The object lesson concerns the passage from the foundational aims for which various branches of modern logic were originally developed to the discovery of areas and problems for which logical methods are effective tools. The main point stressed here is that this passage did not consist of successive refinements, a gradual evolution by adaptation as it were, but required radical changes of direction, to be compared to evolution by migration. These conflicts are illustrated by reference to set theory, (...) model theory, recursion theory, and proof theory. At the end there is a brief autobiographical note, including the touchy point to what extent the original aims of logical foundations are adequate for the broad question of the heroic tradition in the philosophy of mathematics concerned with the nature of the latter or, in modern jargon, with the architecture of mathematics and our intuitive resonances to it. (shrink)
Ausgangspunkt dieses Artikels ist die Einsicht, die auch von Wittgenstein und der "schweigenden Mehrheit" geteilt wird, daß die meisten sogenannten fundamentalen Begriffe und Probleme der Philosophie erkenntnistheoretisch unrentabel sind, insbesondere der Begriff der Gültigkeit (von Beweis- und Rechenregeln) und seine traditionelle Problematik. Im Gegensatz zu Wittgenstein wird diese Einsicht aber nicht auf "Sinnlosigkeit", d.h. Präzisionsunfähigkeitjener Problematik, sondern auf ihre Oberflächl d.h. unangemessene Allgemeinheit, zurückgeführt.
IN Hilbert's theory of the foundations of any given branch of mathematics the main problem is to establish the consistency (of a suitable formalisation) of this branch. Since the (intuitionist) criticisms of classical logic, which Hilbert's theory was intended to meet, never even alluded to inconsistencies (in classical arithmetic), and since the investigations of Hilbert's school have always established much more than mere consistency, it is natural to formulate another general problem in the foundations of mathematics: to translate statements of (...) theorems and proofs in the branch considered into those of some preferred system, where the translation must satisfy certain appropriate conditions (interpretation). The problem is relative to the choice of preferred system, as is Hilbert's consistency problem since he required the consistency to be established by particular methods (finitist ones). A finitist interpretation of classical number theory, which has been published in full detail elsewhere, is here described by means of typical examples. Partial results on analysis (theory of arbitrary functions whose arguments and values are the non-negative integers) are here presented for the first time. One of these results is restricted to functions whose values are bounded; its interest derives from the fact that real numbers may be represented by such functions. It is hoped that diverse general observations and comments, which would bore the specialist, may be of help to the general reader. The specialist may find some points of interest in the last two sections of the main text and in the notes following it. (shrink)