120 ideas
14122 | Analysis gives us nothing but the truth - but never the whole truth [Russell] |
14109 | The study of grammar is underestimated in philosophy [Russell] |
14165 | Analysis falsifies, if when the parts are broken down they are not equivalent to their sum [Russell] |
14115 | Definition by analysis into constituents is useless, because it neglects the whole [Russell] |
14159 | In mathematics definitions are superfluous, as they name classes, and it all reduces to primitives [Russell] |
14148 | Infinite regresses have propositions made of propositions etc, with the key term reappearing [Russell] |
18002 | As well as a truth value, propositions have a range of significance for their variables [Russell] |
14102 | What is true or false is not mental, and is best called 'propositions' [Russell] |
14176 | "The death of Caesar is true" is not the same proposition as "Caesar died" [Russell] |
9987 | An aggregate in which order does not matter I call a 'set' [Bolzano] |
14113 | The null class is a fiction [Russell] |
15894 | Russell invented the naïve set theory usually attributed to Cantor [Russell, by Lavine] |
14126 | Order rests on 'between' and 'separation' [Russell] |
14127 | Order depends on transitive asymmetrical relations [Russell] |
14121 | The part-whole relation is ultimate and indefinable [Russell] |
14106 | Implication cannot be defined [Russell] |
14108 | It would be circular to use 'if' and 'then' to define material implication [Russell] |
14167 | The only classes are things, predicates and relations [Russell] |
14105 | There seem to be eight or nine logical constants [Russell] |
18722 | Negations are not just reversals of truth-value, since that can happen without negation [Wittgenstein on Russell] |
14104 | Constants are absolutely definite and unambiguous [Russell] |
14114 | Variables don't stand alone, but exist as parts of propositional functions [Russell] |
14137 | 'Any' is better than 'all' where infinite classes are concerned [Russell] |
14149 | The Achilles Paradox concerns the one-one correlation of infinite classes [Russell] |
15895 | Russell discovered the paradox suggested by Burali-Forti's work [Russell, by Lavine] |
14151 | Pure geometry is deductive, and neutral over what exists [Russell] |
14152 | In geometry, Kant and idealists aimed at the certainty of the premisses [Russell] |
14154 | Geometry throws no light on the nature of actual space [Russell] |
14153 | In geometry, empiricists aimed at premisses consistent with experience [Russell] |
14155 | Two points have a line joining them (descriptive), a distance (metrical), and a whole line (projective) [Russell, by PG] |
18254 | Russell's approach had to treat real 5/8 as different from rational 5/8 [Russell, by Dummett] |
14144 | Ordinals result from likeness among relations, as cardinals from similarity among classes [Russell] |
14128 | Some claim priority for the ordinals over cardinals, but there is no logical priority between them [Russell] |
14129 | Ordinals presuppose two relations, where cardinals only presuppose one [Russell] |
14132 | Properties of numbers don't rely on progressions, so cardinals may be more basic [Russell] |
14139 | Transfinite ordinals don't obey commutativity, so their arithmetic is quite different from basic arithmetic [Russell] |
14142 | Ordinals are types of series of terms in a row, rather than the 'nth' instance [Russell] |
14141 | Ordinals are defined through mathematical induction [Russell] |
14145 | For Cantor ordinals are types of order, not numbers [Russell] |
14146 | We aren't sure if one cardinal number is always bigger than another [Russell] |
14135 | Real numbers are a class of rational numbers (and so not really numbers at all) [Russell] |
14123 | Some quantities can't be measured, and some non-quantities are measurable [Russell] |
14158 | Quantity is not part of mathematics, where it is replaced by order [Russell] |
14120 | Counting explains none of the real problems about the foundations of arithmetic [Russell] |
14118 | We can define one-to-one without mentioning unity [Russell] |
14133 | There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal) [Russell] |
10856 | A truly infinite quantity does not need to be a variable [Bolzano] |
14119 | We do not currently know whether, of two infinite numbers, one must be greater than the other [Russell] |
14134 | Infinite numbers are distinguished by disobeying induction, and the part equalling the whole [Russell] |
14143 | ω names the whole series, or the generating relation of the series of ordinal numbers [Russell] |
14138 | You can't get a new transfinite cardinal from an old one just by adding finite numbers to it [Russell] |
14140 | For every transfinite cardinal there is an infinite collection of transfinite ordinals [Russell] |
14124 | Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater [Russell] |
7530 | Russell tried to replace Peano's Postulates with the simple idea of 'class' [Russell, by Monk] |
18246 | Dedekind failed to distinguish the numbers from other progressions [Shapiro on Russell] |
14125 | Finite numbers, unlike infinite numbers, obey mathematical induction [Russell] |
14147 | Denying mathematical induction gave us the transfinite [Russell] |
14116 | Numbers were once defined on the basis of 1, but neglected infinities and + [Russell] |
14117 | Numbers are properties of classes [Russell] |
9977 | Ordinals can't be defined just by progression; they have intrinsic qualities [Russell] |
14162 | Mathematics doesn't care whether its entities exist [Russell] |
14103 | Pure mathematics is the class of propositions of the form 'p implies q' [Russell] |
