This book challenges the views put forward by Pierre Cartier, one of the anchors of the famous Bourbaki group, and Cédric Villani, one of the most brilliant mathematicians of his generation, who received the Fields Medal in 2010. Jean Dhombres, mathematician and science historian, and Gerhard Heinzmann, philosopher of science and also a specialist in mathematics engage in a fruitful dialogue with the two mathematicians, prompting readers to reflect on mathematical activity and its social consequences in history as well as in the modern world. Cédric Villani's popular success proves once again that a common awareness has developed, albeit in a very confused way, of the major role of mathematics in the construction and efficiency of natural sciences, which are at the origin of our technologies. Despite this, the idea that mathematics cannot be shared remains firmly entrenched, a perceived failing that has even been branded a lack of culture by vocal forces in the media as well as cultural and political establishment.  The authors explore three major directions in their dialogue: the highly complex relationship between mathematics and reality, the subject of many debates and opposing viewpoints; the freedom that the construction of mathematics has given humankind by enabling them to develop the natural sciences as well as mathematical research; and the responsibility with which the scientific community and governments should address the role of mathematics in research and education policies.

PIERRE CARTIER is academic by training, emeritus research director at the Centre national de la recherche scientifique (CNRS), and research scientist and a visitor (for indeterminate term) at the Institut des Hautes Etudes Scientifiques (IHES) of Bures-sur-Yvette. He is currently emeritus research director at the University Paris-Diderot. He is former member of the Bourbaki group. Born in 1932 in Sedan, Professor Carier, despite his provincial background, followed the path of excellence in his studies: a secondary school in his home town, followed by the lycée Saint-Louis in Paris and then the École normale superieure, and he obtained an advanced teaching degree and a doctorate in mathematics (1958). He also went to Princeton for two years; there he became acquainted with legendary figures such as Robert Oppenheimer and Andre Weil (brother of philosopher Simone Weil). Last but not least, he carried out a long military service in the marine corps against the backdrop of the Algerian war. He was then professor at the University of Strasbourg from 1961 to 1971. The years 1950-1975 were the hey-day of the Bourbaki group, of which he is one of the pillars; there he became friends with famous people like Cartan, Schwartz, Dieudonné, Chevalley, Weil, and with younger ones.  A "mathematician without borders", Professor Cartier is active within the framework of the cooperation agency of the French Mathematical Society (Cimpa). He had been working in the Paris region since 1971, and he roamed around between Parisian academic institutions: University Paris-Sud, École polytechnique, École normale supérieure. He is not a member of the French Academy of Sciences (a voluntary choice). His scientific interests are quite diverse (even eclectic), but centered around group theory and mathematical physics. He often cited thesis relates to algebraic geometry, by he also contributed to differential geometry, number theory, combinatorics, numerical analysis, probability and mathematical physics. He wrote a reference book on Feynman Integrals (in collaboration with C. DeWitt-Morette, Functional Integration, Action and symmetries (Cambridge University Press, 2004). He is also one of the editors of Mathematics in the 21st Century: 6th World Conference, Lahore, March 2013 (Springer, 2014). He has supervised more than 40 doctoral theses, and he continues to work with some of my students on multiple zeta values and the Galois theory of differential equations. JEAN DHOMBRES is mathematician and historian of science, emeritus CNRS research director, director of studies at the Alexandre Koyre Centre of the École des hautes études en sciences sociales (EHESS), specialist of the mathematics of functional equations and of their applications, of epistemology of mathematics, and of the history of scientific communities and of the spread of scholarly ideas. Born in Paris in 1942, he studied at the École polytechnique and chose to take a chance with mathematical research instead of graduating as an engineer. He worked in functional analysis and through a series of encounters, he joined a small international group of scientists focused on functional equations. In fact it was in Bangkok that his first book was published as a result of a talk he had given at Chulalongkorn University in 1971. Besides many articles, he is happy to have been able to work with Janos Aczael, the uncontested leader in this field, having published with him a book on the subject, which has now become a reference, in Encyclopaedia of mathematics (University of Cambridge Press, republished in 2008). Probably as a reaction to his family background, as well as to a scientific training centered on France, he wished to see the world, by learning Chinese, by participating in the establishment of experimental classes in Wu Han in China, then by becoming scientific adviser in a French embassy for three years. H