Jan Wolenski

The author opposes two kinds of rationalism to irrationalism. The first kind of rationalism, approved by the author, considers as rational these cognitive procedures that can be communicated and verified intersubjectively. The author calls this view – after K. Ajdukiewicz – anti-irrationalism. On the other hand, irrationalism is a position that negates the need of ability to communicate intersubjectively and to verify cognitive procedures. The views of mystics or of H. Bergson could be an example of irrationalism. The second kind of irrationalism is apriorism, i.e. the position that ascribes certainty to cognitive results achieved by intuition. Data coming from intuition are not intersubjectively communicable, intersubjectively verified and they are not subject to falsification, hence the second kind of rationalism, i.e. apriorism, is in fact irrationalism. Phenomenology, stressing the importance of apriori cognition, would then have many things in common with irrationalism.

The relevance of metamathematical researches for philosophy of math- ematics is an indubitable matter. In the paper I shall speak about impli- cations of metamathematics for general philosophy, especially for classical epistemological problems. Let us start with a historical observation con- cerning Hilbert's programme, the rst research programme in metamathe- matics as a separate study of formal systems. This programme was strongly in uence by epistemological considerations. In fact, Hilbert wanted to se- cure all classical mathematics against inconsistencies and this aim had be achieved with the help of nitary consistency proof. Hilbert's claim is like the Cartesian project of reduction of all concepts to clarae et distinctae ideae. The epistemological commitment of original Hilbert's programme may be treated as a prelim- inary heuristic motivation for looking for epistemological implications of metamathematical results. It seems reasonable to examine in this respect three so-called limitative theorems: the rst Godel's incompleteness theo- rem , the second Godel's incom- pleteness theorem consistency of S, cannot be proved in S, providing that S is consistent) and Tarski's undenability theorem ; \S" stands for formal system containing elementary number theory. From the above-mentioned theorems we obtain an important, from philosophical point of view, conclusion: semantics of S cannot be expressed in S. Now, I shall present three applications of limitative theorems to the analysis of classical epistemological problems