73 ideas
| 9542 | The best known axiomatization of PL is Whitehead/Russell, with four axioms and two rules [Russell/Whitehead, by Hughes/Cresswell] |
| Full Idea: The best known axiomatization of PL is Whitehead/Russell. There are four axioms: (p∨p)→p, q→(p∨q), (p→q)→(q∨p), and (q→r)→((p∨q)→(p∨r)), plus Substitution and Modus Ponens rules. | |
| From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by GE Hughes/M Cresswell - An Introduction to Modal Logic Ch.1 |
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| Full Idea: By Gödel's First Incompleteness Theorem, there cannot be a negation-complete set theory. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 01.3) | |
| A reaction: This means that we can never prove all the truths of a system of set theory. |
| 21720 | Russell saw Reducibility as legitimate for reducing classes to logic [Linsky,B on Russell/Whitehead] |
| Full Idea: The axiom of Reducibility ...is crucial in the reduction of classes to logic, ...and seems to be a quite legitimate logical notion for Russell. | |
| From: comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by Bernard Linsky - Russell's Metaphysical Logic 6.4 | |
| A reaction: This is an unusual defence of the axiom, which is usually presumed to have been kicked into the long grass by Quine. If one could reduce classes to logic, that would destroy the opposition to logicism in a single neat coup. |
| 10044 | Russell denies extensional sets, because the null can't be a collection, and the singleton is just its element [Russell/Whitehead, by Shapiro] |
| Full Idea: Russell adduces two reasons against the extensional view of classes, namely the existence of the null class (which cannot very well be a collection), and the unit classes (which would have to be identical with their single elements). | |
| From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Stewart Shapiro - Structure and Ontology p.459 | |
| A reaction: Gödel believes in the reality of classes. I have great sympathy with Russell, when people start to claim that sets are not just conveniences to help us think about things, but actual abstract entities. Is the singleton of my pencil is on this table? |
| 18208 | We regard classes as mere symbolic or linguistic conveniences [Russell/Whitehead] |
| Full Idea: Classes, so far as we introduce them, are merely symbolic or linguistic conveniences, not genuine objects. | |
| From: B Russell/AN Whitehead (Principia Mathematica [1913], p.72), quoted by Penelope Maddy - Naturalism in Mathematics III.2 |
| 10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
| Full Idea: Going second-order in arithmetic enables us to prove new first-order arithmetical sentences that we couldn't prove before. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 23.4) | |
| A reaction: The wages of Satan, perhaps. We can prove things about objects by proving things about their properties and sets and functions. Smith says this fact goes all the way up the hierarchy. |
| 8204 | Lewis's 'strict implication' preserved Russell's confusion of 'if...then' with implication [Quine on Russell/Whitehead] |
| Full Idea: Russell call 'if...then' implication, when the material conditional is a much better account; C.I.Lewis (in founding modern modal logic) preserved Russell's confusion by creating 'strict implication', and called that implication. | |
| From: comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by Willard Quine - Reply to Professor Marcus p.177 | |
| A reaction: [A compession of Quine's paragraph]. All of this assumes that logicians can give an accurate account of what if...then means, when ordinary usage is broad and vague. Strict implication seems to drain all the normal meaning out of 'if...then'. |
| 9359 | Russell's implication means that random sentences imply one another [Lewis,CI on Russell/Whitehead] |
| Full Idea: In Mr Russell's idea of implication, if twenty random sentences from a newspaper were put in a hat, and two of them drawn at random, one will certainly imply the other, and it is an even bet the implication will be mutual. | |
| From: comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by C.I. Lewis - A Pragmatic Conception of the A Priori p.366 | |
| A reaction: This sort of lament leads modern logicians to suggest 'relevance' as an important criterion. It certainly seems odd that so-called 'classical logic' should contain a principle so at variance with everyday reasoning. |
| 21707 | Russell unusually saw logic as 'interpreted' (though very general, and neutral) [Russell/Whitehead, by Linsky,B] |
| Full Idea: Russell did not view logic as an uninterpreted calculus awaiting interpretations [the modern view]. Rather, logic is a single 'interpreted' body of a priori truths, of propositions rather than sentence forms - but maximally general and topic neutral. | |
| From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Bernard Linsky - Russell's Metaphysical Logic 1 | |
| A reaction: This is the view which Wittgenstein challenged, saying logic is just conventional. Linsky claims that Russell's logicism is much more plausible, once you understand his view of logic. |
| 10075 | A 'partial function' maps only some elements to another set [Smith,P] |
| Full Idea: A 'partial function' is one which maps only some elements of a domain to elements in another set. For example, the reciprocal function 1/x is not defined for x=0. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1 n1) |
