Moritz Schlick: In 1926, Schlick gathered around him a group of philosophers known as the Vienna Circle, which included Rudolf Carnap, Otto Neurath and the mathematicians and scientists Kurt Gödel, Philipp Frank, and Hans Hahn. Influenced by Ernst Mach and Ludwig Boltzmann, the Circle also drew on the work Bertrand Russell and Ludwig Wittgenstein. The Circle aimed to apply modern symbolic logic to further develop the views associated with Ernst Mach. and developed what has been known as Logical Positivism of Logical Empiricism. The Vienna Circle was characterised by an hostility to what they called “metaphysics”, by faith in the techniques of modern symbolic logic, and by belief that the future of philosophy lay in becoming the handmaiden of natural science.
Rudolph Carnap: studied at Jena 1910-14 where he attended Frege’s lectures and joined Schlick’s circle in 1926, and collaborated with a group of Positivist Empiricists in Berlin led by Hans Reichenbach. Carnap was not concerned with the problem of how people arrived at an understanding of the world, which was relegated to psychology, but sought to develop a logical grounding for empirical knowledge. Carnap’s approach was to view the natural language expressing empirical experience. By substituting more extensive logical expressions for words and phrases of the natural language, with symbols indicating immediate sense-data, Carnap aimed to show that all empirical statements are fully translatable into statements about immediate experiences which are subject to logical analysis. Later his methods moved more towards operational rather than empirical reduction. Sentences not subject to such reduction and therefore not subject to logical analysis were declared to be meaningless. All sentences concerning observable physical objects are translatable into the vocabulary of physics and thus Carnap hoped to establish a method of testing the consistency of physical theories.
To avoid the Nazi threat, in 1936 Carnap moved to the US and joined discussions with Bertrand Russell (Logical Positivism), Alfred Tarski and the Pragmatists Willard Quine (Constructivism) and Charles Morris (Semiology).
Carnap and other Logical Empiricists held that the statements of logic and mathematics, unlike those of empirical science, may be established a priori. Some, including Quine, argued that the attempt to delimit a class of statements that are true a priori should be abandoned as misguided. From 1945, Carnap turned his efforts to problems of inductive reasoning and of rational belief and decision to construct a formal system of inductive logic, centred on probabilistic implication.
However, the fact is that this whole school which based itself on the Logicist project initiated by GottlobFrege was transcended with Gödel’s theorem in 1931. Formal logic is a finite instrument, subsumed within mathematics. It is a wonderful thing about mathematics, that a mistaken view can be so shown to be so, so decisively and irrevocably.