The question “what is a number?” is one which has been often asked but correctly answered in our time .Actually it answered in 1884 by Frege in an arithmetic essay. Although its essay was quite short, not difficult and has high importance almost attract no attention. So definition of the number was practically unknown until it was rediscovered by the present author in 1901.
In seeking a definition of the number the first thing to be clear about is what we may call the grammar of our inquiry .many philosophers when attempting to define the number setting to work to define plurality ,which is quite a different thing .number is what is characteristic of numbers ,as man is what is characteristic of men .A plurality is not an instance of number ,but of some particular number .A trio of men ,for example is an instance of number 3,that is an instance of number .But trio isn’t an instance of number. This point may seem elementary and scarcely worth mentioning; yet it has proved too subtle for philosophers, with few exception.
A particular number isn’t identical with any collection of terms having that number :The number 3 is not identical with the trio consisting of Brown ,James, and Robinson .The number 3 is something which all trios have in common ,and which distinguishes them from other collections .The number is something that characterizes certain collections ,namely ,those have that number.
Instead of speaking of “collection,” we shall as a rule speak of a “class,” or sometimes a “set.” Other words used in mathematics for the same thing are “aggregate” and “manifold.” We shall have too much to say later on about classes. But there are some remarks that they must be made immediately.
A class or collection may be defined in two ways that at first sight seem quite distinct. We may enumerate members, as when we say “Collection I mean is Brown, Jones, and Robinson.” Or we may mention a defining property as when we speak of “mankind” or “inhabitant of London.” The definition which enumerates is called a definition by “extension,” and one which mentions a defining property is called a definition by “intension.” Of these two kinds of definition the one by intension is logically more fundamental. This shown by two considerations: 1.That the extensional definition always can be reduced to an intensional one.2.That the intentional one often cannot be reduced even theoretically to the extensional one. Each of these points needs a word of explanation.
1) Brown, Jones, and Robinson all of them possess a certain property which is possessed by nothing else in the whole universe, namely, the property of being either Brow or Jones or Robinson. This property can be used to give a definition by intension of the class consisting of Brown and Jones and Robinson. Consider such a formula as “x is Brown or x is Jones or x is Robinson.” This formula will be true just for three x’s , namely, Brown and Jones and Robinson. In this respect it resembles a cubic equation with its roots. It may be taken as assigning a property common to the members of the class consisting of these three men, and peculiar to them. A similar treatment can obviously be applied to any other class given in extension.
2) It is obvious that in practice we can often know a great deal about class without being able to enumerate its members. No man can actually enumerates all men, or even all inhabitants of London, yet great deal is known about each of these classes. This is enough to show that definition by extension isn’t necessary to knowledge about a class. But when we come to consider infinite classes, we find that enumeration isn’t even theoretically possible for beings who only live for finite time. We cannot enumerate all the natural numbers: they are 0, 1, 2, and so on. At some point we can content ourselves with “and so on.” We cannot enumerate all fractions, or all irrational numbers, or all of any other infinite collection. Thus our knowledge in regard to all such collections can only be derived from a definition by intension.