Volume 37, 1998
Philosophy of Science
The Heuristic Function of the Axiomatic Method
This lecture will deal with the heuristic power of the deductive method and its contributions to the scientific task of finding new knowledge. I will argue for a new reading of the term 'deductive method.' It will be presented as an architectural scheme for the reconstruction of the processes of gaining reliable scientific knowledge. This scheme combines the activities of doing science ('context of discovery') with the activities of presenting scientific results ('context of justification'). It combines the heuristic and the deductive side of science. The heuristic side is represented, e.g., by the creative methods to find the 'best' hypotheses (abduction), to design experimental systems for empirical research in order to formulate general laws (induction), or to create axiomatic systems. The other side consists of the production of deductive knowledge. This combination leads to a clear hierarchy: the heuristic side provides the basic presuppositions from which the deductive side takes off. The former is used to make deductions possible. The deductive method can be presented as an analysis-synthesis scheme as it can be found, e.g., in the tradition of Kant, Jakob Friedrich Fries, and Leonard Nelson. Nelson's critical philosophy can be seen as a key for understanding the philosophy behind David Hilbert's early axiomatic method. This axiomatic method is usually restricted to a non-philosophical approach to pure mathematics ('formalism'). But Hilbert was not an exclusive formalist; he proposed a mathesis universalis in the Cartesian-Leibnizian sense according to which mathematics is the syntactical tool for a general philosophy of science, applicable to all scientific disciplines. In this function, mathematics takes its problems from the sciences. Hilbert did not deny that mathematics should play a role in explaining the world. The analysis-synthesis scheme helps to provide a consistent interpretation of these two sides of Hilbert's attitude towards his working field.