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Philosophical problem of proving of mathematical knowledge: from absolutism to fallibilism

Annotation

It is criticized that the dominant in philosophy of mathematics in the absolutist notion that any mathematical truth is completely justified, and therefore infallible, and mathematics is perhaps the only form of undeniable, objective knowledge. This theoretical position is correlated with the opposite fallibilist point of view. For the latter is characterized by the belief that mathematical truth is subject to change and should not be regarded as irreparably verification procedures and/or corrections. Absolutist approach to mathematical knowledge has been challenged in the early 20th century, after the identified logical paradoxes and contradictions in Cantor's set theory and logical-mathematical system of Frege. The data obtained have had serious consequences for the absolutist conception of mathematical knowledge. The conviction that the antinomies are found due to the presence of errors in the very foundations of mathematics is considered. The crisis has resulted in the development of several areas of philosophy of mathematics, whose objectives are to study mathematics and the restoration of its status as an absolutely reliable science. Three major programs the foundations of mathematics – logicism, formalism and constructivism (closely related to intuitionism) – are considered. In each of the programs established that it ensures a solid foundation of absolute truth: the axioms of logic (logicism); intuitively valid principles of meta-mathematics (formalism); intuitively self-evident axioms and rules (intuitionism). The programs reviewed justification of mathematical knowledge using deductive logic to demonstrate the truth of mathematical theorems. However, none of the programs are not able to establish the absolute validity of mathematical truth. With the accumulation of knowledge about the basics of mathematics became clear that the absolutist view is just an idealization, more myth than reality. Experience in understanding the problem of justification of mathematical knowledge in the 20th century brings us to the conclusion that an absolutist approach, typical of traditional mathematics, is generated as a result of the idealization of nature mathematical truth. Mathematicians and philosophers, coming from the Platonic idea of the immutability, reliability and absolute nature of mathematical knowledge, paying little attention to the history of mathematics and the actual mathematical practice. Therefore, the position of fallibilism should be considered as a realistic view of mathematical knowledge and mathematical activities.

Keywords

philosophy of mathematics; foundation of mathematical knowledge; absolutes-Semitism; fallibilism; naturalism; empiricism; formalism; intuitionism; constructivism

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UDC

1:001; 001.8

Pages

20-33

References

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Received

2014-07-01

Section of issue

Questions of theory and methodology

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