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Definition of Number
My mother is a graduate mathematics student, but isnt good with internet and essay writing (or doesnt want to be anyway 
), so I help her when it comes to papers. I have recently been doing research on Bertrand Russell's definition of a number. I have a paper from her 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 any one 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 am so far from mathematics, guys. I did have a spark somewhere in my brain (
), but I wanna hear opinions of others as well. What do you think of this definition? How do you understand it?
), so I help her when it comes to papers. I have recently been doing research on Bertrand Russell's definition of a number. I have a paper from her 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 any one 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 am so far from mathematics, guys. I did have a spark somewhere in my brain (
), but I wanna hear opinions of others as well. What do you think of this definition? How do you understand it?
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