The second half of the 20 th century witnessed a kind of revolution in the history and philosophy of science with the edition of T. S. Kuhn’s book Structure of Scientific Revolutions, published in 1962, which view of science is generally labeled « historical philosophy of science ».

In my presentation I will try to argue whether or not elements of the « historical philosophy of science » can be applied to the field of mathematics.
By presenting, notions (object level and meta-level) from one very well known example from the bibliography concerning Non-Euclidean Geometry by using the analyses of Zheng and Dunmore we will try to apply these notions into the field of arithmetic during the middle Ages in Europe. Our object by studying the question if the point of view of T. S. Kuhn for the scientific revolutions can be
applied in the context of mathematics come from our study of the development of our arithmetical system and the methods for doing the operation of multiplication during the Middle ages in Europe.

Especially by studying the way we have passed from the arithmetic of pebbles to the foundation ofmodern arithmetic, via Fibonacci and Pacioli, helped by the translation in latin of al-Khwarizmi’s treatise.