| 10074 | A 'total function' maps every element to one element in another set [Smith,P] |
| Full Idea: A 'total function' is one which maps every element of a domain to exactly one corresponding value in another set. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) |
| 10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
| Full Idea: If a function f maps the argument a back to a itself, so that f(a) = a, then a is said to be a 'fixed point' for f. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 20.5) |
| 10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
| Full Idea: The 'range' of a function is the set of elements in the output set that are values of the function for elements in the original set. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) | |
| A reaction: In other words, the range is the set of values that were created by the function. |
| 10605 | Two functions are the same if they have the same extension [Smith,P] |
| Full Idea: We count two functions as being the same if they have the same extension, i.e. if they pair up arguments with values in the same way. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 11.3) | |
| A reaction: So there's only one way to skin a cat in mathematical logic. |
| 10036 | In 'Principia' a new abstract theory of relations appeared, and was applied [Russell/Whitehead, by Gödel] |
| Full Idea: In 'Principia' a young science was enriched with a new abstract theory of relations, ..and not only Cantor's set theory but also ordinary arithmetic and the theory of measurement are treated from this abstract relational standpoint. | |
| From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Kurt Gödel - Russell's Mathematical Logic p.448 | |
| A reaction: I presume this is accounting for relations in terms of ordered sets. |
| 10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
| Full Idea: The so-called Comprehension Schema ∃X∀x(Xx ↔ φ(x)) says that there is a property which is had by just those things which satisfy the condition φ. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 22.3) |
| 10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
| Full Idea: 'Theorem': given a derivation of the sentence φ from the axioms of the theory T using the background logical proof system, we will say that φ is a 'theorem' of the theory. Standard abbreviation is T |- φ. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) |
| 10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
| Full Idea: A 'natural deduction system' will have no logical axioms but may rules of inference. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 09.1) | |
| A reaction: He contrasts this with 'Hilbert-style systems', which have many axioms but few rules. Natural deduction uses many assumptions which are then discharged, and so tree-systems are good for representing it. |
| 10613 | No nice theory can define truth for its own language [Smith,P] |
| Full Idea: No nice theory can define truth for its own language. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 21.5) | |
| A reaction: This leads on to Tarski's account of truth. |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
| Full Idea: An 'injective' function is 'one-to-one' - each element of the output set results from a different element of the original set. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) | |
| A reaction: That is, two different original elements cannot lead to the same output element. |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| Full Idea: A 'surjective' function is 'onto' - the whole of the output set results from the function being applied to elements of the original set. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| Full Idea: A 'bijective' function has 'one-to-one correspondence' - it is both surjective and injective, so that every element in each of the original and the output sets has a matching element in the other. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) | |
| A reaction: Note that 'injective' is also one-to-one, but only in the one direction. |
| 10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
| Full Idea: If everything that a theory proves must be true, then it is a 'sound' theory. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 01.1) |
| 10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
| Full Idea: Soundness is normally a matter of having true axioms and a truth-preserving proof system. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) | |
| A reaction: The only exception I can think of is if a theory consisted of nothing but the axioms. |
| 10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
| Full Idea: A theory is 'sound' iff every theorem of it is true (i.e. true on the interpretation built into its language). Soundness is normally a matter of having true axioms and a truth-preserving proof system. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) |
| 10598 | A theory is 'negation complete' if it proves all sentences or their negation [Smith,P] |
| Full Idea: A theory is 'negation complete' if it decides every sentence of its language (either the sentence, or its negation). | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) |
| 10597 | 'Complete' applies both to whole logics, and to theories within them [Smith,P] |
