The concept of number

Summary

Each individual number is an independent object

55. Having recognized that a statement of number is an assertion about a concept, we can attempt to supplement the leibnizian definitions of the individual numbers by means of the definitions of 0 and of 1.

Right away we might say: the number 0 applies to a concept, if no object falls under that concept. Here, however, “no” appears to have been substituted for 0, with which it is synonymous. Therefore the following wording is preferable: the number 0 applies to a concept if, no matter what a might be, the statement always holds that a does not fall under this concept.

Similarly we could say: the number 1 applies to a concept F if it is not the case that no matter what a is, a does not fall under F, and if from the statement

‘a falls under F’ and ‘b falls under F’

it always follows that a and b are the same.

We must still define in general the transition from one number to the next. We will try the following formulation: the number (n + 1) applies to the concept F if there is an object a which falls under F and such that the number n applies to the concept “falling under F but not [identical with] a”.

56. These definitions appear so natural, following our previous results, that an explanation is called for to show why they cannot satisfy us.