display all the ideas for this combination of philosophers
22 ideas
14113 | The null class is a fiction [Russell] |
Full Idea: The null class is a fiction. | |
From: Bertrand Russell (The Principles of Mathematics [1903], §079) | |
A reaction: This does not commit him to regarding all classes as fictions - though he seems to have eventually come to believe that. The null class seems to have a role something like 'Once upon a time...' in story-telling. You can then tell truth or fiction. |
6103 | Normally a class with only one member is a problem, because the class and the member are identical [Russell] |
Full Idea: With the ordinary view of classes you would say that a class that has only one member was the same as that one member; that will land you in terrible difficulties, because in that case that one member is a member of that class, namely, itself. | |
From: Bertrand Russell (The Philosophy of Logical Atomism [1918], §VII) | |
A reaction: The problem (I think) is that classes (sets) were defined by Frege as being identical with their members (their extension). With hindsight this may have been a mistake. The question is always 'why is that particular a member of that set?' |
14427 | We can enumerate finite classes, but an intensional definition is needed for infinite classes [Russell] |
Full Idea: We know a great deal about a class without enumerating its members …so definition by extension is not necessary to knowledge about a class ..but enumeration of infinite classes is impossible for finite beings, so definition must be by intension. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II) | |
A reaction: Presumably mathematical induction (which keeps apply the rule to extend the class) will count as an intension here. |
14428 | Members define a unique class, whereas defining characteristics are numerous [Russell] |
Full Idea: There is only one class having a given set of members, whereas there are always many different characteristics by which a given class may be defined. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II) |
14440 | We may assume that there are infinite collections, as there is no logical reason against them [Russell] |
Full Idea: There is no logical reason against infinite collections, and we are therefore justified, in logic, in investigating the hypothesis that there are such collections. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VIII) |
14447 | Infinity says 'for any inductive cardinal, there is a class having that many terms' [Russell] |
Full Idea: The Axiom of Infinity may be enunciated as 'If n be any inductive cardinal number, there is at least one class of individuals having n terms'. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XIII) | |
A reaction: So for every possible there exists a set of terms for it. Notice that they are 'terms', not 'objects'. We must decide whether we are allowed terms which don't refer to real objects. |
14443 | The British parliament has one representative selected from each constituency [Russell] |
Full Idea: We have a class of representatives, who make up our Parliament, one being selected out of each constituency. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII) | |
A reaction: You can rely on Russell for the clearest illustrations of these abstract ideas. He calls the Axiom of Choice the 'Multiplicative' Axiom. |
14445 | Choice shows that if any two cardinals are not equal, one must be the greater [Russell] |
Full Idea: The [Axiom of Choice] is also equivalent to the assumption that of any two cardinals which are not equal, one must be the greater. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII) | |
A reaction: It is illuminating for the uninitiated to learn that this result can't be taken for granted (with infinite cardinals). |
14444 | Choice is equivalent to the proposition that every class is well-ordered [Russell] |
Full Idea: Zermelo has shown that [the Axiom of Choice] is equivalent to the proposition that every class is well-ordered, i.e. can be arranged in a series in which every sub-class has a first term (except, of course, the null class). | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII) | |
A reaction: Russell calls Choice the 'Multiplicative' Axiom. |
14446 | We can pick all the right or left boots, but socks need Choice to insure the representative class [Russell] |
Full Idea: Among boots we distinguish left and right, so we can choose all the right or left boots; with socks no such principle suggests itself, and we cannot be sure, without the [Axiom of Choice], that there is a class consisting of one sock from each pair. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII) | |
A reaction: A deservedly famous illustration of a rather tricky part of set theory. |
18130 | Axiom of Reducibility: there is always a function of the lowest possible order in a given level [Russell, by Bostock] |
Full Idea: Russell's Axiom of Reducibility states that to any propositional function of any order in a given level, there corresponds another which is of the lowest possible order in the level. There corresponds what he calls a 'predicative' function of that level. | |
From: report of Bertrand Russell (Substitutional Classes and Relations [1906]) by David Bostock - Philosophy of Mathematics 8.2 |