| Full Idea: There is an annoying double-use of 'complete': a logic may be semantically complete, but there may be an incomplete theory expressed in it. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) |
| 10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P] |
| Full Idea: Logicians say that a theory T is '(negation) complete' if, for every sentence φ in the language of the theory, either φ or ¬φ is deducible in T's proof system. If this were the case, then truth could be equated with provability. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 01.1) | |
| A reaction: The word 'negation' seems to be a recent addition to the concept. Presumable it might be the case that φ can always be proved, but not ¬φ. |
| 10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
| Full Idea: There are two routes to Incompleteness results. One goes via the semantic assumption that we are dealing with sound theories, using a result about what they can express. The other uses the syntactic notion of consistency, with stronger notions of proof. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 18.1) |
| 10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P] |
| Full Idea: An 'effectively decidable' (or 'computable') algorithm will be step-by-small-step, with no need for intuition, or for independent sources, with no random methods, possible for a dumb computer, and terminates in finite steps. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.2) | |
| A reaction: [a compressed paragraph] |
| 10087 | A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P] |
| Full Idea: A theory is 'decidable' iff there is a mechanical procedure for determining whether any sentence of its language can be proved. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) | |
| A reaction: Note that it doesn't actually have to be proved. The theorems of the theory are all effectively decidable. |
| 10088 | Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P] |
| Full Idea: Any consistent, axiomatized, negation-complete formal theory is decidable. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.6) |
| 10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
| Full Idea: A set is 'enumerable' iff either the set is empty, or there is a surjective function to the set from the set of natural numbers, so that the set is in the range of that function. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.3) |
| 10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
| Full Idea: A set is 'effectively enumerable' if an (idealised) computer could be programmed to generate a list of its members such that any member will eventually be mentioned (even if the list is empty, or without end, or contains repetitions). | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.4) |
| 10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
| Full Idea: A finite set of finitely specifiable objects is always effectively enumerable (for example, the prime numbers). | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.4) |
| 10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
| Full Idea: The set of ordered pairs of natural numbers (i,j) is effectively enumerable, as proven by listing them in an array (across: <0,0>, <0,1>, <0,2> ..., and down: <0,0>, <1,0>, <2,0>...), and then zig-zagging. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.5) |
| 10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |
| Full Idea: The theorems of any properly axiomatized theory can be effectively enumerated. However, the truths of any sufficiently expressive arithmetic can't be effectively enumerated. Hence the theorems and truths of arithmetic cannot be the same. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 05 Intro) |
| 10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
| Full Idea: Whether a property is 'expressible' in a given theory depends on the richness of the theory's language. Whether the property can be 'captured' (or 'represented') by the theory depends on the richness of the axioms and proof system. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 04.7) |
| 10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
| Full Idea: For prime numbers we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))). That is, the only way to multiply two numbers and a get a prime is if one of them is 1. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 04.5) |
| 10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
| Full Idea: It has been proved (by Tarski) that the real numbers R is a complete theory. But this means that while the real numbers contain the natural numbers, the pure theory of real numbers doesn't contain the theory of natural numbers. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 18.2) |
| 18248 | A real number is the class of rationals less than the number [Russell/Whitehead, by Shapiro] |
| Full Idea: For Russell the real number 2 is the class of rationals less than 2 (i.e. 2/1). ...Notice that on this definition, real numbers are classes of rational numbers. | |
| From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Stewart Shapiro - Thinking About Mathematics 5.2 |
| 10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
| Full Idea: The truths of arithmetic are just the true equations involving particular numbers, and universally quantified versions of such equations. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 27.7) | |
| A reaction: Must each equation be universally quantified? Why can't we just universally quantify over the whole system? |
