PHENOMENOLOGICAL EXTENSION OF THE FORMAL PROOF THEORY

In the paper, I investigate phenomenological foundations of the finitary proof theory. The main motive of the investigation is the fact that the finitary proof theory contains some interesting phenomenological reflections. The phenomenological description is expected to promote progress in logical investigations and mathematical applications. The convergence between symbolic logic, mathematical reflections and phenomenological researches will be good for mathematical foundations. In this case, the term "foundations of mathematics" is used in the extended sense. It includes not only logical components of mathematical structures but also phenomenological descriptions of proving. For that reason, I attempted in the paper to combine the efforts of logicism, mathematics and phenomenology. I am guided here by experiences of Gottlob Frege, David Hilbert, Edmund Husserl, and Kurt Godel. The initial point is Frege's approach to formalization of mathematical theories on the basis of logical fundamental laws. Further consideration of the matter leads to the proposition that the finitary reasoning in mathematical logic is closely connected with phenomenological descriptions. The rigorous phenomenological description also offered the opportunity to define the space of proofs. The space of proofs is the specific form of thinking suitable for representations of logical propositions. In the present definition of "space of proofs", I make with Emil Post's definition of a "symbol space" in which any work leading from problem to answer is to be carried out. I will also follow Alan Turing's thesis concerning the "behaviour of the computer" and his "state of mind" at any moment. According to Godel, any mathematical theories are incomplete. However, these theories can be extended with the help of logical and phenomenological procedures. These procedures require a deep understanding of cognitive processes. If we want to find the common ground between logic, mathematics and phenomenology, we must answer the following questions. How Godel's appeal to Husserlian phenomenology should be understood? What did he want to achieve using phenomenological methods in logical and mathematical investigations? Godel held that phenomenology is a very important method of investigation of cognitive processes.