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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...

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...l proofs for someone who accepts the axioms from which they begin." Those axioms are continually being challenged, but if they are to be justified, it shall not be within the context of mathematical activities. Now we must turn to the philosophy of mathematics, "to the great debates between the formalists, the intuitionists, and the Platonists." These debates cannot be settled by mathematical proofs, however. The certainty of mathematics is merely conditional; it rests upon assumptions that cannot be proven within mathematics, but only within the philosophy of mathematics. Exactly the same problem applies with respect to the primary problems of philosophy. We can easily give practical arguments that seem very convincing, but when we analyze these arguments philosophically, we often find that the simple conventions of ordinary argument cannot be regarded as adequate.
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