| 10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
| Full Idea: All numbers are related to zero by the ancestral of the successor relation. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 23.5) | |
| A reaction: The successor relation only ties a number to the previous one, not to the whole series. Ancestrals are a higher level of abstraction. |
| 10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
| Full Idea: The number of Fs is the 'successor' of the number of Gs if there is an object which is an F, and the remaining things that are F but not identical to the object are equinumerous with the Gs. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 14.1) |
| 10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
| Full Idea: Baby Arithmetic 'knows' the addition of particular numbers and multiplication, but can't express general facts about numbers, because it lacks quantification. It has a constant '0', a function 'S', and functions '+' and 'x', and identity and negation. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 08.1) |
| 10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
| Full Idea: Baby Arithmetic is negation complete, so it can prove every claim (or its negation) that it can express, but it is expressively extremely impoverished. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 08.3) |
| 10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
| Full Idea: We can beef up Baby Arithmetic into Robinson Arithmetic (referred to as 'Q'), by restoring quantifiers and variables. It has seven generalised axioms, plus standard first-order logic. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 08.3) |
| 10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
| Full Idea: Robinson Arithmetic (Q) is not negation complete | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 08.4) |
| 10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
| Full Idea: The sequence of natural numbers starts from zero, and each number has just one immediate successor; the sequence continues without end, never circling back on itself, and there are no 'stray' numbers, lurking outside the sequence. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 01.1) | |
| A reaction: These are the characteristics of the natural numbers which have to be pinned down by any axiom system, such as Peano's, or any more modern axiomatic structures. We are in the territory of Gödel's theorems. |
| 10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
| Full Idea: If the logic of arithmetic doesn't have second-order quantifiers to range over properties of numbers, how can it handle induction? | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 10.1) |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| Full Idea: Multiplication in itself isn't is intractable. In 1929 Skolem showed a complete theory for a first-order language with multiplication but lacking addition (or successor). Multiplication together with addition and successor produces incompleteness. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 10.7 n8) |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| Full Idea: Putting multiplication together with addition and successor in the language of arithmetic produces incompleteness. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 10.7) | |
| A reaction: His 'Baby Arithmetic' has all three and is complete, but lacks quantification (p.51) |
| 18152 | Russell takes numbers to be classes, but then reduces the classes to numerical quantifiers [Russell/Whitehead, by Bostock] |
| Full Idea: Although Russell takes numbers to be certain classes, his 'no-class' theory then eliminates all mention of classes in favour of the 'propositional functions' that define them; and in the case of the numbers these just are the numerical quantifiers. | |
| From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by David Bostock - Philosophy of Mathematics 9.B.4 |
| 10025 | Russell and Whitehead took arithmetic to be higher-order logic [Russell/Whitehead, by Hodes] |
| Full Idea: Russell and Whitehead took arithmetic to be higher-order logic, ..and came close to identifying numbers with numerical quantifiers. | |
| From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.148 | |
| A reaction: The point here is 'higher-order'. |
| 8683 | Russell and Whitehead were not realists, but embraced nearly all of maths in logic [Russell/Whitehead, by Friend] |
| Full Idea: Unlike Frege, Russell and Whitehead were not realists about mathematical objects, and whereas Frege thought that only arithmetic and analysis are branches of logic, they think the vast majority of mathematics (including geometry) is essentially logical. | |
| From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Michčle Friend - Introducing the Philosophy of Mathematics 3.1 | |
| A reaction: If, in essence, Descartes reduced geometry to algebra (by inventing co-ordinates), then geometry ought to be included. It is characteristic of Russell's hubris to want to embrace everything. |
| 10037 | 'Principia' lacks a precise statement of the syntax [Gödel on Russell/Whitehead] |
| Full Idea: What is missing, above all, in 'Principia', is a precise statement of the syntax of the formalism. | |
| From: comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by Kurt Gödel - Russell's Mathematical Logic p.448 |
