Contemporary computer technology is so advanced that it leads some mathematicians to suggestions that computers can enrich the already-known forms of mathematical theses proving. Some mathematicians openly speak about computer method of proving mathematical theorems. The paper is an attempt to consider notion of computer proof. The author describes various roles that computer can play in mathematics, compares the notion of experiment in the sciences and mathematics and then indicates possible conceptual difficulties with application of computers to proving mathematical theorems. Finally, the author formulates a key questions for the considered issue: Can computer prove mathematical theses in a different sense than a formalized proof does?; and: Can the application of computer in the procedure of proving the theorems undermine the dominating position of the mathematician?
The watersheds in the development of mathematics that lay bare the ambiguities and contradictions among its very basic notions and conceptions are commonly described as 'crises' in mathematics. They coincide with periods of intense mathematical research, often inspired by philosophical doctrines which help to clarify the notions and methods with the use of which many fundamental theorems are proved. The paper presents three best known and most widely discussed crises in the foundations of mathematics, reviews the causes of their appearance and discusses the difficulties that they have led to as well as possible ways of their resolution. The three crises arose in connection with the discovery of the incommensurable line segments in the Pythagorean School, in connection with the operations on the infinitesimals by the creators of the differential calculus (17th-18th centuries) and with respect to the foundations of mathematics connected with operations on the actually infinite sets (19th and 20th centuries).
With reference to certain ideas of Reuben Hersh, the paper attempts to present and confront two approaches concerning the essence of mathematics. On one hand mathematics is presented as an inflexible stronghold of truth in which what is established is considered final. However, on the other hand, as R. Hersh points, it does not differ from other forms of scientific research; it is burdened with uncertainty and it can be modified. Appreciating the importance of Hersh's social-mathematical observations, the author indicates that mathematics simply has two different facets. In one mathematics is a process of achieving the truth, in the other, it is a generally accepted way of presenting this truth.
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