In an attempt to express certain basic concepts of mathematics precisely, one should consider a handful of different accepted and developed conceptions. Pythagoras, in the Fifth Century B.C., believed that the ultimate elements of reality were numbers; therefore the explanation for the existence of any object could only be explained in number. Gottlob Frege stated, in an idea referred to as logicism, that mathematics could in some sense be reduced to logic. The views of Plato state that we "know" these rules of mathematics at the intuitive level rather than the conscious level. Plato also believed that these forms existed previously in their perfect forms; humans know them in their imperfect forms through concept and imagination. Humans did not invent mathematics, but rediscovered these transcendent but real forms.
Almost a century ago, Bertrand Russell wrote in The Problems of Philosophy that "philosophy should not be studied 'for the sake of definite answers to its questions, since no definite answers can, as a rule, be known to be true.'" For the problems mentioned here, however, it seems possible to give and justify answers. Certainly the effort should be made. Perhaps, through Pythagorean ideas, logicism and Platonism, a firmer understanding can be known of the grasp that mathematics has on the world.
Due to the secrecy of the society in which Pythagoras, it is difficult to distinguish between any original works of Pythagoras from those of his followers. However, it is not the author that is important, but rather the notions presented. According to the view of the Pythagoreans that "all is number," the first four numbers have a special significance in that their sum accounts for all possible dimensions. In dot notation, "one" forms a point, which can be defined as the generator of dimension. "Two" forms a line of one dimension, "three" forms a triangle of two dimensions, and "four" forms a tetrahedron of three dimensions. The sum of these numbers represents the sum of all objects. The number ten, the tetractys, therefore represents the universe. Some contend that this early argument for the number ten is why, along with the number of hands and toes we have, we conduct mathematics in base ten.
The landmark work on mathematical logic and the foundations of mathematics is Principia Mathematica, written in 1910 by Alfred North Whitehead and Bertrand Russell in defense of the logic of Gottlob Frege. Whitehead and Russell owe an amazing debt to Frege, whose Grundgesetze der Arithmetik ('Basic Laws of Arithmetic) provided the stepping stone for their collaborative work. Whitehead's and Russell's work succeeds in providing its intended purpose, but two ideas in particular are arguably non-logical in character: the idea of infinity and the idea of reducibility. The axiom of infinity states that there exists an infinity of objects, an assumption generally thought to be empirical rather than logical in...