Understanding Bertrand Russell's Definition of Number

[Barbie]

Hi, guys! I am looking for help understanding Russell's definition of a number for a grad math class. Any help will be much appreciated.

I have a paper from class that I can't find online to post here, but this is the actual "definition": (Ignore the circular appearance of the wording)

"The number of a class is the class of all those classes that are similar to it." thus "A number is anything which is the number of some class." -Bertrand Russell

(He is only talking about whole cardinal numbers here of course)

Here is an explanation I found online for you guys since I can't show you my paper:

According to Russell, the goal of the logicist programme is to show that

all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles (Russell 1903: v).

That is to say, pure mathematics is defined as a class of propositions asserting formal implications and containing only logical constants. The logical constants are: implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and the notions that are involved in formal implication, that is, truth, propositional function, class, denoting, and any or every term (Russell 1903: 106). According to Russell, the above apparatus of general logical notions is sufficient to establish "the whole theory of cardinal integers as a special branch of logic" (Russell 1903: 111). In his view, the "irreproachable" definition of number in purely logical terms is to define number as a class of classes. Two classes have the same number when their terms can be correlated one to one so that anyone term of either class corresponds to one and only one term of the other class. When the two classes have the same number, Russell calls them similar. The number of a class is the class of all classes similar to the given class. The Membership of this class of classes is a common property of all the similar classes and no others (Russell 1903: 115).

*** I did have a spark somewhere in my brain ( ), but I want to hear opinions of others as well. What do you think of this definition? How do you understand it?
__________________

 

[guigus]

Hi, guys! I am looking for help understanding Russell's definition of a number for a grad math class. Any help will be much appreciated.

I have a paper from class that I can't find online to post here, but this is the actual "definition": (Ignore the circular appearance of the wording)

"The number of a class is the class of all those classes that are similar to it." thus "A number is anything which is the number of some class." -Bertrand Russell

(He is only talking about whole cardinal numbers here of course)

Here is an explanation I found online for you guys since I can't show you my paper:

According to Russell, the goal of the logicist programme is to show that

all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles (Russell 1903: v).

That is to say, pure mathematics is defined as a class of propositions asserting formal implications and containing only logical constants. The logical constants are: implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and the notions that are involved in formal implication, that is, truth, propositional function, class, denoting, and any or every term (Russell 1903: 106). According to Russell, the above apparatus of general logical notions is sufficient to establish "the whole theory of cardinal integers as a special branch of logic" (Russell 1903: 111). In his view, the "irreproachable" definition of number in purely logical terms is to define number as a class of classes. Two classes have the same number when their terms can be correlated one to one so that anyone term of either class corresponds to one and only one term of the other class. When the two classes have the same number, Russell calls them similar. The number of a class is the class of all classes similar to the given class. The Membership of this class of classes is a common property of all the similar classes and no others (Russell 1903: 115).

*** I did have a spark somewhere in my brain ( ), but I want to hear opinions of others as well. What do you think of this definition? How do you understand it?
__________________

Russell's definition is not circular, despite appearances, but only when you succeed in formulating it without resorting to the concept of a number, which is a class of similar classes. Now you need a definition of similar classes, which are any two classes with a one-to-one correspondence between their elements. However, this latter definition still depends on the numbers two (any two classes) and one (one-to-one correspondence). We need a number-free definition of similar classes. But for now this will help you better understand Russell's definition.