# Mathematics Essay

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

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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 nature. The axiom of reducibility is introduced as a means of overcoming the theory of types, discussed in Principia, and to avoid the paradoxes (such as Russell's Paradox discovered in 1901). Although technically feasible, the axiom was thought by many to be too convenient and exclusive to be justified philosophically. "There is often a possibility of human