1. Platonism - the Philosophy of Working Mathematicians
Charles Hermite has said once he is convinced that numbers and functions are not mere inventions of mathematicians, that they do exist independently of us, as do things in our everyday practice. Some time ago in the former USSR this proposition was quoted as the evidence for "the naive materialism of outstanding scientists".
But such propositions stated by mathematicians are evidences not for their naive materialism, but for their naive Platonism. Platonist attitude of mathematicians to objects of their investigations, as will be shown below, is determined by the very nature of the mathematical method.
Our main conclusion is the following: everyday work is permanently moving mathematicians to Platonism (and, as a creative method, this Platonism is extremely efficient), but passing to methodology we must reject such a philosophy deliberately. Most essays on philosophy of mathematics disregard this problem.
Naive set theory led to paradoxes. Axiomatic set theory was defined to avoid these paradoxes. Goedels theory says interesting things about axiomatic set theory.
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