What is a Number?

The concept of number is the most basic and fundamental in the world of science and mathematics. Yet a satisfactory answer to what a number is was attained only in 1884 A.D. by Gottlob Fregé [1], the founder of modern mathematical logic. His answer remained unknown to the world until Bertrand Russell, the English mathematician and logician, in his attempt to base all of mathematics in terms of the concept of sets, rediscovered the concept of number.

The concept of number is associated with the concept of a "set." As of 2005 A.D., mankind possessed two ways of explaining the concept of number. One was the Von Neuman method and the other the Fregé concept.

A "set" is a term like the term "point" is in Geometry where it is not defined but taken as a "primitive" whose meaning is brought out by the axioms. Similarly, a set is described by the axioms of set theory. Informally, set means a collection of definite and separate objects of any kind for which we can decide whether or not a given object belongs. So to exhibit a set, you show all the objects - popularly called elements - in the collection by either exhibiting each individual element in the collection or precisely describing which elements belong. An instance of the former is the set comprised of, say, three names, Bob, Mary, and Marg; the set is exhibited by providing a list of its elements inside two curly brackets thus: {Bob,Mary,Marg}. The order in which the elements are listed inside the brackets is not relevant. When a set contains a very large number of elements, it is inconvenient or impractical to position each element inside the curly brackets and so the second method is used. That is, we describe precisely those elements that belong to the set. We can use ordinary language when that will do - for example, the set of all people on earth. Or, we can also use curly brackets thus: {x | x is a person of the Earth}; where the vertical line means "satisfying the condition that," or simply "such that."

Von Neumann [1923] proposed that all numbers could be bootstrapped out of the empty set by the operations of the mind as follows.

0 = {} (empty set)
1 = {0} = { {} }
2 = {0,1} = { {}, { {} } }
3 = {0,1,2} = {{}, { {} }, { {}, { {} } }}
4 = {0,1,2,3} = { {}, { {} }, { {}, { {} } }, {{}, { {} }, { {}, { {} } }} } ....

This construction is wonderful and simple and shows why, for instance, 1 is less than 2, or in general why given any two distinct numbers a and b, either a < b or a > b. There are many other properties of this scheme. However, the one shortcoming of the scheme is that it is an artifice of construction and does not tell us what a number is except in terms of the construction. For instance, 0 is the empty set, 1 the set consisting of the empty set, 2 is the set whose elements are the empty set and the set consisting of the empty set, and so on.

To understand this scheme, we would have to go to the concept enshrined in it, namely the Theory of Concepts advanced by Fregé. And it would explain why von Neumann chose the empty set to represent zero.

Numeral
Numeral is the symbol for the idea called number. Put another way, the number is the idea we think of when we see the numeral or when we see or hear the word for a numeral.

Suppose there is a person named Jim. This person has the name Jim because he was named so. It is very convenient! A numeral is like the name Jim.

Now, if someone says number 3, we know what really is meant. 3 is the numeral for the number the person wishes to communicate to us. Since this is to be always understood, we just say "number 3."

An alien coming to earth might be amused to note that we have given this number the name 3. A computer on earth would have to be told that this number is 11, because 11 is 3 in binary notation. 11 in binary and 3 in decimal notation are called "numerals." As you know, III is the Roman numeral for 3.

On the web page Numeration Systems, we discuss the various systems like the binary, decimal and others for writing numerals.

ORIGINAL SOURCES
[1] Gottlob Fregé, Die Grundlagen der Arithmetik (1884)