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14109 | The study of grammar is underestimated in philosophy |

14122 | Analysis gives us nothing but the truth - but never the whole truth |

14165 | Analysis falsifies, if when the parts are broken down they are not equivalent to their sum |

14115 | Definition by analysis into constituents is useless, because it neglects the whole |

14159 | In mathematics definitions are superfluous, as they name classes, and it all reduces to primitives |

14148 | Infinite regresses have propositions made of propositions etc, with the key term reappearing |

18002 | As well as a truth value, propositions have a range of significance for their variables |

14102 | What is true or false is not mental, and is best called 'propositions' |

14176 | "The death of Caesar is true" is not the same proposition as "Caesar died" |

14113 | The null class is a fiction |

15894 | Russell invented the naïve set theory usually attributed to Cantor |

14126 | Order rests on 'between' and 'separation' |

14127 | Order depends on transitive asymmetrical relations |

14121 | The part-whole relation is ultimate and indefinable |

14108 | It would be circular to use 'if' and 'then' to define material implication |

14106 | Implication cannot be defined |

14167 | The only classes are things, predicates and relations |

14105 | There seem to be eight or nine logical constants |

14104 | Constants are absolutely definite and unambiguous |

14114 | Variables don't stand alone, but exist as parts of propositional functions |

14137 | 'Any' is better than 'all' where infinite classes are concerned |

14150 | Plato's 'Parmenides' is perhaps the best collection of antinomies ever made |

14149 | The Achilles Paradox concerns the one-one correlation of infinite classes |

15895 | Russell discovered the paradox suggested by Burali-Forti's work |

14123 | Some quantities can't be measured, and some non-quantities are measurable |

14158 | Quantity is not part of mathematics, where it is replaced by order |

18254 | Russell's approach had to treat real 5/8 as different from rational 5/8 |

14144 | Ordinals result from likeness among relations, as cardinals from similarity among classes |

14129 | Ordinals presuppose two relations, where cardinals only presuppose one |

14128 | Some claim priority for the ordinals over cardinals, but there is no logical priority between them |

14132 | Properties of numbers don't rely on progressions, so cardinals may be more basic |

14139 | Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic |

14142 | Ordinals are types of series of terms in a row, rather than than the 'nth' instance |

14145 | For Cantor ordinals are types of order, not numbers |

14131 | Dedekind's ordinals are just members of any progression whatever |

14141 | Ordinals are defined through mathematical induction |

14146 | We aren't sure if one cardinal number is always bigger than another |

14135 | Real numbers are a class of rational numbers (and so not really numbers at all) |

14118 | We can define one-to-one without mentioning unity |

14120 | Counting explains none of the real problems about the foundations of arithmetic |

14119 | We do not currently know whether, of two infinite numbers, one must be greater than the other |

14133 | There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal) |

14134 | Infinite numbers are distinguished by disobeying induction, and the part equalling the whole |

14143 | ω names the whole series, or the generating relation of the series of ordinal numbers |

14138 | You can't get a new transfinite cardinal from an old one just by adding finite numbers to it |

14140 | For every transfinite cardinal there is an infinite collection of transfinite ordinals |

14155 | Two points have a line joining them (descriptive), a distance (metrical), and a whole line (projective) |

14152 | In geometry, Kant and idealists aimed at the certainty of the premisses |

14154 | Geometry throws no light on the nature of actual space |

14151 | Pure geometry is deductive, and neutral over what exists |

14153 | In geometry, empiricists aimed at premisses consistent with experience |

14124 | Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater |

7530 | Russell tried to replace Peano's Postulates with the simple idea of 'class' |

18246 | Dedekind failed to distinguish the numbers from other progressions |

14125 | Finite numbers, unlike infinite numbers, obey mathematical induction |

14147 | Denying mathematical induction gave us the transfinite |

14130 | Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer |

14116 | Numbers were once defined on the basis of 1, but neglected infinities and + |

14117 | Numbers are properties of classes |

9977 | Ordinals can't be defined just by progression; they have intrinsic qualities |

14162 | Mathematics doesn't care whether its entities exist |

14103 | Pure mathematics is the class of propositions of the form 'p implies q' |

18003 | In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless |

11010 | Being is what belongs to every possible object of thought |

14161 | Many things have being (as topics of propositions), but may not have actual existence |

14173 | What exists has causal relations, but non-existent things may also have them |

14163 | Four classes of terms: instants, points, terms at instants only, and terms at instants and points |

10586 | 'Reflexiveness' holds between a term and itself, and cannot be inferred from symmetry and transitiveness |

10585 | Symmetrical and transitive relations are formally like equality |

7781 | I call an object of thought a 'term'. This is a wide concept implying unity and existence. |

14166 | Unities are only in propositions or concepts, and nothing that exists has unity |

14164 | The only unities are simples, or wholes composed of parts |

14112 | A set has some sort of unity, but not enough to be a 'whole' |

14170 | Change is obscured by substance, a thing's nature, subject-predicate form, and by essences |

14107 | Terms are identical if they belong to all the same classes |

10583 | Abstraction principles identify a common property, which is some third term with the right relation |

10582 | The principle of Abstraction says a symmetrical, transitive relation analyses into an identity |

10584 | A certain type of property occurs if and only if there is an equivalence relation |

14110 | Proposition contain entities indicated by words, rather than the words themselves |

19157 | Russell said the proposition must explain its own unity - or else objective truth is impossible |

14111 | A proposition is a unity, and analysis destroys it |

14160 | Space is the extension of 'point', and aggregates of points seem necessary for geometry |

14156 | Mathematicians don't distinguish between instants of time and points on a line |

14172 | Moments and points seem to imply other moments and points, but don't cause them |

14175 | We can drop 'cause', and just make inferences between facts |

14174 | The laws of motion and gravitation are just parts of the definition of a kind of matter |

14168 | Occupying a place and change are prior to motion, so motion is just occupying places at continuous times |

14171 | Force is supposed to cause acceleration, but acceleration is a mathematical fiction. |

14169 | The 'universe' can mean what exists now, what always has or will exist |