choice to insure the representative class [Russell] |
14459 | Reducibility: a family of functions is equivalent to a single type of function [Russell] |
14461 | Propositions about classes can be reduced to propositions about their defining functions [Russell] |
8469 | Russell's proposal was that only meaningful predicates have sets as their extensions [Russell, by Orenstein] |
8745 | Classes are logical fictions, and are not part of the ultimate furniture of the world [Russell] |
10706 | Dedekind originally thought more in terms of mereology than of sets [Dedekind, by Potter] |
14452 | All the propositions of logic are completely general [Russell] |
14462 | In modern times, logic has become mathematical, and mathematics has become logical [Russell] |
14464 | Logic can be known a priori, without study of the actual world [Russell] |
12444 | Logic is concerned with the real world just as truly as zoology [Russell] |
10057 | Logic can only assert hypothetical existence [Russell] |
10450 | Russell admitted that even names could also be used as descriptions [Russell, by Bach] |
14458 | Asking 'Did Homer exist?' is employing an abbreviated description [Russell] |
14457 | Names are really descriptions, except for a few words like 'this' and 'that' [Russell] |
7311 | The only genuine proper names are 'this' and 'that' [Russell] |
14455 | 'I met a unicorn' is meaningful, and so is 'unicorn', but 'a unicorn' is not [Russell] |
14442 | If straight lines were like ratios they might intersect at a 'gap', and have no point in common [Russell] |
9823 | Numbers are free creations of the human mind, to understand differences [Dedekind] |
14438 | New numbers solve problems: negatives for subtraction, fractions for division, complex for equations [Russell] |
10090 | Dedekind defined the integers, rationals and reals in terms of just the natural numbers [Dedekind, by George/Velleman] |
7524 | Order, not quantity, is central to defining numbers [Dedekind, by Monk] |
17452 | Ordinals can define cardinals, as the smallest ordinal that maps the set [Dedekind, by Heck] |
13510 | Could a number just be something which occurs in a progression? [Russell, by Hart,WD] |
14131 | Dedekind's ordinals are just members of any progression whatever [Dedekind, by Russell] |
14436 | A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum [Russell] |
14437 | Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Dedekind, by Russell] |
18094 | Dedekind says each cut matches a real; logicists say the cuts are the reals [Dedekind, by Bostock] |
14439 | A complex number is simply an ordered couple of real numbers [Russell] |
14421 | Discovering that 1 is a number was difficult [Russell] |
9824 | In counting we see the human ability to relate, correspond and represent [Dedekind] |
14424 | Numbers are needed for counting, so they need a meaning, and not just formal properties [Russell] |
14441 | The formal laws of arithmetic are the Commutative, the Associative and the Distributive [Russell] |
14420 | Infinity and continuity used to be philosophy, but are now mathematics [Russell] |
9826 | A system S is said to be infinite when it is similar to a proper part of itself [Dedekind] |
13508 | Dedekind gives a base number which isn't a successor, then adds successors and induction [Dedekind, by Hart,WD] |
14431 | The definition of order needs a transitive relation, to leap over infinite intermediate terms [Russell] |
18096 | Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Dedekind, by Bostock] |
14422 | Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms [Russell] |
14423 | '0', 'number' and 'successor' cannot be defined by Peano's axioms [Russell] |
18841 | Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt on Dedekind] |
14130 | Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Dedekind, by Russell] |
14425 | A number is something which characterises collections of the same size [Russell] |
8924 | Dedekind originated the structuralist conception of mathematics [Dedekind, by MacBride] |
14434 | What matters is the logical interrelation of mathematical terms, not their intrinsic nature [Russell] |
9153 | Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Dedekind, by Fine,K] |
14465 | Maybe numbers are adjectives, since 'ten men' grammatically resembles 'white men' [Russell] |
13414 | For Russell, numbers are sets of equivalent sets [Russell, by Benacerraf] |
14449 | There is always something psychological about inference [Russell] |
14463 | Existence can only be asserted of something described, not of something named [Russell] |
14429 | Classes are logical fictions, made from defining characteristics [Russell] |
14430 | If a relation is symmetrical and transitive, it has to be reflexive [Russell] |
14432 | 'Asymmetry' is incompatible with its converse; a is husband of b, so b can't be husband of a [Russell] |
9825 | A thing is completely determined by all that can be thought concerning it [Dedekind] |
14435 | The essence of individuality is beyond description, and hence irrelevant to science [Russell] |
12197 | Inferring q from p only needs p to be true, and 'not-p or q' to be true [Russell] |
14450 | All forms of implication are expressible as truth-functions [Russell] |
14460 | If something is true in all possible worlds then it is logically necessary [Russell] |
14433 | Mathematically expressed propositions are true of the world, but how to interpret them? [Russell] |
9189 | Dedekind said numbers were abstracted from systems of objects, leaving only their position [Dedekind, by Dummett] |
9827 | We derive the natural numbers, by neglecting everything of a system except distinctness and order [Dedekind] |
9979 | Dedekind has a conception of abstraction which is not psychologistic [Dedekind, by Tait] |
14451 | Propositions are mainly verbal expressions of true or false, and perhaps also symbolic thoughts [Russell] |
5996 | Critolaus redefined Aristotle's moral aim as fulfilment instead of happiness [Critolaus, by White,SA] |