21555 | For 'x is a u' to be meaningful, u must be one range of individuals (or 'type') higher than x [Russell] |
18003 | In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless [Russell, by Magidor] |
11010 | Being is what belongs to every possible object of thought [Russell] |
14161 | Many things have being (as topics of propositions), but may not have actual existence [Russell] |
19482 | Current physics says matter and antimatter should have reduced to light at the big bang [New Sci.] |
19483 | CP violation shows a decay imbalance in matter and antimatter, leading to matter's dominance [New Sci.] |
14173 | What exists has causal relations, but non-existent things may also have them [Russell] |
14163 | Four classes of terms: instants, points, terms at instants only, and terms at instants and points [Russell] |
21341 | Philosophers of logic and maths insisted that a vocabulary of relations was essential [Russell, by Heil] |
10586 | 'Reflexiveness' holds between a term and itself, and cannot be inferred from symmetry and transitiveness [Russell] |
10585 | Symmetrical and transitive relations are formally like equality [Russell] |
7781 | I call an object of thought a 'term'. This is a wide concept implying unity and existence. [Russell] |
14166 | Unities are only in propositions or concepts, and nothing that exists has unity [Russell] |
14164 | The only unities are simples, or wholes composed of parts [Russell] |
14112 | A set has some sort of unity, but not enough to be a 'whole' [Russell] |
14170 | Change is obscured by substance, a thing's nature, subject-predicate form, and by essences [Russell] |
14107 | Terms are identical if they belong to all the same classes [Russell] |
11849 | It at least makes sense to say two objects have all their properties in common [Wittgenstein on Russell] |
22303 | It makes no sense to say that a true proposition could have been false [Russell] |
19737 | A system can infer the structure of the world by making predictions about it [New Sci.] |
19736 | Neural networks can extract the car-ness of a car, or the chair-ness of a chair [New Sci.] |
16419 | No one has yet devised a rationality test [New Sci.] |
16418 | People can be highly intelligent, yet very stupid [New Sci.] |
16417 | About a third of variation in human intelligence is environmental [New Sci.] |
19484 | Psychologists measure personality along five dimensions [New Sci.] |
10583 | Abstraction principles identify a common property, which is some third term with the right relation [Russell] |
10582 | The principle of Abstraction says a symmetrical, transitive relation analyses into an identity [Russell] |
10584 | A certain type of property occurs if and only if there is an equivalence relation [Russell] |
14110 | Proposition contain entities indicated by words, rather than the words themselves [Russell] |
19164 | If propositions are facts, then false and true propositions are indistinguishable [Davidson on Russell] |
19157 | Russell said the proposition must explain its own unity - or else objective truth is impossible [Russell, by Davidson] |
14111 | A proposition is a unity, and analysis destroys it [Russell] |
14175 | We can drop 'cause', and just make inferences between facts [Russell] |
14172 | Moments and points seem to imply other moments and points, but don't cause them [Russell] |
14174 | The laws of motion and gravitation are just parts of the definition of a kind of matter [Russell] |
14168 | Occupying a place and change are prior to motion, so motion is just occupying places at continuous times [Russell] |
14171 | Force is supposed to cause acceleration, but acceleration is a mathematical fiction [Russell] |
19950 | Entropy is the only time-asymmetric law, so time may be linked to entropy [New Sci.] |
19478 | Light moves at a constant space-time speed, but its direction is in neither space nor time [New Sci.] |
19474 | Quantum states are measured by external time, of unknown origin [New Sci.] |
19473 | The Schrödinger equation describes the evolution of an object's wave function in Hilbert space [New Sci.] |
19476 | String theory needs at least 10 space-time dimensions [New Sci.] |
19953 | In string theory space-time has a grainy indivisible substructure [New Sci.] |
19954 | It is impossible for find a model of actuality among the innumerable models in string theory [New Sci.] |
19947 | Hilbert Space is an abstraction representing all possible states of a quantum system [New Sci.] |
14160 | Space is the extension of 'point', and aggregates of points seem necessary for geometry [Russell] |
19955 | Space-time may be a geometrical manifestation of quantum entanglement [New Sci.] |
19948 | Einstein's merging of time with space has left us confused about the nature of time [New Sci.] |
19475 | Relativity makes time and space jointly basic; quantum theory splits them, and prioritises time [New Sci.] |
19949 | Quantum theory relies on a clock outside the system - but where is it located? [New Sci.] |
19951 | Entropy is puzzling, so we may need to build new laws which include time directionality [New Sci.] |
14156 | Mathematicians don't distinguish between instants of time and points on a line [Russell] |
14169 | The 'universe' can mean what exists now, what always has or will exist [Russell] |
19477 | General relativity predicts black holes, as former massive stars, and as galaxy centres [New Sci.] |
19952 | Black holes have entropy, but general relativity says they are unstructured, and lack entropy [New Sci.] |
16420 | 84.5 percent of the universe is made of dark matter [New Sci.] |
17604 | We are halfway to synthesising any molecule we want [New Sci.] |
17603 | Chemistry just needs the periodic table, and protons, electrons and neutrinos [New Sci.] |