HISTORY OF FORMALISM: FROM ARISTOTLE TO GÖDEL
HISTORY OF FORMALISM: FROM ARISTOTLE TO GÖDEL

Summary/Abstract: Using formal means for developing scientific theories became a tradition from the times of Aristotle’s Analytics. Ernst Schröder built the complete algebraic theory of inferences by the end of the 19th century. The idea of a complete formalization emerged as a way for eliminating paradoxes in foundations of mathematics that Bertrand Russell has revealed at the very start of the 20th century. Bertrand Russell and Alfred North Whitehead developed the first completely formalized theory in the three volumes of Principia Mathematica (1910 - 1913). David Hilbert enhanced the formation of metatheoretical approach to axiomatic theories by his call for proving the consistency of mathematics by using only finitary means. All of a sudden, in this atmosphere of steady axiomatic studies, a young mathematical genius Kurt Gödel published his famous theorem, which proved the incompleteness of a formal arithmetic system. Gödel’s theorem raised a huge wave of metatheoretical studies of formal systems. His main instrument, called Gödel’s numbering, was a special type of self-referential expressions that caused paradoxes just in foundations of mathematics. An aspect of Gödel's approach, that may raise discussions, is the formalization of metalogic itself, which actually may eliminate the idea of metatheory.

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