MATHEMATICS CONCEPTS

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."