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In Jena, Carnap was a student of Gottlob Frege and through his lectures he was first introduced to modern logic. Carnap was highly impressed by Frege’s system- a formulation system built from two dimensions- propositional logic (connectives), and first-order and second-order logic (laws of mathematics). this is an extensional relation system that aims to provide a logical foundation for all mathematics.[[CiteRef:: Gottfried (2008)]]
When Carnap returned to Jena after the war, he discovered another logician- Bertrand Russell. Carnap studied Russell’s Principia Mathematica and his symbolic logic of relations. Russell’s theory offered a solution to a contradiction to Frege’s system previously revealed by Russell (“Russell’s paradox”). Carnap was also influenced by Russell’s philosophical views which centered logic as method to conduct philosophy .[[CiteRef::Carnap (111963c)|pp.3-20]]
Carnap’s early ideas utilized logic tools provided by Frege and Russell to reconstruct a system of axioms that will give foundation not only to mathematics, but also to empirical science. The main factor stimulating Carnap to justify science was the difficulties Kantianism (link to Kant) and neo-Kantianism faced in early 20th Century- as Einstien’s theory of relativity got accepted .[[CiteRef::Cirera (31994)|pp.1-42]]
Immanuel Kant directed philosophy of science to a new theory of empiricism in the 18th Century. He coined the distinction between analytic and synthetic propositions to provide an alternative infallible foundation for geometry and Newtonian physics. For Kant, analytic proposition predicts the containment of objects and synthetic propositions predicts contradiction or no connection between objects. In that sense, statements in arithmetic, geometry, physics and philosophy are all synthetic per Kant (13). He also defined ‘a priori’ and ‘a posteriori’ knowledge. A priori knowledge is true independent of experience and is infallible- analytical pre-contained statements always a priori by definition. While a posteriori knowledge is based on experience and by the problem of induction and sensation cannot be absolutely certain (14). Furthermore, Kant makes a distinction between the world is it (noumena) and the world as we perceive it (phenomena), and claims that only the world of phenomena is knowable. Kant argued that the world of phenomena is structured by a priori forms such as sensibilities and cognitive faculties (15). In particular, Euclidian geometry is an a priori form that shapes how we perceive space. By working within the realm of phenomena, and structuring it by a priori form, Kant claims that synthetic propositions can be justified a priori as they are deducible from the a priori forms, and accordingly that synthetic proposition of mathematics and physical science are absolutely.
