Philosophy must not forget mathematics. The everyday spectacle of scientific popularization and of scientific discoveries is meant for effect and chooses what is easiest to understand. It is well known that mathematics is difficult to develop and difficult to understand, just as it is difficult to reason about its status and its objects (philosophers were always aware of this hard task).

Mathematics and Knowledge

Mathematics maintains a central role in the production of modern knowledge. When we realize how mathematical models are integrated in scientific theories and in experimental sciences, economics, engineering, and also in philosophical thought, then we discover in amazement its vast congnitive fertility. What is more, in this light mathematics seems to be the only vehicle to produce “rational” knowledge: philosophy has always been aware of this thesis, as shown by the following historical example. From Plato to Kant to present times, geometry has been a constant and favourite landmark for philosophical reflection, engaged in elaborating theories of knowledge, theories of scientific rationality and Weltanschauungen (see L. Magnani, Philosophy and Geometry. Theoretical and Historical Issues, Kluwer Academic, Dordrecht, 2001). Many great philosophers of the past have studied mathematics because mathematics helped them to reflect upon knowledge: mathematics has often stimulated the philosopher to create new powerful theories of knowledge. Hence, in the history of western culture the relationships between philosophy and mathematics have to be considered as privileged. Kant himself, in a passage of the Critique of Pure Reason, magnificently states that “The science of mathematics presents the most brilliant example of the extension of the sphere of pure reason without the aid of experience” (A712-B740). In its turn philosophy has productively interacted with the sciences and mathematics itself: the imagination of a great mathematician like Riemann has benefited by the study of the Herbart's well-known philosophy of “reals”.

Mathematics and philosophy have generated modern logic. At present logic is a very powerful tool, useful i) for explaining some problems related to the foundations of mathematics itself; ii) as a method able to provide models and representations of knowledge and reasoning (for instance in artificial intelligence); iii) as a method for clarifying many philosophical and epistemological problems. When we look at mathematics as a kind of knowledge distributed in different cultural contexts, its extraordinary relevance as a way of producing new knowledge and new rational models clearly emerges. Hence, mathematics demonstrates itself to be an indefatigable discipline which is continually able to elaborate new changes in order to make the world intelligible (See L. Magnani and R. Gennari, Manuale di logica. Logica classica e del senso comune, Guerini & Associati, Milan, 1997, in Italian).

Epistemology, History, and Mathematics

Epistemologists, at the beginning of neopositivism, have found in mathematical knowledge many methodological, logical, and conceptual tools that have constituted their rational framework for analyzing science. The historian of science has shown, in analyzing the dynamics of mathematical thought, how it is integrated in sciences and cultures.

Through philosophy, epistemology, and history, and dealing with the problems related to the various scientific theories, logic, probability, automatic reasoning in artificial intelligence, complexity, the articles of the book L. Magnani, ed., Conoscenza e matematica, Marcos y Marcos, Milan, 1990, propose considering mathematics not only, traditionally, as the privileged discipline that continually produces rationality in various kind of knowledge, but also as a complete and exemplary “cultural” object, endowed with unsuspected and often neglected “expressive” attitudes. Another book edited by L. Magnani, ed, Le geometrie non euclidee, Zanichelli, Bologna, 1978, gives an instructive example of a historical research devoted to illustrate this complex role played by mathematics in different cultural and scientific places, showing the numerous and interesting mutual interactions.

References about Philosophy of Science and Computational Methods

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