21 ideas
| 18776 | Contextual definitions eliminate descriptions from contexts [Linsky,B] |
| Full Idea: A 'contextual' definition shows how to eliminate a description from a context. | |
| From: Bernard Linsky (Quantification and Descriptions [2014], 2) | |
| A reaction: I'm trying to think of an example, but what I come up with are better described as 'paraphrases' than as 'definitions'. |
| 21704 | 'Impredictative' definitions fix a class in terms of the greater class to which it belongs [Linsky,B] |
| Full Idea: The ban on 'impredicative' definitions says you can't define a class in terms of a totality to which that class must be seen as belonging. | |
| From: Bernard Linsky (Russell's Metaphysical Logic [1999], 1) | |
| A reaction: So that would be defining 'citizen' in terms of the community to which the citizen belongs? If you are asked to define 'community' and 'citizen' together, where do you start? But how else can it be done? Russell's Reducibility aimed to block this. |
| 21705 | Reducibility says any impredicative function has an appropriate predicative replacement [Linsky,B] |
| Full Idea: The Axiom of Reducibility avoids impredicativity, by asserting that for any predicate of given arguments defined by quantifying over higher-order functions or classes, there is another co-extensive but predicative function of the same type of arguments. | |
| From: Bernard Linsky (Russell's Metaphysical Logic [1999], 1) | |
| A reaction: Eventually the axiom seemed too arbitrary, and was dropped. Linsky's book explores it. |
| 18774 | Definite descriptions, unlike proper names, have a logical structure [Linsky,B] |
| Full Idea: Definite descriptions seem to have a logical structure in a way that proper names do not. | |
| From: Bernard Linsky (Quantification and Descriptions [2014], 1.1.1) | |
| A reaction: Thus descriptions have implications which plain names do not. |
| 21727 | Definite descriptions theory eliminates the King of France, but not the Queen of England [Linsky,B] |
| Full Idea: The theory of definite descriptions may eliminate apparent commitment to such entities as the present King of France, but certainly not to the present Queen of England. | |
| From: Bernard Linsky (Russell's Metaphysical Logic [1999], 7.3) |
| 21719 | Extensionalism means what is true of a function is true of coextensive functions [Linsky,B] |
| Full Idea: With the principle of extensionality anything true of one propositional functions will be true of every coextensive one. | |
| From: Bernard Linsky (Russell's Metaphysical Logic [1999], 6.3) |
| 21554 | Sets always exceed terms, so all the sets must exceed all the sets [Lackey] |
| Full Idea: Cantor proved that the number of sets in a collection of terms is larger than the number of terms. Hence Cantor's Paradox says the number of sets in the collection of all sets must be larger than the number of sets in the collection of all sets. | |
| From: Douglas Lackey (Intros to Russell's 'Essays in Analysis' [1973], p.127) | |
| A reaction: The sets must count as terms in the next iteration, but that is a normal application of the Power Set axiom. |
| 21553 | It seems that the ordinal number of all the ordinals must be bigger than itself [Lackey] |
| Full Idea: The ordinal series is well-ordered and thus has an ordinal number, and a series of ordinals to a given ordinal exceeds that ordinal by 1. So the series of all ordinals has an ordinal number that exceeds its own ordinal number by 1. | |
| From: Douglas Lackey (Intros to Russell's 'Essays in Analysis' [1973], p.127) | |
| A reaction: Formulated by Burali-Forti in 1897. |
| 10245 | One geometry cannot be more true than another [Poincaré] |
| Full Idea: One geometry cannot be more true than another; it can only be more convenient. | |
| From: Henri Poincaré (Science and Method [1908], p.65), quoted by Stewart Shapiro - Philosophy of Mathematics | |
| A reaction: This is the culminating view after new geometries were developed by tinkering with Euclid's parallels postulate. |
| 15923 | Poincaré rejected the actual infinite, claiming definitions gave apparent infinity to finite objects [Poincaré, by Lavine] |
| Full Idea: Poincaré rejected the actual infinite. He viewed mathematics that is apparently concerned with the actual infinite as actually concerning the finite linguistic definitions the putatively describe actually infinite objects. | |
| From: report of Henri Poincaré (On the Nature of Mathematical Reasoning [1894]) by Shaughan Lavine - Understanding the Infinite |
| 10180 | Mathematicians do not study objects, but relations between objects [Poincaré] |
| Full Idea: Mathematicians do not study objects, but relations between objects; it is a matter of indifference if the objects are replaced by others, provided the relations do not change. They are interested in form alone, not matter. | |
| From: Henri Poincaré (Science and Hypothesis [1902], p.20), quoted by E Reck / M Price - Structures and Structuralism in Phil of Maths §6 | |
| A reaction: This connects modern structuralism with Aritotle's interest in the 'form' of things. Contrary to the views of the likes of Frege, it is hard to see that the number '7' has any properties at all, apart from its relations. A daffodil would do just as well. |
| 21723 | The task of logicism was to define by logic the concepts 'number', 'successor' and '0' [Linsky,B] |