| 10093 | The ramified theory of types used propositional functions, and covered bound variables [Russell/Whitehead, by George/Velleman] |
| Full Idea: Russell and Whitehead's ramified theory of types worked not with sets, but with propositional functions (similar to Frege's concepts), with a more restrictive assignment of variables, insisting that bound, as well as free, variables be of lower type. | |
| From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.3 | |
| A reaction: I don't fully understand this (and no one seems much interested any more), but I think variables are a key notion, and there is something interesting going on here. I am intrigued by ordinary language which behaves like variables. |
| 8691 | The Russell/Whitehead type theory was limited, and was not really logic [Friend on Russell/Whitehead] |
| Full Idea: The Russell/Whitehead type theory reduces mathematics to a consistent founding discipline, but is criticised for not really being logic. They could not prove the existence of infinite sets, and introduced a non-logical 'axiom of reducibility'. | |
| From: comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by Michčle Friend - Introducing the Philosophy of Mathematics 3.6 | |
| A reaction: To have reduced most of mathematics to a founding discipline sounds like quite an achievement, and its failure to be based in pure logic doesn't sound too bad. However, it seems to reduce some maths to just other maths. |
| 10305 | In 'Principia Mathematica', logic is exceeded in the axioms of infinity and reducibility, and in the domains [Bernays on Russell/Whitehead] |
| Full Idea: In the system of 'Principia Mathematica', it is not only the axioms of infinity and reducibility which go beyond pure logic, but also the initial conception of a universal domain of individuals and of a domain of predicates. | |
| From: comment on B Russell/AN Whitehead (Principia Mathematica [1913], p.267) by Paul Bernays - On Platonism in Mathematics p.267 | |
| A reaction: This sort of criticism seems to be the real collapse of the logicist programme, rather than Russell's paradox, or Gödel's Incompleteness Theorems. It just became impossible to stick strictly to logic in the reduction of arithmetic. |
| 8684 | Russell and Whitehead consider the paradoxes to indicate that we create mathematical reality [Russell/Whitehead, by Friend] |
| Full Idea: Russell and Whitehead are particularly careful to avoid paradox, and consider the paradoxes to indicate that we create mathematical reality. | |
| From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Michčle Friend - Introducing the Philosophy of Mathematics 3.1 | |
| A reaction: This strikes me as quite a good argument. It is certainly counterintuitive that reality, and abstractions from reality, would contain contradictions. The realist view would be that we have paradoxes because we have misdescribed the facts. |
| 8746 | To avoid vicious circularity Russell produced ramified type theory, but Ramsey simplified it [Russell/Whitehead, by Shapiro] |
| Full Idea: Russell insisted on the vicious circle principle, and thus rejected impredicative definitions, which resulted in an unwieldy ramified type theory, with the ad hoc axiom of reducibility. Ramsey's simpler theory was impredicative and avoided the axiom. | |
| From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Stewart Shapiro - Thinking About Mathematics 5.2 | |
| A reaction: Nowadays the theory of types seems to have been given up, possibly because it has no real attraction if it lacks the strict character which Russell aspired to. |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| Full Idea: The 'ancestral' of a relation is that relation which holds when there is an indefinitely long chain of things having the initial relation. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 23.5) | |
| A reaction: The standard example is spotting the relation 'ancestor' from the receding relation 'parent'. This is a sort of abstraction derived from a relation which is not equivalent (parenthood being transitive but not reflexive). The idea originated with Frege. |
| 12033 | An object is identical with itself, and no different indiscernible object can share that [Russell/Whitehead, by Adams,RM] |
| Full Idea: Trivially, the Identity of Indiscernibles says that two individuals, Castor and Pollux, cannot have all properties in common. For Castor must have the properties of being identical with Castor and not being identical with Pollux, which Pollux can't share. | |
| From: report of B Russell/AN Whitehead (Principia Mathematica [1913], I p.57) by Robert Merrihew Adams - Primitive Thisness and Primitive Identity 2 | |
| A reaction: I suspect that either the property of being identical with itself is quite vacuous, or it is parasytic on primitive identity, or it is the criterion which is actually used to define identity. Either way, I don't find this claim very illuminating. |
| 10040 | Russell showed, through the paradoxes, that our basic logical intuitions are self-contradictory [Russell/Whitehead, by Gödel] |
| Full Idea: By analyzing the paradoxes to which Cantor's set theory had led, ..Russell brought to light the amazing fact that our logical intuitions (concerning such notions as truth, concept, being, class) are self-contradictory. | |
| From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Kurt Gödel - Russell's Mathematical Logic p.452 | |
