The Heuristic Function of the Axiomatic Method
Educating humanity is a big task for philosophy which should stand as a guiding idea behind all philosophical activities. In epistemology and in the philosophy of science the idea would be served by considering at least the following three interrelated questions:
(1) How is knowledge possible?
(2) How can we grant the reliability of acquired knowledge?
(3) How can we acquire new knowledge?
The lecture will deal with the third question. It will especially deal with the heuristic power of deductive methods. If the meaning of the term "deductive method" is restricted to "deductive inference" there are some good arguments to view these inferences as simply preserving the truth of their antecedents, as tautologies, or even to dispute their scientific character as Peirce did when comparing them to abductive and inductive inferences.
The aim of the lecture is to argue for a new reading of the term "deductive method." The deductive method 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 "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 propositions from which the deduction takes off. It is used to make deductions possible.
The deductive method can be presented as an analysis-synthesis scheme. The scientific process starts with a problem, a conjecture or a hypothesis of science or even everyday life. The problem will be analyzed in order to isolate the presuppositions which underlie the problem, the conjecture or the hypothesis. This analytic branch of the deductive method proceeds in a regressive manner from an instance to the general case. Once the list of presuppositions is complete, its items will be examined in order to determine whether they are justified or not. The revised list will serve as the basis for the deduction, i.e., the synthetic branch of the deductive method. It will provide reliable knowledge if the initial presuppositions are reliable. The heuristic power has its place in the analytic branch, i.e. in the creative activities for formulating systems of initial ("first") propositions, but also in the combination of the analytic and the synthetic branch: the insight that a hypothesis can be proved under the condition of a certain set of initial propositions is new knowledge, and also the insight that a certain field of accepted knowledge can be structured in a specific way.
Usually the analytic and the synthetic branch are treated as distinct ways to approach science. This leads to a couple of confusions. Take mathematics as an example. Some authors claim that mathematicians proceed in a dogmatic way, i.e., they start their deductions from a fixed set of axioms stated without justification. But what are the criteria for choosing just these and no other axioms? Other commentators (e.g., Herbert Mehrtens, following some speculations by Georg Cantor, David Hilbert and Felix Hausdorff ) interpret this dogmatism as an expression of the mathematician's liberty which is only restricted by the capacity of his mind. But this is a strong restriction: the mathematician is not able to act against the laws of logic. This restriction unmasks in epistemological respects the mathematician's liberty as pure ideology. One can only speak of the mathematician's liberty in the practical respect that at least the pure mathematician is free from any considerations concerning possible applications of his theories. His constructions are not designed to have applications (although they might find them, but this is not his concern). In his practical work the mathematician is not really free to choose a topic, his choice is influenced by open problems, by vivid and controversial contemporary discussions, by mainstream mathematics, by known and accepted knowledge in the mathematics of his time. The free mathematician is working in the context of his time, and he is influenced by this context. These considerations show that it is ideological to restrict mathematics to the deductive side, whereas a good deal of mathematics takes place in the analytic branch. On the other hand, some authors exaggerate the analytic branch. In their criticism, e.g., against closed axiomatic systems (Cellucci) or against the a priori character of mathematics (Gillies) they argue that mathematics is an empirical activity, using, among others, the trial and error method. Again mathematics is identified only with one branch of the scheme, the analytical branch, which is by no means methodologically restricted. But to prove that there are certain empirical activities in mathematics does not prove that mathematics as such is an empirical science. The analytic-synthetic scheme is able to combine these different views on mathematics.
It is also able to explain some peculiarities in the history of philosophy, especially in its rationalistic branches. Of course, it is modeled upon the Cartesian succession of methodological scepticism and the deductive construction of undoubtable knowledge from a firm base. The scheme helps to explain Leibniz's and Wolff's striving for new knowledge with the help of an ars inveniendi whose most important techniques are combinatorics and syllogism, i.e. deductive means. It furthermore models the relation between the analytical, critical and regressive method and the synthetical, dogmatic and constructive method in Kant, Jakob Friedrich Fries and Leonard Nelson.
Leonard 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 Descartes-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 scheme helps to provide a consistent interpretation of these two sides in Hilbert's attitude towards his working field.