THE PROBLEM OF CORRELATION OF INDUCTION AND DEDUCTION IN THE HISTORY OF MATHEMATICS AND MATHEMATICAL EDUCATION
Authors: Lutfullin M. 
Published: December 2022

The analysis of the scientists’ views on the role of induction and deduction in the development of mathematics is carried out. There are no substantial difference in the estimation of the importance of deductive substantiation of mathematical concepts, theorems and theories in these views. But we had to establish the big differences and even absolutely opposite views on the role of induction in the process of generation, extension and integration of mathematical knowledge. The most conclusive are the opinions of the thing that an induction is a match for a deduction (L. Euler, F. Klein, G. Polya and others). The importance of inductive method of research in the scientific development, especially in mathematics, was even greater emphasized by L. de Broglie. Absolutely opposite are the views of those scientists who consider the mathematics to be an absolutely deductive science (T. Huxley, J. Murray, B. Pierce and others). The most categorical point of such view was expressed by J. Murray.In the mathematical education the false opinion of the thing that this science is supposedly to be absolutely deductive inevitably lead to pedagogical mistakes. The threat of these mistakes concerns mainly the geometrical training. The deductive character of statement of proving the geometrical theorems in the manuals and at the lessons leads to the pupils’ material difficulties in the process of digestion of knowledge. Inductive method of education may lighten the process of digestion of geometry. Not of less importance this method is at the lessons of algebra.If the importance of role of the deduction in the mathematical development is old-confirmed by the perfection of deductive proofs of Euclid, the consciousness of the importance of induction in the mathematical researches is set just in the first quarter of the XVII century with the publication of “Novum Organum” by Francis Bacon. The first thinker who consciously combined deduction with induction in his researches was R. Descartes. Indissoluble correlation of induction and deduction is brilliantly shown in many researches of L. Euler.In the second half of the XVIII century the induction comes to gaining practical application in the mathematical education. For the first time the inductive method was realized in the “Universal arithmetic” manual the author of which was an outstanding educator N. G. Kurganov. This manual was notable for the simplicity of teaching materialexposition.The increasing of attention to the mathematical education by inductive method is founded in the manuals, methodical works and educational activity of F. I. Busse, P. S. Hurjev, O. M. Strannoliubskiy,S. I. Shohor-Trotskiy. The valuable contribution to this problem development is issued to K. F. Lebedintsev, the author of specific inductive method of mathematical education. In the second half of the XX century the many lines of the application of induction in the secondary and higher mathematical education became a subject of long-term fundamental investigation by G. Polya.Hence, it is established that the requisite condition of the improving of mathematical education is following the principle of indissoluble correlation of induction and deduction in the educational process. This principle claims active energies of the manuals authors and teachers of mathematics at general secondary schools and higher educational establishments which provide practical realization of the unique methodical heritage of K. F. Lebedintsev and G. Polya.