| A reaction: The main intuition that failed was, I take it, that every concept has an extension, that is, there are always objects which will or could fall under the concept. |
| 21725 | The multiple relations theory says assertions about propositions are about their ingredients [Russell/Whitehead, by Linsky,B] |
| Full Idea: The multiple relations theory of judgement proposes that assertions about propositions are dependent upon genuine facts involving belief and other attitude relations, subjects of those attitudes, and the constituents of the belief. | |
| From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Bernard Linsky - Russell's Metaphysical Logic 7.2 | |
| A reaction: This seems to require a commitment to universals (especially relations) with which we can be directly acquainted. I prefer propositions, but as mental entities, not platonic entities. |
| 23474 | A judgement is a complex entity, of mind and various objects [Russell/Whitehead] |
| Full Idea: When a judgement occurs, there is a certain complex entity, composed of the mind and the various objects of the judgement. | |
| From: B Russell/AN Whitehead (Principia Mathematica [1913], p.44) | |
| A reaction: This is Russell's multiple-relation theory of judgement, which replaced his earlier belief in unified propositions (now 'false abstractions'). He seems to have accepted Locke's view, that the act of judgement produces the unity. |
| 23455 | The meaning of 'Socrates is human' is completed by a judgement [Russell/Whitehead] |
| Full Idea: When I judge 'Socrates is human', the meaning is completed by the act of judging. | |
| From: B Russell/AN Whitehead (Principia Mathematica [1913], p.44), quoted by Michael Morris - Guidebook to Wittgenstein's Tractatus | |
| A reaction: Morris says this is Russell's multiple-relations theory of judgement. The theory accompanies the rejection of the concept of the unified proposition. When I hear 'Socrates had a mole on his shoulder' I get the meaning without judging. |
| 23480 | The multiple relation theory of judgement couldn't explain the unity of sentences [Morris,M on Russell/Whitehead] |
| Full Idea: When Russell moved to his multiple relation theory of judgement …he then faced difficulties making sense of the unity of sentences. | |
| From: comment on B Russell/AN Whitehead (Principia Mathematica [1913], p.44) by Michael Morris - Guidebook to Wittgenstein's Tractatus 3A | |
| A reaction: Roughly, he seems committed to saying that there is only unity if you think there is unity; there is no unity in a sentence prior to the act of judgement. |
| 18275 | Only the act of judging completes the meaning of a statement [Russell/Whitehead] |
| Full Idea: When I judge 'Socrates is human', the meaning is completed by the act of judging, and we no longer have an incomplete symbol. | |
| From: B Russell/AN Whitehead (Principia Mathematica [1913], p.44), quoted by J. Alberto Coffa - The Semantic Tradition from Kant to Carnap | |
| A reaction: Personally I would have thought that you needed to know the meaning properly before you could make the judgement, but then he is Bertrand Russell and I'm not. |
| 23453 | Propositions as objects of judgement don't exist, because we judge several objects, not one [Russell/Whitehead] |
| Full Idea: A 'proposition', in the sense in which a proposition is supposed to be the object of a judgement, is a false abstraction, because a judgement has several objects, not one. | |
| From: B Russell/AN Whitehead (Principia Mathematica [1913], p.44), quoted by Michael Morris - Guidebook to Wittgenstein's Tractatus 2E | |
| A reaction: This is the rejection of the 'Russellian' theory of propositions, in favour of his multiple-relations theory of judgement. But why don't the related objects add up to a proposition about a state of affairs? |
| 6000 | The goal is rationality in the selection of things according to nature [Diogenes of Babylon, by Blank] |
| Full Idea: Diogenes of Babylon defined the goal to be rationality in the selection and rejection of the things according to nature. | |
| From: report of Diogenes (Bab) (fragments/reports [c.180 BCE]) by D.L. Blank - Diogenes of Babylon | |
| A reaction: This captures the central Stoic idea quite nicely. 'Live according to nature', but this always meant 'live according to reason', because that is (as Aristotle had taught) the essence of our nature. This only makes sense if reason and nature coincide. |
| 5999 | The good is what is perfect by nature [Diogenes of Babylon, by Blank] |
| Full Idea: Diogenes of Babylon defined the good as what is perfect by nature. | |
| From: report of Diogenes (Bab) (fragments/reports [c.180 BCE]) by D.L. Blank - Diogenes of Babylon | |
| A reaction: This might come close to G.E. Moore's Ideal Utilitarianism, but its dependence on the rather uneasy of concept of 'perfection' makes it questionable. Personally I find it appealing. I wish we had Diogenes' explanation. |
| 6001 | Justice is a disposition to distribute according to desert [Diogenes of Babylon, by Blank] |
| Full Idea: Diogenes of Babylon defined justice as the disposition which distributes to everyone what he deserves. | |
| From: report of Diogenes (Bab) (fragments/reports [c.180 BCE]) by D.L. Blank - Diogenes of Babylon | |
| A reaction: The questions that arise would be 'what does a new-born baby deserve?', and 'what do animals deserve?', and 'does the lowest and worst of criminals deserve anything at all?' |