| Full Idea: The problem for logicism was to find definitions of the primitive notions of Peano's theory, number, successor and 0, in terms of logical notions, so that the postulates could then be derived by logic alone. | |
| From: Bernard Linsky (Russell's Metaphysical Logic [1999], 7) | |
| A reaction: Both Frege and Russell defined numbers as equivalence classes. Successor is easily defined (in various ways) in set theory. An impossible set can exemplify zero. The trouble for logicism is this all relies on sets. |
| 21721 | Higher types are needed to distinguished intensional phenomena which are coextensive [Linsky,B] |
| Full Idea: The higher types are needed for intensional phenomena, cases where the same class is picked out by distinct propositional functions. | |
| From: Bernard Linsky (Russell's Metaphysical Logic [1999], 6.4) | |
| A reaction: I take it that in this way 'x is renate' can be distinguished from 'x is cordate', a task nowadays performed by possible worlds. |
| 21703 | Types are 'ramified' when there are further differences between the type of quantifier and its range [Linsky,B] |
| Full Idea: The types is 'ramified' because there are further differences between the type of a function defined in terms of a quantifier ranging over other functions and the type of those other functions, despite the functions applying to the same simple type. | |
| From: Bernard Linsky (Russell's Metaphysical Logic [1999], 1) | |
| A reaction: Not sure I understand this, but it evidently created difficulties for dealing with actual mathematics, and Ramsey showed how you could manage without the ramifications. |
| 21714 | The ramified theory subdivides each type, according to the range of the variables [Linsky,B] |
| Full Idea: The original ramified theory of types ...furthern subdivides each of the types of the 'simple' theory according to the range of the bound variables used in the definition of each propositional function. | |
| From: Bernard Linsky (Russell's Metaphysical Logic [1999], 6) | |
| A reaction: For a non-intiate like me it certainly sounds disappointing that such a bold and neat theory because a tangle of complications. Ramsey and Russell in the 1920s seem to have dropped the ramifications. |
| 21713 | Did logicism fail, when Russell added three nonlogical axioms, to save mathematics? [Linsky,B] |
| Full Idea: It is often thought that Logicism was a failure, because after Frege's contradiction, Russell required obviously nonlogical principles, in order to develop mathematics. The axioms of Reducibility, Infinity and Choice are cited. | |
| From: Bernard Linsky (Russell's Metaphysical Logic [1999], 6) | |
| A reaction: Infinity and Choice remain as axioms of the standard ZFC system of set theory, which is why set theory is always assumed to be 'up to its neck' in ontological commitments. Linsky argues that Russell saw ontology in logic. |
| 21715 | For those who abandon logicism, standard set theory is a rival option [Linsky,B] |
| Full Idea: ZF set theory is seen as a rival to logicism as a foundational scheme. Set theory is for those who have given up the project of reducing mathematics to logic. | |
| From: Bernard Linsky (Russell's Metaphysical Logic [1999], 6.1) | |
| A reaction: Presumably there are other rivals. Set theory has lots of ontological commitments. One could start at the other end, and investigate the basic ontological commitments of arithmetic. I have no idea what those might be. |
| 9916 | Convention, yes! Arbitrary, no! [Poincaré, by Putnam] |
| Full Idea: Poincaré once exclaimed, 'Convention, yes! Arbitrary, no!'. | |
| From: report of Henri Poincaré (talk [1901]) by Hilary Putnam - Models and Reality | |
| A reaction: An interesting view. It mustn't be assumed that conventions are not rooted in something. Maybe a sort of pragmatism is implied. |
| 18203 | Avoid non-predicative classifications and definitions [Poincaré] |
| Full Idea: Never consider any objects but those capable of being defined in a finite number of word ...Avoid non-predicative classifications and definitions. | |
| From: Henri Poincaré (The Logic of Infinity [1909], p.63), quoted by Penelope Maddy - Naturalism in Mathematics II.4 |
| 21729 | Construct properties as sets of objects, or say an object must be in the set to have the property [Linsky,B] |
| Full Idea: Rather than directly constructing properties as sets of objects and proving neat facts about properties by proxy, we can assert biconditionals, such as that an object has a property if and only if it is in a certain set. | |
| From: Bernard Linsky (Russell's Metaphysical Logic [1999], 7.6) | |
| A reaction: Linsky is describing Russell's method of logical construction. I'm not clear what is gained by this move, but at least it is a variant of the usual irritating expression of properties as sets of objects. |
| 15877 | The aim of science is just to create a comprehensive, elegant language to describe brute facts [Poincaré, by Harré] |
| Full Idea: In Poincaré's view, we try to construct a language within which the brute facts of experience are expressed as comprehensively and as elegantly as possible. The job of science is the forging of a language precisely suited to that purpose. | |
| From: report of Henri Poincaré (The Value of Science [1906], Pt III) by Rom Harré - Laws of Nature 2 | |
| A reaction: I'm often struck by how obscure and difficult our accounts of self-evident facts can be. Chairs are easy, and the metaphysics of chairs is hideous. Why is that? I'm a robust realist, but I like Poincaré's idea. He permits facts. |