In the article, philosophical and methodological analysis of the program of Hilbert’s formalism as a really working direction for consideration of the bases of modern mathematics is presented. For the professional mathematicians methodological advantages of the program of formalism advanced by David Hilbert, consist primarily in the fact that the highest possible level of theoretical rigor of modern mathematical theories was practically represented there. To resolve the fundamental difficulties of the problem of bases of mathematics, according to Hilbert, the theory (...) of mathematical proof is needed, but contrary to popular belief rigorous formalization of the proof is not a synonym of reliability and rigor of mathematical reasoning from the point of view of the philosophy of the foundations of mathematics. In fact, the consistency of the theories is "more important" than their logical consistency because not every statement, which does not contradict to the reasonable ones, can be attributed to a true statement. However, for working mathematicians, Hilbert is logical and consistent and the axiomatic method and formalism are an essential part of their rules of thinking. (shrink)

It is well known that the largest philosophers differently explain the origin of mathematics. This question was investigated in antiquity, a substantial and decisive role in this respect was played by the Platonic doctrine. Therefore, discussing this issue the problem of interaction of philosophy and mathematics in the teachings of Plato should be taken into consideration. Many mathematicians believe that abstract mathematical objects belong in a certain sense to the world of ideas and that consistency of objects and theories really (...) describes mathematical reality, as Plato quite clearly expressed his views on math, according to which mathematical concepts objectively exist as distinct entities between the world of ideas and the world of material things. In the context of foundations of mathematics, so called ‘Gödel’s Platonism‘ is of particular interest. It is shown in the article how Platonic objectification of mathematical concepts contributes to the development of modern mathematics by revealing philosophical understanding of the nature of abstraction. To substantiate his point of view, the author draws the works of contemporary experts in the field of philosophy of mathematics. (shrink)

The paper is devoted to the analysis and identification of new philosophical aspects of the problem of justification of modern mathematics according to which to the end of the 20th century the most exact of sciences had experienced new shocks associated with the crisis of excessive complexity of the mathematical theories. In the context of justification of mathematics philosophical conclusion consists in the fact that from a methodological point of view for general assessment of whether mathematics is developed or not (...) just the lack of crises is more negative phenomenon than its too long trouble-free state. Accomplished fact is that a computer can be regarded as one of the serious revolutionary technological inventions of the last century, which has had a huge influence on mathematics. In this development of progress of contemporary mathematics, more precisely, in its instrumental technologies the new philosophical problems are revealed - this is the role of computers and the new standards of mathematical thinking. In addition, with the development of contemporary mathematics and the emergence of increasingly complicated and long proofs, they lose their methodological advantage - the property of visibility and convincingness. Within the philosophy of mathematics this problem has not been discussed yet, possibly because the leading mathematicians haven’t expressed their opinions on the subject. After that, we can wait for a methodologically important period clarifying details, which can make the process itself foundations of mathematics philosophical consummation as the crisis complexity; in mathematics it is not a ‘systemic crisis‘ carrying the potential danger of the destruction of all mathematical knowledge. (shrink)

The concept of truth of the mathematical theory plays an important methodological role in philosophy of mathematics in the context of a link between ontological and gnoseological aspects of a problem of mathematics substantiation. In the article, the author analyzes how the features of mathematical knowledge are reflected in the understanding of concept of truth in the contemporary mathematics. The author claims that the concept of truth of mathematical offers and theorems is metamathematical concept; therefore, underestimation of distinction between mathematics (...) and metamathematics can lead to philosophical misunderstanding. They consist primarily in the fact that theoretically possible level of rigor of contemporary mathematical theories has been virtually represented in the science. In order to resolve the fundamental philosophical and methodological difficulties of the problem of mathematics substantiation, the formal provability in logically consistent theory is necessary. However, contrary to the widespread belief, a rigid formalization of the proof is still not the synonymous with reliability and rigor of mathematical reasoning from the standpoint of the philosophy of contemporary mathematics substantiation. (shrink)

The author of the work proposes a philosophical and methodological interpretation of the mathematical objects, using the system triad of the main directions of substantiation of mathematics: the formalism of Hilbert, Brouwer’s intuitionism and Godel’s Platonism. The need for these directions in the concept of substantiation of mathematics from the point of view of the current state of the philosophy of mathematics is shown on the mathematical examples. The philosophical and methodological analysis of objects of mathematics has never been unambiguous, (...) therefore in this paper the results of studies of philosophers, logicians and mathematicians, in which the problem of substantiation is explicated in the context of trends in the development of mathematics, are used. Their professional view on philosophical characteristics of the objects of mathematics contributes to the identification of the unity of all mathematical knowledge maintaining the initial mathematical base of knowledge and revealing new ways of integrating the directions of substantiation in the philosophy of mathematics. The practical problem of substantiation of mathematics is realized through the elaboration of metatheoretical knowledge under the paradigm shift in the philosophy of mathematics to the productive direction from analysis to synthesis. (shrink)