14459 | Reducibility: a family of functions is equivalent to a single type of function [Russell] |
Full Idea: The Axiom of Reducibility says 'There is a type of a-functions such that, given any a-function, it is formally equivalent to some function of the type in question'. ..It involves all that is really essential in the theory of classes. But is it true? | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVII) | |
A reaction: I take this to say that in the theory of types, it is possible to reduce each level of type down to one type. |
21563 | The 'no classes' theory says the propositions just refer to the members [Russell] |
Full Idea: The contention of the 'no classes' theory is that all significant propositions concerning classes can be regarded as propositions about all or some of their members. | |
From: Bertrand Russell (On 'Insolubilia' and their solution [1906], p.200) | |
A reaction: Apparently this theory has not found favour with later generations of theorists. I see it in terms of Russell trying to get ontology down to the minimum, in the spirit of Goodman and Quine. |
14461 | Propositions about classes can be reduced to propositions about their defining functions [Russell] |
Full Idea: It is right (in its main lines) to say that there is a reduction of propositions nominally about classes to propositions about their defining functions. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVII) | |
A reaction: The defining functions will involve the theory of types, in order to avoid the paradoxes of naïve set theory. This is Russell's strategy for rejecting the existence of sets. |
15894 | Russell invented the naïve set theory usually attributed to Cantor [Russell, by Lavine] |
Full Idea: Russell was the inventor of the naïve set theory so often attributed to Cantor. | |
From: report of Bertrand Russell (The Principles of Mathematics [1903]) by Shaughan Lavine - Understanding the Infinite I |
14126 | Order rests on 'between' and 'separation' [Russell] |
Full Idea: The two sources of order are 'between' and 'separation'. | |
From: Bertrand Russell (The Principles of Mathematics [1903], §204) |
14127 | Order depends on transitive asymmetrical relations [Russell] |
Full Idea: All order depends upon transitive asymmetrical relations. | |
From: Bertrand Russell (The Principles of Mathematics [1903], §208) |
8469 | Russell's proposal was that only meaningful predicates have sets as their extensions [Russell, by Orenstein] |
Full Idea: Russell's solution (in the theory of types) consists of restricting the principle that every predicate has a set as its extension so that only meaningful predicates have sets as their extensions. | |
From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919]) by Alex Orenstein - W.V. Quine Ch.3 | |
A reaction: There might be a chicken-and-egg problem here. How do you decide the members of a set (apart from ostensively) without deciding the predicate(s) that combine them? |
11064 | Classes can be reduced to propositional functions [Russell, by Hanna] |
Full Idea: Russell held that classes can be reduced to propositional functions. | |
From: report of Bertrand Russell (Mathematical logic and theory of types [1908]) by Robert Hanna - Rationality and Logic 2.4 | |
A reaction: The exact nature of a propositional function is disputed amongst Russell scholars (though it is roughly an open sentence of the form 'x is red'). |
7548 | Classes, grouped by a convenient property, are logical constructions [Russell] |
Full Idea: Classes or series of particulars, collected together on account of some property which makes it convenient to be able to speak of them as wholes, are what I call logical constructions or symbolic fictions. | |
From: Bertrand Russell (The Ultimate Constituents of Matter [1915], p.125) | |
A reaction: When does a construction become 'logical' instead of arbitrary? What is it about a property that makes it 'convenient'? At this point Russell seems to have built his ontology on classes, and the edifice was crumbling, thanks to Wittgenstein. |
8745 | Classes are logical fictions, and are not part of the ultimate furniture of the world [Russell] |
Full Idea: The symbols for classes are mere conveniences, not representing objects called 'classes'. Classes are in fact logical fictions; they cannot be regarded as part of the ultimate furniture of the world. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], Ch.18), quoted by Stewart Shapiro - Thinking About Mathematics 5.2 | |
A reaction: I agree. For 'logical fictions' read 'abstractions'. To equate abstractions with fictions is to underline the fact that they are a human creation. They are either that or platonic objects - there is no middle way. |
6436 | I gradually replaced classes with properties, and they ended as a symbolic convenience [Russell] |
Full Idea: My original use of classes was gradually more and more replaced by properties, and in the end disappeared except as a symbolic convenience. | |
From: Bertrand Russell (My Philosophical Development [1959], Ch.14) | |
A reaction: I wish I knew what properties are. On the whole, though, I agree with this, because it is more naturalistic. We may place things in classes because of their properties, and this means there are natural classes, but classes can't have a life of their own. |